[4977] | 1 | /*************************************************************************
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| 2 | Copyright (c) Sergey Bochkanov (ALGLIB project).
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| 3 |
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| 4 | >>> SOURCE LICENSE >>>
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| 5 | This program is free software; you can redistribute it and/or modify
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| 6 | it under the terms of the GNU General Public License as published by
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| 7 | the Free Software Foundation (www.fsf.org); either version 2 of the
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| 8 | License, or (at your option) any later version.
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| 9 |
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| 10 | This program is distributed in the hope that it will be useful,
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| 11 | but WITHOUT ANY WARRANTY; without even the implied warranty of
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| 12 | MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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| 13 | GNU General Public License for more details.
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| 14 |
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| 15 | A copy of the GNU General Public License is available at
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| 16 | http://www.fsf.org/licensing/licenses
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| 17 | >>> END OF LICENSE >>>
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| 18 | *************************************************************************/
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| 19 | #pragma warning disable 162
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| 20 | #pragma warning disable 219
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| 21 | using System;
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| 22 |
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| 23 | public partial class alglib
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| 24 | {
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| 25 |
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| 26 |
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| 27 | /*************************************************************************
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| 28 | Integration report:
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| 29 | * TerminationType = completetion code:
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| 30 | * -5 non-convergence of Gauss-Kronrod nodes
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| 31 | calculation subroutine.
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| 32 | * -1 incorrect parameters were specified
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| 33 | * 1 OK
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| 34 | * Rep.NFEV countains number of function calculations
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| 35 | * Rep.NIntervals contains number of intervals [a,b]
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| 36 | was partitioned into.
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| 37 | *************************************************************************/
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| 38 | public class autogkreport
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| 39 | {
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| 40 | //
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| 41 | // Public declarations
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| 42 | //
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| 43 | public int terminationtype { get { return _innerobj.terminationtype; } set { _innerobj.terminationtype = value; } }
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| 44 | public int nfev { get { return _innerobj.nfev; } set { _innerobj.nfev = value; } }
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| 45 | public int nintervals { get { return _innerobj.nintervals; } set { _innerobj.nintervals = value; } }
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| 46 |
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| 47 | public autogkreport()
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| 48 | {
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| 49 | _innerobj = new autogk.autogkreport();
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| 50 | }
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| 51 |
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| 52 | //
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| 53 | // Although some of declarations below are public, you should not use them
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| 54 | // They are intended for internal use only
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| 55 | //
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| 56 | private autogk.autogkreport _innerobj;
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| 57 | public autogk.autogkreport innerobj { get { return _innerobj; } }
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| 58 | public autogkreport(autogk.autogkreport obj)
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| 59 | {
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| 60 | _innerobj = obj;
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| 61 | }
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| 62 | }
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| 63 |
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| 64 |
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| 65 | /*************************************************************************
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| 66 | This structure stores state of the integration algorithm.
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| 67 |
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| 68 | Although this class has public fields, they are not intended for external
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| 69 | use. You should use ALGLIB functions to work with this class:
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| 70 | * autogksmooth()/AutoGKSmoothW()/... to create objects
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| 71 | * autogkintegrate() to begin integration
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| 72 | * autogkresults() to get results
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| 73 | *************************************************************************/
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| 74 | public class autogkstate
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| 75 | {
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| 76 | //
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| 77 | // Public declarations
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| 78 | //
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| 79 | public bool needf { get { return _innerobj.needf; } set { _innerobj.needf = value; } }
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| 80 | public double x { get { return _innerobj.x; } set { _innerobj.x = value; } }
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| 81 | public double xminusa { get { return _innerobj.xminusa; } set { _innerobj.xminusa = value; } }
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| 82 | public double bminusx { get { return _innerobj.bminusx; } set { _innerobj.bminusx = value; } }
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| 83 | public double f { get { return _innerobj.f; } set { _innerobj.f = value; } }
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| 84 |
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| 85 | public autogkstate()
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| 86 | {
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| 87 | _innerobj = new autogk.autogkstate();
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| 88 | }
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| 89 |
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| 90 | //
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| 91 | // Although some of declarations below are public, you should not use them
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| 92 | // They are intended for internal use only
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| 93 | //
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| 94 | private autogk.autogkstate _innerobj;
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| 95 | public autogk.autogkstate innerobj { get { return _innerobj; } }
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| 96 | public autogkstate(autogk.autogkstate obj)
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| 97 | {
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| 98 | _innerobj = obj;
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| 99 | }
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| 100 | }
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| 101 |
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| 102 | /*************************************************************************
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| 103 | Integration of a smooth function F(x) on a finite interval [a,b].
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| 104 |
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| 105 | Fast-convergent algorithm based on a Gauss-Kronrod formula is used. Result
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| 106 | is calculated with accuracy close to the machine precision.
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| 107 |
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| 108 | Algorithm works well only with smooth integrands. It may be used with
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| 109 | continuous non-smooth integrands, but with less performance.
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| 110 |
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| 111 | It should never be used with integrands which have integrable singularities
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| 112 | at lower or upper limits - algorithm may crash. Use AutoGKSingular in such
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| 113 | cases.
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| 114 |
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| 115 | INPUT PARAMETERS:
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| 116 | A, B - interval boundaries (A<B, A=B or A>B)
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| 117 |
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| 118 | OUTPUT PARAMETERS
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| 119 | State - structure which stores algorithm state
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| 120 |
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| 121 | SEE ALSO
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| 122 | AutoGKSmoothW, AutoGKSingular, AutoGKResults.
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| 123 |
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| 124 |
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| 125 | -- ALGLIB --
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| 126 | Copyright 06.05.2009 by Bochkanov Sergey
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| 127 | *************************************************************************/
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| 128 | public static void autogksmooth(double a, double b, out autogkstate state)
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| 129 | {
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| 130 | state = new autogkstate();
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| 131 | autogk.autogksmooth(a, b, state.innerobj);
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| 132 | return;
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| 133 | }
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| 134 |
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| 135 | /*************************************************************************
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| 136 | Integration of a smooth function F(x) on a finite interval [a,b].
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| 137 |
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| 138 | This subroutine is same as AutoGKSmooth(), but it guarantees that interval
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| 139 | [a,b] is partitioned into subintervals which have width at most XWidth.
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| 140 |
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| 141 | Subroutine can be used when integrating nearly-constant function with
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| 142 | narrow "bumps" (about XWidth wide). If "bumps" are too narrow, AutoGKSmooth
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| 143 | subroutine can overlook them.
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| 144 |
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| 145 | INPUT PARAMETERS:
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| 146 | A, B - interval boundaries (A<B, A=B or A>B)
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| 147 |
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| 148 | OUTPUT PARAMETERS
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| 149 | State - structure which stores algorithm state
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| 150 |
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| 151 | SEE ALSO
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| 152 | AutoGKSmooth, AutoGKSingular, AutoGKResults.
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| 153 |
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| 154 |
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| 155 | -- ALGLIB --
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| 156 | Copyright 06.05.2009 by Bochkanov Sergey
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| 157 | *************************************************************************/
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| 158 | public static void autogksmoothw(double a, double b, double xwidth, out autogkstate state)
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| 159 | {
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| 160 | state = new autogkstate();
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| 161 | autogk.autogksmoothw(a, b, xwidth, state.innerobj);
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| 162 | return;
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| 163 | }
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| 164 |
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| 165 | /*************************************************************************
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| 166 | Integration on a finite interval [A,B].
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| 167 | Integrand have integrable singularities at A/B.
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| 168 |
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| 169 | F(X) must diverge as "(x-A)^alpha" at A, as "(B-x)^beta" at B, with known
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| 170 | alpha/beta (alpha>-1, beta>-1). If alpha/beta are not known, estimates
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| 171 | from below can be used (but these estimates should be greater than -1 too).
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| 172 |
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| 173 | One of alpha/beta variables (or even both alpha/beta) may be equal to 0,
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| 174 | which means than function F(x) is non-singular at A/B. Anyway (singular at
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| 175 | bounds or not), function F(x) is supposed to be continuous on (A,B).
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| 176 |
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| 177 | Fast-convergent algorithm based on a Gauss-Kronrod formula is used. Result
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| 178 | is calculated with accuracy close to the machine precision.
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| 179 |
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| 180 | INPUT PARAMETERS:
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| 181 | A, B - interval boundaries (A<B, A=B or A>B)
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| 182 | Alpha - power-law coefficient of the F(x) at A,
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| 183 | Alpha>-1
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| 184 | Beta - power-law coefficient of the F(x) at B,
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| 185 | Beta>-1
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| 186 |
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| 187 | OUTPUT PARAMETERS
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| 188 | State - structure which stores algorithm state
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| 189 |
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| 190 | SEE ALSO
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| 191 | AutoGKSmooth, AutoGKSmoothW, AutoGKResults.
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| 192 |
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| 193 |
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| 194 | -- ALGLIB --
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| 195 | Copyright 06.05.2009 by Bochkanov Sergey
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| 196 | *************************************************************************/
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| 197 | public static void autogksingular(double a, double b, double alpha, double beta, out autogkstate state)
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| 198 | {
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| 199 | state = new autogkstate();
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| 200 | autogk.autogksingular(a, b, alpha, beta, state.innerobj);
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| 201 | return;
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| 202 | }
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| 203 |
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| 204 | /*************************************************************************
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| 205 | This function provides reverse communication interface
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| 206 | Reverse communication interface is not documented or recommended to use.
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| 207 | See below for functions which provide better documented API
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| 208 | *************************************************************************/
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| 209 | public static bool autogkiteration(autogkstate state)
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| 210 | {
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| 211 |
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| 212 | bool result = autogk.autogkiteration(state.innerobj);
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| 213 | return result;
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| 214 | }
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| 215 |
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| 216 |
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| 217 | /*************************************************************************
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| 218 | This function is used to launcn iterations of ODE solver
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| 219 |
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| 220 | It accepts following parameters:
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| 221 | diff - callback which calculates dy/dx for given y and x
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| 222 | obj - optional object which is passed to diff; can be NULL
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| 223 |
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| 224 |
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| 225 | -- ALGLIB --
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| 226 | Copyright 07.05.2009 by Bochkanov Sergey
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| 227 |
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| 228 | *************************************************************************/
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| 229 | public static void autogkintegrate(autogkstate state, integrator1_func func, object obj)
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| 230 | {
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| 231 | if( func==null )
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| 232 | throw new alglibexception("ALGLIB: error in 'autogkintegrate()' (func is null)");
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| 233 | while( alglib.autogkiteration(state) )
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| 234 | {
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| 235 | if( state.needf )
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| 236 | {
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| 237 | func(state.innerobj.x, state.innerobj.xminusa, state.innerobj.bminusx, ref state.innerobj.f, obj);
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| 238 | continue;
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| 239 | }
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| 240 | throw new alglibexception("ALGLIB: unexpected error in 'autogksolve'");
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| 241 | }
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| 242 | }
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| 243 |
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| 244 | /*************************************************************************
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| 245 | Adaptive integration results
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| 246 |
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| 247 | Called after AutoGKIteration returned False.
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| 248 |
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| 249 | Input parameters:
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| 250 | State - algorithm state (used by AutoGKIteration).
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| 251 |
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| 252 | Output parameters:
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| 253 | V - integral(f(x)dx,a,b)
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| 254 | Rep - optimization report (see AutoGKReport description)
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| 255 |
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| 256 | -- ALGLIB --
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| 257 | Copyright 14.11.2007 by Bochkanov Sergey
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| 258 | *************************************************************************/
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| 259 | public static void autogkresults(autogkstate state, out double v, out autogkreport rep)
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| 260 | {
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| 261 | v = 0;
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| 262 | rep = new autogkreport();
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| 263 | autogk.autogkresults(state.innerobj, ref v, rep.innerobj);
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| 264 | return;
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| 265 | }
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| 266 |
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| 267 | }
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| 268 | public partial class alglib
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| 269 | {
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| 270 |
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| 271 |
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| 272 | /*************************************************************************
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| 273 | Computation of nodes and weights for a Gauss quadrature formula
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| 274 |
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| 275 | The algorithm generates the N-point Gauss quadrature formula with weight
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| 276 | function given by coefficients alpha and beta of a recurrence relation
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| 277 | which generates a system of orthogonal polynomials:
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| 278 |
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| 279 | P-1(x) = 0
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| 280 | P0(x) = 1
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| 281 | Pn+1(x) = (x-alpha(n))*Pn(x) - beta(n)*Pn-1(x)
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| 282 |
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| 283 | and zeroth moment Mu0
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| 284 |
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| 285 | Mu0 = integral(W(x)dx,a,b)
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| 286 |
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| 287 | INPUT PARAMETERS:
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| 288 | Alpha array[0..N-1], alpha coefficients
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| 289 | Beta array[0..N-1], beta coefficients
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| 290 | Zero-indexed element is not used and may be arbitrary.
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| 291 | Beta[I]>0.
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| 292 | Mu0 zeroth moment of the weight function.
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| 293 | N number of nodes of the quadrature formula, N>=1
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| 294 |
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| 295 | OUTPUT PARAMETERS:
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| 296 | Info - error code:
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| 297 | * -3 internal eigenproblem solver hasn't converged
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| 298 | * -2 Beta[i]<=0
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| 299 | * -1 incorrect N was passed
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| 300 | * 1 OK
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| 301 | X - array[0..N-1] - array of quadrature nodes,
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| 302 | in ascending order.
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| 303 | W - array[0..N-1] - array of quadrature weights.
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| 304 |
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| 305 | -- ALGLIB --
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| 306 | Copyright 2005-2009 by Bochkanov Sergey
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| 307 | *************************************************************************/
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| 308 | public static void gqgeneraterec(double[] alpha, double[] beta, double mu0, int n, out int info, out double[] x, out double[] w)
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| 309 | {
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| 310 | info = 0;
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| 311 | x = new double[0];
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| 312 | w = new double[0];
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| 313 | gq.gqgeneraterec(alpha, beta, mu0, n, ref info, ref x, ref w);
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| 314 | return;
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| 315 | }
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| 316 |
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| 317 | /*************************************************************************
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| 318 | Computation of nodes and weights for a Gauss-Lobatto quadrature formula
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| 319 |
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| 320 | The algorithm generates the N-point Gauss-Lobatto quadrature formula with
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| 321 | weight function given by coefficients alpha and beta of a recurrence which
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| 322 | generates a system of orthogonal polynomials.
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| 323 |
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| 324 | P-1(x) = 0
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| 325 | P0(x) = 1
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| 326 | Pn+1(x) = (x-alpha(n))*Pn(x) - beta(n)*Pn-1(x)
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| 327 |
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| 328 | and zeroth moment Mu0
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| 329 |
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| 330 | Mu0 = integral(W(x)dx,a,b)
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| 331 |
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| 332 | INPUT PARAMETERS:
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| 333 | Alpha array[0..N-2], alpha coefficients
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| 334 | Beta array[0..N-2], beta coefficients.
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| 335 | Zero-indexed element is not used, may be arbitrary.
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| 336 | Beta[I]>0
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| 337 | Mu0 zeroth moment of the weighting function.
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| 338 | A left boundary of the integration interval.
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| 339 | B right boundary of the integration interval.
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| 340 | N number of nodes of the quadrature formula, N>=3
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| 341 | (including the left and right boundary nodes).
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| 342 |
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| 343 | OUTPUT PARAMETERS:
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| 344 | Info - error code:
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| 345 | * -3 internal eigenproblem solver hasn't converged
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| 346 | * -2 Beta[i]<=0
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| 347 | * -1 incorrect N was passed
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| 348 | * 1 OK
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| 349 | X - array[0..N-1] - array of quadrature nodes,
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| 350 | in ascending order.
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| 351 | W - array[0..N-1] - array of quadrature weights.
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| 352 |
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| 353 | -- ALGLIB --
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| 354 | Copyright 2005-2009 by Bochkanov Sergey
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| 355 | *************************************************************************/
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| 356 | public static void gqgenerategausslobattorec(double[] alpha, double[] beta, double mu0, double a, double b, int n, out int info, out double[] x, out double[] w)
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| 357 | {
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| 358 | info = 0;
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| 359 | x = new double[0];
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| 360 | w = new double[0];
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| 361 | gq.gqgenerategausslobattorec(alpha, beta, mu0, a, b, n, ref info, ref x, ref w);
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| 362 | return;
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| 363 | }
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| 364 |
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| 365 | /*************************************************************************
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| 366 | Computation of nodes and weights for a Gauss-Radau quadrature formula
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| 367 |
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| 368 | The algorithm generates the N-point Gauss-Radau quadrature formula with
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| 369 | weight function given by the coefficients alpha and beta of a recurrence
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| 370 | which generates a system of orthogonal polynomials.
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| 371 |
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| 372 | P-1(x) = 0
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| 373 | P0(x) = 1
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| 374 | Pn+1(x) = (x-alpha(n))*Pn(x) - beta(n)*Pn-1(x)
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| 375 |
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| 376 | and zeroth moment Mu0
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| 377 |
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| 378 | Mu0 = integral(W(x)dx,a,b)
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| 379 |
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| 380 | INPUT PARAMETERS:
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| 381 | Alpha array[0..N-2], alpha coefficients.
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| 382 | Beta array[0..N-1], beta coefficients
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| 383 | Zero-indexed element is not used.
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| 384 | Beta[I]>0
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| 385 | Mu0 zeroth moment of the weighting function.
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| 386 | A left boundary of the integration interval.
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| 387 | N number of nodes of the quadrature formula, N>=2
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| 388 | (including the left boundary node).
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| 389 |
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| 390 | OUTPUT PARAMETERS:
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| 391 | Info - error code:
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| 392 | * -3 internal eigenproblem solver hasn't converged
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| 393 | * -2 Beta[i]<=0
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| 394 | * -1 incorrect N was passed
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| 395 | * 1 OK
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| 396 | X - array[0..N-1] - array of quadrature nodes,
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| 397 | in ascending order.
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| 398 | W - array[0..N-1] - array of quadrature weights.
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| 399 |
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| 400 |
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| 401 | -- ALGLIB --
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| 402 | Copyright 2005-2009 by Bochkanov Sergey
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| 403 | *************************************************************************/
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| 404 | public static void gqgenerategaussradaurec(double[] alpha, double[] beta, double mu0, double a, int n, out int info, out double[] x, out double[] w)
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| 405 | {
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| 406 | info = 0;
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| 407 | x = new double[0];
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| 408 | w = new double[0];
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| 409 | gq.gqgenerategaussradaurec(alpha, beta, mu0, a, n, ref info, ref x, ref w);
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| 410 | return;
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| 411 | }
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| 412 |
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| 413 | /*************************************************************************
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| 414 | Returns nodes/weights for Gauss-Legendre quadrature on [-1,1] with N
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| 415 | nodes.
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| 416 |
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| 417 | INPUT PARAMETERS:
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| 418 | N - number of nodes, >=1
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| 419 |
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| 420 | OUTPUT PARAMETERS:
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| 421 | Info - error code:
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| 422 | * -4 an error was detected when calculating
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| 423 | weights/nodes. N is too large to obtain
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| 424 | weights/nodes with high enough accuracy.
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| 425 | Try to use multiple precision version.
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| 426 | * -3 internal eigenproblem solver hasn't converged
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| 427 | * -1 incorrect N was passed
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| 428 | * +1 OK
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| 429 | X - array[0..N-1] - array of quadrature nodes,
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| 430 | in ascending order.
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| 431 | W - array[0..N-1] - array of quadrature weights.
|
---|
| 432 |
|
---|
| 433 |
|
---|
| 434 | -- ALGLIB --
|
---|
| 435 | Copyright 12.05.2009 by Bochkanov Sergey
|
---|
| 436 | *************************************************************************/
|
---|
| 437 | public static void gqgenerategausslegendre(int n, out int info, out double[] x, out double[] w)
|
---|
| 438 | {
|
---|
| 439 | info = 0;
|
---|
| 440 | x = new double[0];
|
---|
| 441 | w = new double[0];
|
---|
| 442 | gq.gqgenerategausslegendre(n, ref info, ref x, ref w);
|
---|
| 443 | return;
|
---|
| 444 | }
|
---|
| 445 |
|
---|
| 446 | /*************************************************************************
|
---|
| 447 | Returns nodes/weights for Gauss-Jacobi quadrature on [-1,1] with weight
|
---|
| 448 | function W(x)=Power(1-x,Alpha)*Power(1+x,Beta).
|
---|
| 449 |
|
---|
| 450 | INPUT PARAMETERS:
|
---|
| 451 | N - number of nodes, >=1
|
---|
| 452 | Alpha - power-law coefficient, Alpha>-1
|
---|
| 453 | Beta - power-law coefficient, Beta>-1
|
---|
| 454 |
|
---|
| 455 | OUTPUT PARAMETERS:
|
---|
| 456 | Info - error code:
|
---|
| 457 | * -4 an error was detected when calculating
|
---|
| 458 | weights/nodes. Alpha or Beta are too close
|
---|
| 459 | to -1 to obtain weights/nodes with high enough
|
---|
| 460 | accuracy, or, may be, N is too large. Try to
|
---|
| 461 | use multiple precision version.
|
---|
| 462 | * -3 internal eigenproblem solver hasn't converged
|
---|
| 463 | * -1 incorrect N/Alpha/Beta was passed
|
---|
| 464 | * +1 OK
|
---|
| 465 | X - array[0..N-1] - array of quadrature nodes,
|
---|
| 466 | in ascending order.
|
---|
| 467 | W - array[0..N-1] - array of quadrature weights.
|
---|
| 468 |
|
---|
| 469 |
|
---|
| 470 | -- ALGLIB --
|
---|
| 471 | Copyright 12.05.2009 by Bochkanov Sergey
|
---|
| 472 | *************************************************************************/
|
---|
| 473 | public static void gqgenerategaussjacobi(int n, double alpha, double beta, out int info, out double[] x, out double[] w)
|
---|
| 474 | {
|
---|
| 475 | info = 0;
|
---|
| 476 | x = new double[0];
|
---|
| 477 | w = new double[0];
|
---|
| 478 | gq.gqgenerategaussjacobi(n, alpha, beta, ref info, ref x, ref w);
|
---|
| 479 | return;
|
---|
| 480 | }
|
---|
| 481 |
|
---|
| 482 | /*************************************************************************
|
---|
| 483 | Returns nodes/weights for Gauss-Laguerre quadrature on [0,+inf) with
|
---|
| 484 | weight function W(x)=Power(x,Alpha)*Exp(-x)
|
---|
| 485 |
|
---|
| 486 | INPUT PARAMETERS:
|
---|
| 487 | N - number of nodes, >=1
|
---|
| 488 | Alpha - power-law coefficient, Alpha>-1
|
---|
| 489 |
|
---|
| 490 | OUTPUT PARAMETERS:
|
---|
| 491 | Info - error code:
|
---|
| 492 | * -4 an error was detected when calculating
|
---|
| 493 | weights/nodes. Alpha is too close to -1 to
|
---|
| 494 | obtain weights/nodes with high enough accuracy
|
---|
| 495 | or, may be, N is too large. Try to use
|
---|
| 496 | multiple precision version.
|
---|
| 497 | * -3 internal eigenproblem solver hasn't converged
|
---|
| 498 | * -1 incorrect N/Alpha was passed
|
---|
| 499 | * +1 OK
|
---|
| 500 | X - array[0..N-1] - array of quadrature nodes,
|
---|
| 501 | in ascending order.
|
---|
| 502 | W - array[0..N-1] - array of quadrature weights.
|
---|
| 503 |
|
---|
| 504 |
|
---|
| 505 | -- ALGLIB --
|
---|
| 506 | Copyright 12.05.2009 by Bochkanov Sergey
|
---|
| 507 | *************************************************************************/
|
---|
| 508 | public static void gqgenerategausslaguerre(int n, double alpha, out int info, out double[] x, out double[] w)
|
---|
| 509 | {
|
---|
| 510 | info = 0;
|
---|
| 511 | x = new double[0];
|
---|
| 512 | w = new double[0];
|
---|
| 513 | gq.gqgenerategausslaguerre(n, alpha, ref info, ref x, ref w);
|
---|
| 514 | return;
|
---|
| 515 | }
|
---|
| 516 |
|
---|
| 517 | /*************************************************************************
|
---|
| 518 | Returns nodes/weights for Gauss-Hermite quadrature on (-inf,+inf) with
|
---|
| 519 | weight function W(x)=Exp(-x*x)
|
---|
| 520 |
|
---|
| 521 | INPUT PARAMETERS:
|
---|
| 522 | N - number of nodes, >=1
|
---|
| 523 |
|
---|
| 524 | OUTPUT PARAMETERS:
|
---|
| 525 | Info - error code:
|
---|
| 526 | * -4 an error was detected when calculating
|
---|
| 527 | weights/nodes. May be, N is too large. Try to
|
---|
| 528 | use multiple precision version.
|
---|
| 529 | * -3 internal eigenproblem solver hasn't converged
|
---|
| 530 | * -1 incorrect N/Alpha was passed
|
---|
| 531 | * +1 OK
|
---|
| 532 | X - array[0..N-1] - array of quadrature nodes,
|
---|
| 533 | in ascending order.
|
---|
| 534 | W - array[0..N-1] - array of quadrature weights.
|
---|
| 535 |
|
---|
| 536 |
|
---|
| 537 | -- ALGLIB --
|
---|
| 538 | Copyright 12.05.2009 by Bochkanov Sergey
|
---|
| 539 | *************************************************************************/
|
---|
| 540 | public static void gqgenerategausshermite(int n, out int info, out double[] x, out double[] w)
|
---|
| 541 | {
|
---|
| 542 | info = 0;
|
---|
| 543 | x = new double[0];
|
---|
| 544 | w = new double[0];
|
---|
| 545 | gq.gqgenerategausshermite(n, ref info, ref x, ref w);
|
---|
| 546 | return;
|
---|
| 547 | }
|
---|
| 548 |
|
---|
| 549 | }
|
---|
| 550 | public partial class alglib
|
---|
| 551 | {
|
---|
| 552 |
|
---|
| 553 |
|
---|
| 554 | /*************************************************************************
|
---|
| 555 | Computation of nodes and weights of a Gauss-Kronrod quadrature formula
|
---|
| 556 |
|
---|
| 557 | The algorithm generates the N-point Gauss-Kronrod quadrature formula with
|
---|
| 558 | weight function given by coefficients alpha and beta of a recurrence
|
---|
| 559 | relation which generates a system of orthogonal polynomials:
|
---|
| 560 |
|
---|
| 561 | P-1(x) = 0
|
---|
| 562 | P0(x) = 1
|
---|
| 563 | Pn+1(x) = (x-alpha(n))*Pn(x) - beta(n)*Pn-1(x)
|
---|
| 564 |
|
---|
| 565 | and zero moment Mu0
|
---|
| 566 |
|
---|
| 567 | Mu0 = integral(W(x)dx,a,b)
|
---|
| 568 |
|
---|
| 569 |
|
---|
| 570 | INPUT PARAMETERS:
|
---|
| 571 | Alpha alpha coefficients, array[0..floor(3*K/2)].
|
---|
| 572 | Beta beta coefficients, array[0..ceil(3*K/2)].
|
---|
| 573 | Beta[0] is not used and may be arbitrary.
|
---|
| 574 | Beta[I]>0.
|
---|
| 575 | Mu0 zeroth moment of the weight function.
|
---|
| 576 | N number of nodes of the Gauss-Kronrod quadrature formula,
|
---|
| 577 | N >= 3,
|
---|
| 578 | N = 2*K+1.
|
---|
| 579 |
|
---|
| 580 | OUTPUT PARAMETERS:
|
---|
| 581 | Info - error code:
|
---|
| 582 | * -5 no real and positive Gauss-Kronrod formula can
|
---|
| 583 | be created for such a weight function with a
|
---|
| 584 | given number of nodes.
|
---|
| 585 | * -4 N is too large, task may be ill conditioned -
|
---|
| 586 | x[i]=x[i+1] found.
|
---|
| 587 | * -3 internal eigenproblem solver hasn't converged
|
---|
| 588 | * -2 Beta[i]<=0
|
---|
| 589 | * -1 incorrect N was passed
|
---|
| 590 | * +1 OK
|
---|
| 591 | X - array[0..N-1] - array of quadrature nodes,
|
---|
| 592 | in ascending order.
|
---|
| 593 | WKronrod - array[0..N-1] - Kronrod weights
|
---|
| 594 | WGauss - array[0..N-1] - Gauss weights (interleaved with zeros
|
---|
| 595 | corresponding to extended Kronrod nodes).
|
---|
| 596 |
|
---|
| 597 | -- ALGLIB --
|
---|
| 598 | Copyright 08.05.2009 by Bochkanov Sergey
|
---|
| 599 | *************************************************************************/
|
---|
| 600 | public static void gkqgeneraterec(double[] alpha, double[] beta, double mu0, int n, out int info, out double[] x, out double[] wkronrod, out double[] wgauss)
|
---|
| 601 | {
|
---|
| 602 | info = 0;
|
---|
| 603 | x = new double[0];
|
---|
| 604 | wkronrod = new double[0];
|
---|
| 605 | wgauss = new double[0];
|
---|
| 606 | gkq.gkqgeneraterec(alpha, beta, mu0, n, ref info, ref x, ref wkronrod, ref wgauss);
|
---|
| 607 | return;
|
---|
| 608 | }
|
---|
| 609 |
|
---|
| 610 | /*************************************************************************
|
---|
| 611 | Returns Gauss and Gauss-Kronrod nodes/weights for Gauss-Legendre
|
---|
| 612 | quadrature with N points.
|
---|
| 613 |
|
---|
| 614 | GKQLegendreCalc (calculation) or GKQLegendreTbl (precomputed table) is
|
---|
| 615 | used depending on machine precision and number of nodes.
|
---|
| 616 |
|
---|
| 617 | INPUT PARAMETERS:
|
---|
| 618 | N - number of Kronrod nodes, must be odd number, >=3.
|
---|
| 619 |
|
---|
| 620 | OUTPUT PARAMETERS:
|
---|
| 621 | Info - error code:
|
---|
| 622 | * -4 an error was detected when calculating
|
---|
| 623 | weights/nodes. N is too large to obtain
|
---|
| 624 | weights/nodes with high enough accuracy.
|
---|
| 625 | Try to use multiple precision version.
|
---|
| 626 | * -3 internal eigenproblem solver hasn't converged
|
---|
| 627 | * -1 incorrect N was passed
|
---|
| 628 | * +1 OK
|
---|
| 629 | X - array[0..N-1] - array of quadrature nodes, ordered in
|
---|
| 630 | ascending order.
|
---|
| 631 | WKronrod - array[0..N-1] - Kronrod weights
|
---|
| 632 | WGauss - array[0..N-1] - Gauss weights (interleaved with zeros
|
---|
| 633 | corresponding to extended Kronrod nodes).
|
---|
| 634 |
|
---|
| 635 |
|
---|
| 636 | -- ALGLIB --
|
---|
| 637 | Copyright 12.05.2009 by Bochkanov Sergey
|
---|
| 638 | *************************************************************************/
|
---|
| 639 | public static void gkqgenerategausslegendre(int n, out int info, out double[] x, out double[] wkronrod, out double[] wgauss)
|
---|
| 640 | {
|
---|
| 641 | info = 0;
|
---|
| 642 | x = new double[0];
|
---|
| 643 | wkronrod = new double[0];
|
---|
| 644 | wgauss = new double[0];
|
---|
| 645 | gkq.gkqgenerategausslegendre(n, ref info, ref x, ref wkronrod, ref wgauss);
|
---|
| 646 | return;
|
---|
| 647 | }
|
---|
| 648 |
|
---|
| 649 | /*************************************************************************
|
---|
| 650 | Returns Gauss and Gauss-Kronrod nodes/weights for Gauss-Jacobi
|
---|
| 651 | quadrature on [-1,1] with weight function
|
---|
| 652 |
|
---|
| 653 | W(x)=Power(1-x,Alpha)*Power(1+x,Beta).
|
---|
| 654 |
|
---|
| 655 | INPUT PARAMETERS:
|
---|
| 656 | N - number of Kronrod nodes, must be odd number, >=3.
|
---|
| 657 | Alpha - power-law coefficient, Alpha>-1
|
---|
| 658 | Beta - power-law coefficient, Beta>-1
|
---|
| 659 |
|
---|
| 660 | OUTPUT PARAMETERS:
|
---|
| 661 | Info - error code:
|
---|
| 662 | * -5 no real and positive Gauss-Kronrod formula can
|
---|
| 663 | be created for such a weight function with a
|
---|
| 664 | given number of nodes.
|
---|
| 665 | * -4 an error was detected when calculating
|
---|
| 666 | weights/nodes. Alpha or Beta are too close
|
---|
| 667 | to -1 to obtain weights/nodes with high enough
|
---|
| 668 | accuracy, or, may be, N is too large. Try to
|
---|
| 669 | use multiple precision version.
|
---|
| 670 | * -3 internal eigenproblem solver hasn't converged
|
---|
| 671 | * -1 incorrect N was passed
|
---|
| 672 | * +1 OK
|
---|
| 673 | * +2 OK, but quadrature rule have exterior nodes,
|
---|
| 674 | x[0]<-1 or x[n-1]>+1
|
---|
| 675 | X - array[0..N-1] - array of quadrature nodes, ordered in
|
---|
| 676 | ascending order.
|
---|
| 677 | WKronrod - array[0..N-1] - Kronrod weights
|
---|
| 678 | WGauss - array[0..N-1] - Gauss weights (interleaved with zeros
|
---|
| 679 | corresponding to extended Kronrod nodes).
|
---|
| 680 |
|
---|
| 681 |
|
---|
| 682 | -- ALGLIB --
|
---|
| 683 | Copyright 12.05.2009 by Bochkanov Sergey
|
---|
| 684 | *************************************************************************/
|
---|
| 685 | public static void gkqgenerategaussjacobi(int n, double alpha, double beta, out int info, out double[] x, out double[] wkronrod, out double[] wgauss)
|
---|
| 686 | {
|
---|
| 687 | info = 0;
|
---|
| 688 | x = new double[0];
|
---|
| 689 | wkronrod = new double[0];
|
---|
| 690 | wgauss = new double[0];
|
---|
| 691 | gkq.gkqgenerategaussjacobi(n, alpha, beta, ref info, ref x, ref wkronrod, ref wgauss);
|
---|
| 692 | return;
|
---|
| 693 | }
|
---|
| 694 |
|
---|
| 695 | /*************************************************************************
|
---|
| 696 | Returns Gauss and Gauss-Kronrod nodes for quadrature with N points.
|
---|
| 697 |
|
---|
| 698 | Reduction to tridiagonal eigenproblem is used.
|
---|
| 699 |
|
---|
| 700 | INPUT PARAMETERS:
|
---|
| 701 | N - number of Kronrod nodes, must be odd number, >=3.
|
---|
| 702 |
|
---|
| 703 | OUTPUT PARAMETERS:
|
---|
| 704 | Info - error code:
|
---|
| 705 | * -4 an error was detected when calculating
|
---|
| 706 | weights/nodes. N is too large to obtain
|
---|
| 707 | weights/nodes with high enough accuracy.
|
---|
| 708 | Try to use multiple precision version.
|
---|
| 709 | * -3 internal eigenproblem solver hasn't converged
|
---|
| 710 | * -1 incorrect N was passed
|
---|
| 711 | * +1 OK
|
---|
| 712 | X - array[0..N-1] - array of quadrature nodes, ordered in
|
---|
| 713 | ascending order.
|
---|
| 714 | WKronrod - array[0..N-1] - Kronrod weights
|
---|
| 715 | WGauss - array[0..N-1] - Gauss weights (interleaved with zeros
|
---|
| 716 | corresponding to extended Kronrod nodes).
|
---|
| 717 |
|
---|
| 718 | -- ALGLIB --
|
---|
| 719 | Copyright 12.05.2009 by Bochkanov Sergey
|
---|
| 720 | *************************************************************************/
|
---|
| 721 | public static void gkqlegendrecalc(int n, out int info, out double[] x, out double[] wkronrod, out double[] wgauss)
|
---|
| 722 | {
|
---|
| 723 | info = 0;
|
---|
| 724 | x = new double[0];
|
---|
| 725 | wkronrod = new double[0];
|
---|
| 726 | wgauss = new double[0];
|
---|
| 727 | gkq.gkqlegendrecalc(n, ref info, ref x, ref wkronrod, ref wgauss);
|
---|
| 728 | return;
|
---|
| 729 | }
|
---|
| 730 |
|
---|
| 731 | /*************************************************************************
|
---|
| 732 | Returns Gauss and Gauss-Kronrod nodes for quadrature with N points using
|
---|
| 733 | pre-calculated table. Nodes/weights were computed with accuracy up to
|
---|
| 734 | 1.0E-32 (if MPFR version of ALGLIB is used). In standard double precision
|
---|
| 735 | accuracy reduces to something about 2.0E-16 (depending on your compiler's
|
---|
| 736 | handling of long floating point constants).
|
---|
| 737 |
|
---|
| 738 | INPUT PARAMETERS:
|
---|
| 739 | N - number of Kronrod nodes.
|
---|
| 740 | N can be 15, 21, 31, 41, 51, 61.
|
---|
| 741 |
|
---|
| 742 | OUTPUT PARAMETERS:
|
---|
| 743 | X - array[0..N-1] - array of quadrature nodes, ordered in
|
---|
| 744 | ascending order.
|
---|
| 745 | WKronrod - array[0..N-1] - Kronrod weights
|
---|
| 746 | WGauss - array[0..N-1] - Gauss weights (interleaved with zeros
|
---|
| 747 | corresponding to extended Kronrod nodes).
|
---|
| 748 |
|
---|
| 749 |
|
---|
| 750 | -- ALGLIB --
|
---|
| 751 | Copyright 12.05.2009 by Bochkanov Sergey
|
---|
| 752 | *************************************************************************/
|
---|
| 753 | public static void gkqlegendretbl(int n, out double[] x, out double[] wkronrod, out double[] wgauss, out double eps)
|
---|
| 754 | {
|
---|
| 755 | x = new double[0];
|
---|
| 756 | wkronrod = new double[0];
|
---|
| 757 | wgauss = new double[0];
|
---|
| 758 | eps = 0;
|
---|
| 759 | gkq.gkqlegendretbl(n, ref x, ref wkronrod, ref wgauss, ref eps);
|
---|
| 760 | return;
|
---|
| 761 | }
|
---|
| 762 |
|
---|
| 763 | }
|
---|
| 764 | public partial class alglib
|
---|
| 765 | {
|
---|
| 766 | public class autogk
|
---|
| 767 | {
|
---|
| 768 | /*************************************************************************
|
---|
| 769 | Integration report:
|
---|
| 770 | * TerminationType = completetion code:
|
---|
| 771 | * -5 non-convergence of Gauss-Kronrod nodes
|
---|
| 772 | calculation subroutine.
|
---|
| 773 | * -1 incorrect parameters were specified
|
---|
| 774 | * 1 OK
|
---|
| 775 | * Rep.NFEV countains number of function calculations
|
---|
| 776 | * Rep.NIntervals contains number of intervals [a,b]
|
---|
| 777 | was partitioned into.
|
---|
| 778 | *************************************************************************/
|
---|
| 779 | public class autogkreport
|
---|
| 780 | {
|
---|
| 781 | public int terminationtype;
|
---|
| 782 | public int nfev;
|
---|
| 783 | public int nintervals;
|
---|
| 784 | };
|
---|
| 785 |
|
---|
| 786 |
|
---|
| 787 | public class autogkinternalstate
|
---|
| 788 | {
|
---|
| 789 | public double a;
|
---|
| 790 | public double b;
|
---|
| 791 | public double eps;
|
---|
| 792 | public double xwidth;
|
---|
| 793 | public double x;
|
---|
| 794 | public double f;
|
---|
| 795 | public int info;
|
---|
| 796 | public double r;
|
---|
| 797 | public double[,] heap;
|
---|
| 798 | public int heapsize;
|
---|
| 799 | public int heapwidth;
|
---|
| 800 | public int heapused;
|
---|
| 801 | public double sumerr;
|
---|
| 802 | public double sumabs;
|
---|
| 803 | public double[] qn;
|
---|
| 804 | public double[] wg;
|
---|
| 805 | public double[] wk;
|
---|
| 806 | public double[] wr;
|
---|
| 807 | public int n;
|
---|
| 808 | public rcommstate rstate;
|
---|
| 809 | public autogkinternalstate()
|
---|
| 810 | {
|
---|
| 811 | heap = new double[0,0];
|
---|
| 812 | qn = new double[0];
|
---|
| 813 | wg = new double[0];
|
---|
| 814 | wk = new double[0];
|
---|
| 815 | wr = new double[0];
|
---|
| 816 | rstate = new rcommstate();
|
---|
| 817 | }
|
---|
| 818 | };
|
---|
| 819 |
|
---|
| 820 |
|
---|
| 821 | /*************************************************************************
|
---|
| 822 | This structure stores state of the integration algorithm.
|
---|
| 823 |
|
---|
| 824 | Although this class has public fields, they are not intended for external
|
---|
| 825 | use. You should use ALGLIB functions to work with this class:
|
---|
| 826 | * autogksmooth()/AutoGKSmoothW()/... to create objects
|
---|
| 827 | * autogkintegrate() to begin integration
|
---|
| 828 | * autogkresults() to get results
|
---|
| 829 | *************************************************************************/
|
---|
| 830 | public class autogkstate
|
---|
| 831 | {
|
---|
| 832 | public double a;
|
---|
| 833 | public double b;
|
---|
| 834 | public double alpha;
|
---|
| 835 | public double beta;
|
---|
| 836 | public double xwidth;
|
---|
| 837 | public double x;
|
---|
| 838 | public double xminusa;
|
---|
| 839 | public double bminusx;
|
---|
| 840 | public bool needf;
|
---|
| 841 | public double f;
|
---|
| 842 | public int wrappermode;
|
---|
| 843 | public autogkinternalstate internalstate;
|
---|
| 844 | public rcommstate rstate;
|
---|
| 845 | public double v;
|
---|
| 846 | public int terminationtype;
|
---|
| 847 | public int nfev;
|
---|
| 848 | public int nintervals;
|
---|
| 849 | public autogkstate()
|
---|
| 850 | {
|
---|
| 851 | internalstate = new autogkinternalstate();
|
---|
| 852 | rstate = new rcommstate();
|
---|
| 853 | }
|
---|
| 854 | };
|
---|
| 855 |
|
---|
| 856 |
|
---|
| 857 |
|
---|
| 858 |
|
---|
| 859 | /*************************************************************************
|
---|
| 860 | Integration of a smooth function F(x) on a finite interval [a,b].
|
---|
| 861 |
|
---|
| 862 | Fast-convergent algorithm based on a Gauss-Kronrod formula is used. Result
|
---|
| 863 | is calculated with accuracy close to the machine precision.
|
---|
| 864 |
|
---|
| 865 | Algorithm works well only with smooth integrands. It may be used with
|
---|
| 866 | continuous non-smooth integrands, but with less performance.
|
---|
| 867 |
|
---|
| 868 | It should never be used with integrands which have integrable singularities
|
---|
| 869 | at lower or upper limits - algorithm may crash. Use AutoGKSingular in such
|
---|
| 870 | cases.
|
---|
| 871 |
|
---|
| 872 | INPUT PARAMETERS:
|
---|
| 873 | A, B - interval boundaries (A<B, A=B or A>B)
|
---|
| 874 |
|
---|
| 875 | OUTPUT PARAMETERS
|
---|
| 876 | State - structure which stores algorithm state
|
---|
| 877 |
|
---|
| 878 | SEE ALSO
|
---|
| 879 | AutoGKSmoothW, AutoGKSingular, AutoGKResults.
|
---|
| 880 |
|
---|
| 881 |
|
---|
| 882 | -- ALGLIB --
|
---|
| 883 | Copyright 06.05.2009 by Bochkanov Sergey
|
---|
| 884 | *************************************************************************/
|
---|
| 885 | public static void autogksmooth(double a,
|
---|
| 886 | double b,
|
---|
| 887 | autogkstate state)
|
---|
| 888 | {
|
---|
| 889 | ap.assert(math.isfinite(a), "AutoGKSmooth: A is not finite!");
|
---|
| 890 | ap.assert(math.isfinite(b), "AutoGKSmooth: B is not finite!");
|
---|
| 891 | autogksmoothw(a, b, 0.0, state);
|
---|
| 892 | }
|
---|
| 893 |
|
---|
| 894 |
|
---|
| 895 | /*************************************************************************
|
---|
| 896 | Integration of a smooth function F(x) on a finite interval [a,b].
|
---|
| 897 |
|
---|
| 898 | This subroutine is same as AutoGKSmooth(), but it guarantees that interval
|
---|
| 899 | [a,b] is partitioned into subintervals which have width at most XWidth.
|
---|
| 900 |
|
---|
| 901 | Subroutine can be used when integrating nearly-constant function with
|
---|
| 902 | narrow "bumps" (about XWidth wide). If "bumps" are too narrow, AutoGKSmooth
|
---|
| 903 | subroutine can overlook them.
|
---|
| 904 |
|
---|
| 905 | INPUT PARAMETERS:
|
---|
| 906 | A, B - interval boundaries (A<B, A=B or A>B)
|
---|
| 907 |
|
---|
| 908 | OUTPUT PARAMETERS
|
---|
| 909 | State - structure which stores algorithm state
|
---|
| 910 |
|
---|
| 911 | SEE ALSO
|
---|
| 912 | AutoGKSmooth, AutoGKSingular, AutoGKResults.
|
---|
| 913 |
|
---|
| 914 |
|
---|
| 915 | -- ALGLIB --
|
---|
| 916 | Copyright 06.05.2009 by Bochkanov Sergey
|
---|
| 917 | *************************************************************************/
|
---|
| 918 | public static void autogksmoothw(double a,
|
---|
| 919 | double b,
|
---|
| 920 | double xwidth,
|
---|
| 921 | autogkstate state)
|
---|
| 922 | {
|
---|
| 923 | ap.assert(math.isfinite(a), "AutoGKSmoothW: A is not finite!");
|
---|
| 924 | ap.assert(math.isfinite(b), "AutoGKSmoothW: B is not finite!");
|
---|
| 925 | ap.assert(math.isfinite(xwidth), "AutoGKSmoothW: XWidth is not finite!");
|
---|
| 926 | state.wrappermode = 0;
|
---|
| 927 | state.a = a;
|
---|
| 928 | state.b = b;
|
---|
| 929 | state.xwidth = xwidth;
|
---|
| 930 | state.needf = false;
|
---|
| 931 | state.rstate.ra = new double[10+1];
|
---|
| 932 | state.rstate.stage = -1;
|
---|
| 933 | }
|
---|
| 934 |
|
---|
| 935 |
|
---|
| 936 | /*************************************************************************
|
---|
| 937 | Integration on a finite interval [A,B].
|
---|
| 938 | Integrand have integrable singularities at A/B.
|
---|
| 939 |
|
---|
| 940 | F(X) must diverge as "(x-A)^alpha" at A, as "(B-x)^beta" at B, with known
|
---|
| 941 | alpha/beta (alpha>-1, beta>-1). If alpha/beta are not known, estimates
|
---|
| 942 | from below can be used (but these estimates should be greater than -1 too).
|
---|
| 943 |
|
---|
| 944 | One of alpha/beta variables (or even both alpha/beta) may be equal to 0,
|
---|
| 945 | which means than function F(x) is non-singular at A/B. Anyway (singular at
|
---|
| 946 | bounds or not), function F(x) is supposed to be continuous on (A,B).
|
---|
| 947 |
|
---|
| 948 | Fast-convergent algorithm based on a Gauss-Kronrod formula is used. Result
|
---|
| 949 | is calculated with accuracy close to the machine precision.
|
---|
| 950 |
|
---|
| 951 | INPUT PARAMETERS:
|
---|
| 952 | A, B - interval boundaries (A<B, A=B or A>B)
|
---|
| 953 | Alpha - power-law coefficient of the F(x) at A,
|
---|
| 954 | Alpha>-1
|
---|
| 955 | Beta - power-law coefficient of the F(x) at B,
|
---|
| 956 | Beta>-1
|
---|
| 957 |
|
---|
| 958 | OUTPUT PARAMETERS
|
---|
| 959 | State - structure which stores algorithm state
|
---|
| 960 |
|
---|
| 961 | SEE ALSO
|
---|
| 962 | AutoGKSmooth, AutoGKSmoothW, AutoGKResults.
|
---|
| 963 |
|
---|
| 964 |
|
---|
| 965 | -- ALGLIB --
|
---|
| 966 | Copyright 06.05.2009 by Bochkanov Sergey
|
---|
| 967 | *************************************************************************/
|
---|
| 968 | public static void autogksingular(double a,
|
---|
| 969 | double b,
|
---|
| 970 | double alpha,
|
---|
| 971 | double beta,
|
---|
| 972 | autogkstate state)
|
---|
| 973 | {
|
---|
| 974 | ap.assert(math.isfinite(a), "AutoGKSingular: A is not finite!");
|
---|
| 975 | ap.assert(math.isfinite(b), "AutoGKSingular: B is not finite!");
|
---|
| 976 | ap.assert(math.isfinite(alpha), "AutoGKSingular: Alpha is not finite!");
|
---|
| 977 | ap.assert(math.isfinite(beta), "AutoGKSingular: Beta is not finite!");
|
---|
| 978 | state.wrappermode = 1;
|
---|
| 979 | state.a = a;
|
---|
| 980 | state.b = b;
|
---|
| 981 | state.alpha = alpha;
|
---|
| 982 | state.beta = beta;
|
---|
| 983 | state.xwidth = 0.0;
|
---|
| 984 | state.needf = false;
|
---|
| 985 | state.rstate.ra = new double[10+1];
|
---|
| 986 | state.rstate.stage = -1;
|
---|
| 987 | }
|
---|
| 988 |
|
---|
| 989 |
|
---|
| 990 | /*************************************************************************
|
---|
| 991 |
|
---|
| 992 | -- ALGLIB --
|
---|
| 993 | Copyright 07.05.2009 by Bochkanov Sergey
|
---|
| 994 | *************************************************************************/
|
---|
| 995 | public static bool autogkiteration(autogkstate state)
|
---|
| 996 | {
|
---|
| 997 | bool result = new bool();
|
---|
| 998 | double s = 0;
|
---|
| 999 | double tmp = 0;
|
---|
| 1000 | double eps = 0;
|
---|
| 1001 | double a = 0;
|
---|
| 1002 | double b = 0;
|
---|
| 1003 | double x = 0;
|
---|
| 1004 | double t = 0;
|
---|
| 1005 | double alpha = 0;
|
---|
| 1006 | double beta = 0;
|
---|
| 1007 | double v1 = 0;
|
---|
| 1008 | double v2 = 0;
|
---|
| 1009 |
|
---|
| 1010 |
|
---|
| 1011 | //
|
---|
| 1012 | // Reverse communication preparations
|
---|
| 1013 | // I know it looks ugly, but it works the same way
|
---|
| 1014 | // anywhere from C++ to Python.
|
---|
| 1015 | //
|
---|
| 1016 | // This code initializes locals by:
|
---|
| 1017 | // * random values determined during code
|
---|
| 1018 | // generation - on first subroutine call
|
---|
| 1019 | // * values from previous call - on subsequent calls
|
---|
| 1020 | //
|
---|
| 1021 | if( state.rstate.stage>=0 )
|
---|
| 1022 | {
|
---|
| 1023 | s = state.rstate.ra[0];
|
---|
| 1024 | tmp = state.rstate.ra[1];
|
---|
| 1025 | eps = state.rstate.ra[2];
|
---|
| 1026 | a = state.rstate.ra[3];
|
---|
| 1027 | b = state.rstate.ra[4];
|
---|
| 1028 | x = state.rstate.ra[5];
|
---|
| 1029 | t = state.rstate.ra[6];
|
---|
| 1030 | alpha = state.rstate.ra[7];
|
---|
| 1031 | beta = state.rstate.ra[8];
|
---|
| 1032 | v1 = state.rstate.ra[9];
|
---|
| 1033 | v2 = state.rstate.ra[10];
|
---|
| 1034 | }
|
---|
| 1035 | else
|
---|
| 1036 | {
|
---|
| 1037 | s = -983;
|
---|
| 1038 | tmp = -989;
|
---|
| 1039 | eps = -834;
|
---|
| 1040 | a = 900;
|
---|
| 1041 | b = -287;
|
---|
| 1042 | x = 364;
|
---|
| 1043 | t = 214;
|
---|
| 1044 | alpha = -338;
|
---|
| 1045 | beta = -686;
|
---|
| 1046 | v1 = 912;
|
---|
| 1047 | v2 = 585;
|
---|
| 1048 | }
|
---|
| 1049 | if( state.rstate.stage==0 )
|
---|
| 1050 | {
|
---|
| 1051 | goto lbl_0;
|
---|
| 1052 | }
|
---|
| 1053 | if( state.rstate.stage==1 )
|
---|
| 1054 | {
|
---|
| 1055 | goto lbl_1;
|
---|
| 1056 | }
|
---|
| 1057 | if( state.rstate.stage==2 )
|
---|
| 1058 | {
|
---|
| 1059 | goto lbl_2;
|
---|
| 1060 | }
|
---|
| 1061 |
|
---|
| 1062 | //
|
---|
| 1063 | // Routine body
|
---|
| 1064 | //
|
---|
| 1065 | eps = 0;
|
---|
| 1066 | a = state.a;
|
---|
| 1067 | b = state.b;
|
---|
| 1068 | alpha = state.alpha;
|
---|
| 1069 | beta = state.beta;
|
---|
| 1070 | state.terminationtype = -1;
|
---|
| 1071 | state.nfev = 0;
|
---|
| 1072 | state.nintervals = 0;
|
---|
| 1073 |
|
---|
| 1074 | //
|
---|
| 1075 | // smooth function at a finite interval
|
---|
| 1076 | //
|
---|
| 1077 | if( state.wrappermode!=0 )
|
---|
| 1078 | {
|
---|
| 1079 | goto lbl_3;
|
---|
| 1080 | }
|
---|
| 1081 |
|
---|
| 1082 | //
|
---|
| 1083 | // special case
|
---|
| 1084 | //
|
---|
| 1085 | if( (double)(a)==(double)(b) )
|
---|
| 1086 | {
|
---|
| 1087 | state.terminationtype = 1;
|
---|
| 1088 | state.v = 0;
|
---|
| 1089 | result = false;
|
---|
| 1090 | return result;
|
---|
| 1091 | }
|
---|
| 1092 |
|
---|
| 1093 | //
|
---|
| 1094 | // general case
|
---|
| 1095 | //
|
---|
| 1096 | autogkinternalprepare(a, b, eps, state.xwidth, state.internalstate);
|
---|
| 1097 | lbl_5:
|
---|
| 1098 | if( !autogkinternaliteration(state.internalstate) )
|
---|
| 1099 | {
|
---|
| 1100 | goto lbl_6;
|
---|
| 1101 | }
|
---|
| 1102 | x = state.internalstate.x;
|
---|
| 1103 | state.x = x;
|
---|
| 1104 | state.xminusa = x-a;
|
---|
| 1105 | state.bminusx = b-x;
|
---|
| 1106 | state.needf = true;
|
---|
| 1107 | state.rstate.stage = 0;
|
---|
| 1108 | goto lbl_rcomm;
|
---|
| 1109 | lbl_0:
|
---|
| 1110 | state.needf = false;
|
---|
| 1111 | state.nfev = state.nfev+1;
|
---|
| 1112 | state.internalstate.f = state.f;
|
---|
| 1113 | goto lbl_5;
|
---|
| 1114 | lbl_6:
|
---|
| 1115 | state.v = state.internalstate.r;
|
---|
| 1116 | state.terminationtype = state.internalstate.info;
|
---|
| 1117 | state.nintervals = state.internalstate.heapused;
|
---|
| 1118 | result = false;
|
---|
| 1119 | return result;
|
---|
| 1120 | lbl_3:
|
---|
| 1121 |
|
---|
| 1122 | //
|
---|
| 1123 | // function with power-law singularities at the ends of a finite interval
|
---|
| 1124 | //
|
---|
| 1125 | if( state.wrappermode!=1 )
|
---|
| 1126 | {
|
---|
| 1127 | goto lbl_7;
|
---|
| 1128 | }
|
---|
| 1129 |
|
---|
| 1130 | //
|
---|
| 1131 | // test coefficients
|
---|
| 1132 | //
|
---|
| 1133 | if( (double)(alpha)<=(double)(-1) | (double)(beta)<=(double)(-1) )
|
---|
| 1134 | {
|
---|
| 1135 | state.terminationtype = -1;
|
---|
| 1136 | state.v = 0;
|
---|
| 1137 | result = false;
|
---|
| 1138 | return result;
|
---|
| 1139 | }
|
---|
| 1140 |
|
---|
| 1141 | //
|
---|
| 1142 | // special cases
|
---|
| 1143 | //
|
---|
| 1144 | if( (double)(a)==(double)(b) )
|
---|
| 1145 | {
|
---|
| 1146 | state.terminationtype = 1;
|
---|
| 1147 | state.v = 0;
|
---|
| 1148 | result = false;
|
---|
| 1149 | return result;
|
---|
| 1150 | }
|
---|
| 1151 |
|
---|
| 1152 | //
|
---|
| 1153 | // reduction to general form
|
---|
| 1154 | //
|
---|
| 1155 | if( (double)(a)<(double)(b) )
|
---|
| 1156 | {
|
---|
| 1157 | s = 1;
|
---|
| 1158 | }
|
---|
| 1159 | else
|
---|
| 1160 | {
|
---|
| 1161 | s = -1;
|
---|
| 1162 | tmp = a;
|
---|
| 1163 | a = b;
|
---|
| 1164 | b = tmp;
|
---|
| 1165 | tmp = alpha;
|
---|
| 1166 | alpha = beta;
|
---|
| 1167 | beta = tmp;
|
---|
| 1168 | }
|
---|
| 1169 | alpha = Math.Min(alpha, 0);
|
---|
| 1170 | beta = Math.Min(beta, 0);
|
---|
| 1171 |
|
---|
| 1172 | //
|
---|
| 1173 | // first, integrate left half of [a,b]:
|
---|
| 1174 | // integral(f(x)dx, a, (b+a)/2) =
|
---|
| 1175 | // = 1/(1+alpha) * integral(t^(-alpha/(1+alpha))*f(a+t^(1/(1+alpha)))dt, 0, (0.5*(b-a))^(1+alpha))
|
---|
| 1176 | //
|
---|
| 1177 | autogkinternalprepare(0, Math.Pow(0.5*(b-a), 1+alpha), eps, state.xwidth, state.internalstate);
|
---|
| 1178 | lbl_9:
|
---|
| 1179 | if( !autogkinternaliteration(state.internalstate) )
|
---|
| 1180 | {
|
---|
| 1181 | goto lbl_10;
|
---|
| 1182 | }
|
---|
| 1183 |
|
---|
| 1184 | //
|
---|
| 1185 | // Fill State.X, State.XMinusA, State.BMinusX.
|
---|
| 1186 | // Latter two are filled correctly even if B<A.
|
---|
| 1187 | //
|
---|
| 1188 | x = state.internalstate.x;
|
---|
| 1189 | t = Math.Pow(x, 1/(1+alpha));
|
---|
| 1190 | state.x = a+t;
|
---|
| 1191 | if( (double)(s)>(double)(0) )
|
---|
| 1192 | {
|
---|
| 1193 | state.xminusa = t;
|
---|
| 1194 | state.bminusx = b-(a+t);
|
---|
| 1195 | }
|
---|
| 1196 | else
|
---|
| 1197 | {
|
---|
| 1198 | state.xminusa = a+t-b;
|
---|
| 1199 | state.bminusx = -t;
|
---|
| 1200 | }
|
---|
| 1201 | state.needf = true;
|
---|
| 1202 | state.rstate.stage = 1;
|
---|
| 1203 | goto lbl_rcomm;
|
---|
| 1204 | lbl_1:
|
---|
| 1205 | state.needf = false;
|
---|
| 1206 | if( (double)(alpha)!=(double)(0) )
|
---|
| 1207 | {
|
---|
| 1208 | state.internalstate.f = state.f*Math.Pow(x, -(alpha/(1+alpha)))/(1+alpha);
|
---|
| 1209 | }
|
---|
| 1210 | else
|
---|
| 1211 | {
|
---|
| 1212 | state.internalstate.f = state.f;
|
---|
| 1213 | }
|
---|
| 1214 | state.nfev = state.nfev+1;
|
---|
| 1215 | goto lbl_9;
|
---|
| 1216 | lbl_10:
|
---|
| 1217 | v1 = state.internalstate.r;
|
---|
| 1218 | state.nintervals = state.nintervals+state.internalstate.heapused;
|
---|
| 1219 |
|
---|
| 1220 | //
|
---|
| 1221 | // then, integrate right half of [a,b]:
|
---|
| 1222 | // integral(f(x)dx, (b+a)/2, b) =
|
---|
| 1223 | // = 1/(1+beta) * integral(t^(-beta/(1+beta))*f(b-t^(1/(1+beta)))dt, 0, (0.5*(b-a))^(1+beta))
|
---|
| 1224 | //
|
---|
| 1225 | autogkinternalprepare(0, Math.Pow(0.5*(b-a), 1+beta), eps, state.xwidth, state.internalstate);
|
---|
| 1226 | lbl_11:
|
---|
| 1227 | if( !autogkinternaliteration(state.internalstate) )
|
---|
| 1228 | {
|
---|
| 1229 | goto lbl_12;
|
---|
| 1230 | }
|
---|
| 1231 |
|
---|
| 1232 | //
|
---|
| 1233 | // Fill State.X, State.XMinusA, State.BMinusX.
|
---|
| 1234 | // Latter two are filled correctly (X-A, B-X) even if B<A.
|
---|
| 1235 | //
|
---|
| 1236 | x = state.internalstate.x;
|
---|
| 1237 | t = Math.Pow(x, 1/(1+beta));
|
---|
| 1238 | state.x = b-t;
|
---|
| 1239 | if( (double)(s)>(double)(0) )
|
---|
| 1240 | {
|
---|
| 1241 | state.xminusa = b-t-a;
|
---|
| 1242 | state.bminusx = t;
|
---|
| 1243 | }
|
---|
| 1244 | else
|
---|
| 1245 | {
|
---|
| 1246 | state.xminusa = -t;
|
---|
| 1247 | state.bminusx = a-(b-t);
|
---|
| 1248 | }
|
---|
| 1249 | state.needf = true;
|
---|
| 1250 | state.rstate.stage = 2;
|
---|
| 1251 | goto lbl_rcomm;
|
---|
| 1252 | lbl_2:
|
---|
| 1253 | state.needf = false;
|
---|
| 1254 | if( (double)(beta)!=(double)(0) )
|
---|
| 1255 | {
|
---|
| 1256 | state.internalstate.f = state.f*Math.Pow(x, -(beta/(1+beta)))/(1+beta);
|
---|
| 1257 | }
|
---|
| 1258 | else
|
---|
| 1259 | {
|
---|
| 1260 | state.internalstate.f = state.f;
|
---|
| 1261 | }
|
---|
| 1262 | state.nfev = state.nfev+1;
|
---|
| 1263 | goto lbl_11;
|
---|
| 1264 | lbl_12:
|
---|
| 1265 | v2 = state.internalstate.r;
|
---|
| 1266 | state.nintervals = state.nintervals+state.internalstate.heapused;
|
---|
| 1267 |
|
---|
| 1268 | //
|
---|
| 1269 | // final result
|
---|
| 1270 | //
|
---|
| 1271 | state.v = s*(v1+v2);
|
---|
| 1272 | state.terminationtype = 1;
|
---|
| 1273 | result = false;
|
---|
| 1274 | return result;
|
---|
| 1275 | lbl_7:
|
---|
| 1276 | result = false;
|
---|
| 1277 | return result;
|
---|
| 1278 |
|
---|
| 1279 | //
|
---|
| 1280 | // Saving state
|
---|
| 1281 | //
|
---|
| 1282 | lbl_rcomm:
|
---|
| 1283 | result = true;
|
---|
| 1284 | state.rstate.ra[0] = s;
|
---|
| 1285 | state.rstate.ra[1] = tmp;
|
---|
| 1286 | state.rstate.ra[2] = eps;
|
---|
| 1287 | state.rstate.ra[3] = a;
|
---|
| 1288 | state.rstate.ra[4] = b;
|
---|
| 1289 | state.rstate.ra[5] = x;
|
---|
| 1290 | state.rstate.ra[6] = t;
|
---|
| 1291 | state.rstate.ra[7] = alpha;
|
---|
| 1292 | state.rstate.ra[8] = beta;
|
---|
| 1293 | state.rstate.ra[9] = v1;
|
---|
| 1294 | state.rstate.ra[10] = v2;
|
---|
| 1295 | return result;
|
---|
| 1296 | }
|
---|
| 1297 |
|
---|
| 1298 |
|
---|
| 1299 | /*************************************************************************
|
---|
| 1300 | Adaptive integration results
|
---|
| 1301 |
|
---|
| 1302 | Called after AutoGKIteration returned False.
|
---|
| 1303 |
|
---|
| 1304 | Input parameters:
|
---|
| 1305 | State - algorithm state (used by AutoGKIteration).
|
---|
| 1306 |
|
---|
| 1307 | Output parameters:
|
---|
| 1308 | V - integral(f(x)dx,a,b)
|
---|
| 1309 | Rep - optimization report (see AutoGKReport description)
|
---|
| 1310 |
|
---|
| 1311 | -- ALGLIB --
|
---|
| 1312 | Copyright 14.11.2007 by Bochkanov Sergey
|
---|
| 1313 | *************************************************************************/
|
---|
| 1314 | public static void autogkresults(autogkstate state,
|
---|
| 1315 | ref double v,
|
---|
| 1316 | autogkreport rep)
|
---|
| 1317 | {
|
---|
| 1318 | v = 0;
|
---|
| 1319 |
|
---|
| 1320 | v = state.v;
|
---|
| 1321 | rep.terminationtype = state.terminationtype;
|
---|
| 1322 | rep.nfev = state.nfev;
|
---|
| 1323 | rep.nintervals = state.nintervals;
|
---|
| 1324 | }
|
---|
| 1325 |
|
---|
| 1326 |
|
---|
| 1327 | /*************************************************************************
|
---|
| 1328 | Internal AutoGK subroutine
|
---|
| 1329 | eps<0 - error
|
---|
| 1330 | eps=0 - automatic eps selection
|
---|
| 1331 |
|
---|
| 1332 | width<0 - error
|
---|
| 1333 | width=0 - no width requirements
|
---|
| 1334 | *************************************************************************/
|
---|
| 1335 | private static void autogkinternalprepare(double a,
|
---|
| 1336 | double b,
|
---|
| 1337 | double eps,
|
---|
| 1338 | double xwidth,
|
---|
| 1339 | autogkinternalstate state)
|
---|
| 1340 | {
|
---|
| 1341 |
|
---|
| 1342 | //
|
---|
| 1343 | // Save settings
|
---|
| 1344 | //
|
---|
| 1345 | state.a = a;
|
---|
| 1346 | state.b = b;
|
---|
| 1347 | state.eps = eps;
|
---|
| 1348 | state.xwidth = xwidth;
|
---|
| 1349 |
|
---|
| 1350 | //
|
---|
| 1351 | // Prepare RComm structure
|
---|
| 1352 | //
|
---|
| 1353 | state.rstate.ia = new int[3+1];
|
---|
| 1354 | state.rstate.ra = new double[8+1];
|
---|
| 1355 | state.rstate.stage = -1;
|
---|
| 1356 | }
|
---|
| 1357 |
|
---|
| 1358 |
|
---|
| 1359 | /*************************************************************************
|
---|
| 1360 | Internal AutoGK subroutine
|
---|
| 1361 | *************************************************************************/
|
---|
| 1362 | private static bool autogkinternaliteration(autogkinternalstate state)
|
---|
| 1363 | {
|
---|
| 1364 | bool result = new bool();
|
---|
| 1365 | double c1 = 0;
|
---|
| 1366 | double c2 = 0;
|
---|
| 1367 | int i = 0;
|
---|
| 1368 | int j = 0;
|
---|
| 1369 | double intg = 0;
|
---|
| 1370 | double intk = 0;
|
---|
| 1371 | double inta = 0;
|
---|
| 1372 | double v = 0;
|
---|
| 1373 | double ta = 0;
|
---|
| 1374 | double tb = 0;
|
---|
| 1375 | int ns = 0;
|
---|
| 1376 | double qeps = 0;
|
---|
| 1377 | int info = 0;
|
---|
| 1378 |
|
---|
| 1379 |
|
---|
| 1380 | //
|
---|
| 1381 | // Reverse communication preparations
|
---|
| 1382 | // I know it looks ugly, but it works the same way
|
---|
| 1383 | // anywhere from C++ to Python.
|
---|
| 1384 | //
|
---|
| 1385 | // This code initializes locals by:
|
---|
| 1386 | // * random values determined during code
|
---|
| 1387 | // generation - on first subroutine call
|
---|
| 1388 | // * values from previous call - on subsequent calls
|
---|
| 1389 | //
|
---|
| 1390 | if( state.rstate.stage>=0 )
|
---|
| 1391 | {
|
---|
| 1392 | i = state.rstate.ia[0];
|
---|
| 1393 | j = state.rstate.ia[1];
|
---|
| 1394 | ns = state.rstate.ia[2];
|
---|
| 1395 | info = state.rstate.ia[3];
|
---|
| 1396 | c1 = state.rstate.ra[0];
|
---|
| 1397 | c2 = state.rstate.ra[1];
|
---|
| 1398 | intg = state.rstate.ra[2];
|
---|
| 1399 | intk = state.rstate.ra[3];
|
---|
| 1400 | inta = state.rstate.ra[4];
|
---|
| 1401 | v = state.rstate.ra[5];
|
---|
| 1402 | ta = state.rstate.ra[6];
|
---|
| 1403 | tb = state.rstate.ra[7];
|
---|
| 1404 | qeps = state.rstate.ra[8];
|
---|
| 1405 | }
|
---|
| 1406 | else
|
---|
| 1407 | {
|
---|
| 1408 | i = 497;
|
---|
| 1409 | j = -271;
|
---|
| 1410 | ns = -581;
|
---|
| 1411 | info = 745;
|
---|
| 1412 | c1 = -533;
|
---|
| 1413 | c2 = -77;
|
---|
| 1414 | intg = 678;
|
---|
| 1415 | intk = -293;
|
---|
| 1416 | inta = 316;
|
---|
| 1417 | v = 647;
|
---|
| 1418 | ta = -756;
|
---|
| 1419 | tb = 830;
|
---|
| 1420 | qeps = -871;
|
---|
| 1421 | }
|
---|
| 1422 | if( state.rstate.stage==0 )
|
---|
| 1423 | {
|
---|
| 1424 | goto lbl_0;
|
---|
| 1425 | }
|
---|
| 1426 | if( state.rstate.stage==1 )
|
---|
| 1427 | {
|
---|
| 1428 | goto lbl_1;
|
---|
| 1429 | }
|
---|
| 1430 | if( state.rstate.stage==2 )
|
---|
| 1431 | {
|
---|
| 1432 | goto lbl_2;
|
---|
| 1433 | }
|
---|
| 1434 |
|
---|
| 1435 | //
|
---|
| 1436 | // Routine body
|
---|
| 1437 | //
|
---|
| 1438 |
|
---|
| 1439 | //
|
---|
| 1440 | // initialize quadratures.
|
---|
| 1441 | // use 15-point Gauss-Kronrod formula.
|
---|
| 1442 | //
|
---|
| 1443 | state.n = 15;
|
---|
| 1444 | gkq.gkqgenerategausslegendre(state.n, ref info, ref state.qn, ref state.wk, ref state.wg);
|
---|
| 1445 | if( info<0 )
|
---|
| 1446 | {
|
---|
| 1447 | state.info = -5;
|
---|
| 1448 | state.r = 0;
|
---|
| 1449 | result = false;
|
---|
| 1450 | return result;
|
---|
| 1451 | }
|
---|
| 1452 | state.wr = new double[state.n];
|
---|
| 1453 | for(i=0; i<=state.n-1; i++)
|
---|
| 1454 | {
|
---|
| 1455 | if( i==0 )
|
---|
| 1456 | {
|
---|
| 1457 | state.wr[i] = 0.5*Math.Abs(state.qn[1]-state.qn[0]);
|
---|
| 1458 | continue;
|
---|
| 1459 | }
|
---|
| 1460 | if( i==state.n-1 )
|
---|
| 1461 | {
|
---|
| 1462 | state.wr[state.n-1] = 0.5*Math.Abs(state.qn[state.n-1]-state.qn[state.n-2]);
|
---|
| 1463 | continue;
|
---|
| 1464 | }
|
---|
| 1465 | state.wr[i] = 0.5*Math.Abs(state.qn[i-1]-state.qn[i+1]);
|
---|
| 1466 | }
|
---|
| 1467 |
|
---|
| 1468 | //
|
---|
| 1469 | // special case
|
---|
| 1470 | //
|
---|
| 1471 | if( (double)(state.a)==(double)(state.b) )
|
---|
| 1472 | {
|
---|
| 1473 | state.info = 1;
|
---|
| 1474 | state.r = 0;
|
---|
| 1475 | result = false;
|
---|
| 1476 | return result;
|
---|
| 1477 | }
|
---|
| 1478 |
|
---|
| 1479 | //
|
---|
| 1480 | // test parameters
|
---|
| 1481 | //
|
---|
| 1482 | if( (double)(state.eps)<(double)(0) | (double)(state.xwidth)<(double)(0) )
|
---|
| 1483 | {
|
---|
| 1484 | state.info = -1;
|
---|
| 1485 | state.r = 0;
|
---|
| 1486 | result = false;
|
---|
| 1487 | return result;
|
---|
| 1488 | }
|
---|
| 1489 | state.info = 1;
|
---|
| 1490 | if( (double)(state.eps)==(double)(0) )
|
---|
| 1491 | {
|
---|
| 1492 | state.eps = 1000*math.machineepsilon;
|
---|
| 1493 | }
|
---|
| 1494 |
|
---|
| 1495 | //
|
---|
| 1496 | // First, prepare heap
|
---|
| 1497 | // * column 0 - absolute error
|
---|
| 1498 | // * column 1 - integral of a F(x) (calculated using Kronrod extension nodes)
|
---|
| 1499 | // * column 2 - integral of a |F(x)| (calculated using modified rect. method)
|
---|
| 1500 | // * column 3 - left boundary of a subinterval
|
---|
| 1501 | // * column 4 - right boundary of a subinterval
|
---|
| 1502 | //
|
---|
| 1503 | if( (double)(state.xwidth)!=(double)(0) )
|
---|
| 1504 | {
|
---|
| 1505 | goto lbl_3;
|
---|
| 1506 | }
|
---|
| 1507 |
|
---|
| 1508 | //
|
---|
| 1509 | // no maximum width requirements
|
---|
| 1510 | // start from one big subinterval
|
---|
| 1511 | //
|
---|
| 1512 | state.heapwidth = 5;
|
---|
| 1513 | state.heapsize = 1;
|
---|
| 1514 | state.heapused = 1;
|
---|
| 1515 | state.heap = new double[state.heapsize, state.heapwidth];
|
---|
| 1516 | c1 = 0.5*(state.b-state.a);
|
---|
| 1517 | c2 = 0.5*(state.b+state.a);
|
---|
| 1518 | intg = 0;
|
---|
| 1519 | intk = 0;
|
---|
| 1520 | inta = 0;
|
---|
| 1521 | i = 0;
|
---|
| 1522 | lbl_5:
|
---|
| 1523 | if( i>state.n-1 )
|
---|
| 1524 | {
|
---|
| 1525 | goto lbl_7;
|
---|
| 1526 | }
|
---|
| 1527 |
|
---|
| 1528 | //
|
---|
| 1529 | // obtain F
|
---|
| 1530 | //
|
---|
| 1531 | state.x = c1*state.qn[i]+c2;
|
---|
| 1532 | state.rstate.stage = 0;
|
---|
| 1533 | goto lbl_rcomm;
|
---|
| 1534 | lbl_0:
|
---|
| 1535 | v = state.f;
|
---|
| 1536 |
|
---|
| 1537 | //
|
---|
| 1538 | // Gauss-Kronrod formula
|
---|
| 1539 | //
|
---|
| 1540 | intk = intk+v*state.wk[i];
|
---|
| 1541 | if( i%2==1 )
|
---|
| 1542 | {
|
---|
| 1543 | intg = intg+v*state.wg[i];
|
---|
| 1544 | }
|
---|
| 1545 |
|
---|
| 1546 | //
|
---|
| 1547 | // Integral |F(x)|
|
---|
| 1548 | // Use rectangles method
|
---|
| 1549 | //
|
---|
| 1550 | inta = inta+Math.Abs(v)*state.wr[i];
|
---|
| 1551 | i = i+1;
|
---|
| 1552 | goto lbl_5;
|
---|
| 1553 | lbl_7:
|
---|
| 1554 | intk = intk*(state.b-state.a)*0.5;
|
---|
| 1555 | intg = intg*(state.b-state.a)*0.5;
|
---|
| 1556 | inta = inta*(state.b-state.a)*0.5;
|
---|
| 1557 | state.heap[0,0] = Math.Abs(intg-intk);
|
---|
| 1558 | state.heap[0,1] = intk;
|
---|
| 1559 | state.heap[0,2] = inta;
|
---|
| 1560 | state.heap[0,3] = state.a;
|
---|
| 1561 | state.heap[0,4] = state.b;
|
---|
| 1562 | state.sumerr = state.heap[0,0];
|
---|
| 1563 | state.sumabs = Math.Abs(inta);
|
---|
| 1564 | goto lbl_4;
|
---|
| 1565 | lbl_3:
|
---|
| 1566 |
|
---|
| 1567 | //
|
---|
| 1568 | // maximum subinterval should be no more than XWidth.
|
---|
| 1569 | // so we create Ceil((B-A)/XWidth)+1 small subintervals
|
---|
| 1570 | //
|
---|
| 1571 | ns = (int)Math.Ceiling(Math.Abs(state.b-state.a)/state.xwidth)+1;
|
---|
| 1572 | state.heapsize = ns;
|
---|
| 1573 | state.heapused = ns;
|
---|
| 1574 | state.heapwidth = 5;
|
---|
| 1575 | state.heap = new double[state.heapsize, state.heapwidth];
|
---|
| 1576 | state.sumerr = 0;
|
---|
| 1577 | state.sumabs = 0;
|
---|
| 1578 | j = 0;
|
---|
| 1579 | lbl_8:
|
---|
| 1580 | if( j>ns-1 )
|
---|
| 1581 | {
|
---|
| 1582 | goto lbl_10;
|
---|
| 1583 | }
|
---|
| 1584 | ta = state.a+j*(state.b-state.a)/ns;
|
---|
| 1585 | tb = state.a+(j+1)*(state.b-state.a)/ns;
|
---|
| 1586 | c1 = 0.5*(tb-ta);
|
---|
| 1587 | c2 = 0.5*(tb+ta);
|
---|
| 1588 | intg = 0;
|
---|
| 1589 | intk = 0;
|
---|
| 1590 | inta = 0;
|
---|
| 1591 | i = 0;
|
---|
| 1592 | lbl_11:
|
---|
| 1593 | if( i>state.n-1 )
|
---|
| 1594 | {
|
---|
| 1595 | goto lbl_13;
|
---|
| 1596 | }
|
---|
| 1597 |
|
---|
| 1598 | //
|
---|
| 1599 | // obtain F
|
---|
| 1600 | //
|
---|
| 1601 | state.x = c1*state.qn[i]+c2;
|
---|
| 1602 | state.rstate.stage = 1;
|
---|
| 1603 | goto lbl_rcomm;
|
---|
| 1604 | lbl_1:
|
---|
| 1605 | v = state.f;
|
---|
| 1606 |
|
---|
| 1607 | //
|
---|
| 1608 | // Gauss-Kronrod formula
|
---|
| 1609 | //
|
---|
| 1610 | intk = intk+v*state.wk[i];
|
---|
| 1611 | if( i%2==1 )
|
---|
| 1612 | {
|
---|
| 1613 | intg = intg+v*state.wg[i];
|
---|
| 1614 | }
|
---|
| 1615 |
|
---|
| 1616 | //
|
---|
| 1617 | // Integral |F(x)|
|
---|
| 1618 | // Use rectangles method
|
---|
| 1619 | //
|
---|
| 1620 | inta = inta+Math.Abs(v)*state.wr[i];
|
---|
| 1621 | i = i+1;
|
---|
| 1622 | goto lbl_11;
|
---|
| 1623 | lbl_13:
|
---|
| 1624 | intk = intk*(tb-ta)*0.5;
|
---|
| 1625 | intg = intg*(tb-ta)*0.5;
|
---|
| 1626 | inta = inta*(tb-ta)*0.5;
|
---|
| 1627 | state.heap[j,0] = Math.Abs(intg-intk);
|
---|
| 1628 | state.heap[j,1] = intk;
|
---|
| 1629 | state.heap[j,2] = inta;
|
---|
| 1630 | state.heap[j,3] = ta;
|
---|
| 1631 | state.heap[j,4] = tb;
|
---|
| 1632 | state.sumerr = state.sumerr+state.heap[j,0];
|
---|
| 1633 | state.sumabs = state.sumabs+Math.Abs(inta);
|
---|
| 1634 | j = j+1;
|
---|
| 1635 | goto lbl_8;
|
---|
| 1636 | lbl_10:
|
---|
| 1637 | lbl_4:
|
---|
| 1638 |
|
---|
| 1639 | //
|
---|
| 1640 | // method iterations
|
---|
| 1641 | //
|
---|
| 1642 | lbl_14:
|
---|
| 1643 | if( false )
|
---|
| 1644 | {
|
---|
| 1645 | goto lbl_15;
|
---|
| 1646 | }
|
---|
| 1647 |
|
---|
| 1648 | //
|
---|
| 1649 | // additional memory if needed
|
---|
| 1650 | //
|
---|
| 1651 | if( state.heapused==state.heapsize )
|
---|
| 1652 | {
|
---|
| 1653 | mheapresize(ref state.heap, ref state.heapsize, 4*state.heapsize, state.heapwidth);
|
---|
| 1654 | }
|
---|
| 1655 |
|
---|
| 1656 | //
|
---|
| 1657 | // TODO: every 20 iterations recalculate errors/sums
|
---|
| 1658 | // TODO: one more criterion to prevent infinite loops with too strict Eps
|
---|
| 1659 | //
|
---|
| 1660 | if( (double)(state.sumerr)<=(double)(state.eps*state.sumabs) )
|
---|
| 1661 | {
|
---|
| 1662 | state.r = 0;
|
---|
| 1663 | for(j=0; j<=state.heapused-1; j++)
|
---|
| 1664 | {
|
---|
| 1665 | state.r = state.r+state.heap[j,1];
|
---|
| 1666 | }
|
---|
| 1667 | result = false;
|
---|
| 1668 | return result;
|
---|
| 1669 | }
|
---|
| 1670 |
|
---|
| 1671 | //
|
---|
| 1672 | // Exclude interval with maximum absolute error
|
---|
| 1673 | //
|
---|
| 1674 | mheappop(ref state.heap, state.heapused, state.heapwidth);
|
---|
| 1675 | state.sumerr = state.sumerr-state.heap[state.heapused-1,0];
|
---|
| 1676 | state.sumabs = state.sumabs-state.heap[state.heapused-1,2];
|
---|
| 1677 |
|
---|
| 1678 | //
|
---|
| 1679 | // Divide interval, create subintervals
|
---|
| 1680 | //
|
---|
| 1681 | ta = state.heap[state.heapused-1,3];
|
---|
| 1682 | tb = state.heap[state.heapused-1,4];
|
---|
| 1683 | state.heap[state.heapused-1,3] = ta;
|
---|
| 1684 | state.heap[state.heapused-1,4] = 0.5*(ta+tb);
|
---|
| 1685 | state.heap[state.heapused,3] = 0.5*(ta+tb);
|
---|
| 1686 | state.heap[state.heapused,4] = tb;
|
---|
| 1687 | j = state.heapused-1;
|
---|
| 1688 | lbl_16:
|
---|
| 1689 | if( j>state.heapused )
|
---|
| 1690 | {
|
---|
| 1691 | goto lbl_18;
|
---|
| 1692 | }
|
---|
| 1693 | c1 = 0.5*(state.heap[j,4]-state.heap[j,3]);
|
---|
| 1694 | c2 = 0.5*(state.heap[j,4]+state.heap[j,3]);
|
---|
| 1695 | intg = 0;
|
---|
| 1696 | intk = 0;
|
---|
| 1697 | inta = 0;
|
---|
| 1698 | i = 0;
|
---|
| 1699 | lbl_19:
|
---|
| 1700 | if( i>state.n-1 )
|
---|
| 1701 | {
|
---|
| 1702 | goto lbl_21;
|
---|
| 1703 | }
|
---|
| 1704 |
|
---|
| 1705 | //
|
---|
| 1706 | // F(x)
|
---|
| 1707 | //
|
---|
| 1708 | state.x = c1*state.qn[i]+c2;
|
---|
| 1709 | state.rstate.stage = 2;
|
---|
| 1710 | goto lbl_rcomm;
|
---|
| 1711 | lbl_2:
|
---|
| 1712 | v = state.f;
|
---|
| 1713 |
|
---|
| 1714 | //
|
---|
| 1715 | // Gauss-Kronrod formula
|
---|
| 1716 | //
|
---|
| 1717 | intk = intk+v*state.wk[i];
|
---|
| 1718 | if( i%2==1 )
|
---|
| 1719 | {
|
---|
| 1720 | intg = intg+v*state.wg[i];
|
---|
| 1721 | }
|
---|
| 1722 |
|
---|
| 1723 | //
|
---|
| 1724 | // Integral |F(x)|
|
---|
| 1725 | // Use rectangles method
|
---|
| 1726 | //
|
---|
| 1727 | inta = inta+Math.Abs(v)*state.wr[i];
|
---|
| 1728 | i = i+1;
|
---|
| 1729 | goto lbl_19;
|
---|
| 1730 | lbl_21:
|
---|
| 1731 | intk = intk*(state.heap[j,4]-state.heap[j,3])*0.5;
|
---|
| 1732 | intg = intg*(state.heap[j,4]-state.heap[j,3])*0.5;
|
---|
| 1733 | inta = inta*(state.heap[j,4]-state.heap[j,3])*0.5;
|
---|
| 1734 | state.heap[j,0] = Math.Abs(intg-intk);
|
---|
| 1735 | state.heap[j,1] = intk;
|
---|
| 1736 | state.heap[j,2] = inta;
|
---|
| 1737 | state.sumerr = state.sumerr+state.heap[j,0];
|
---|
| 1738 | state.sumabs = state.sumabs+state.heap[j,2];
|
---|
| 1739 | j = j+1;
|
---|
| 1740 | goto lbl_16;
|
---|
| 1741 | lbl_18:
|
---|
| 1742 | mheappush(ref state.heap, state.heapused-1, state.heapwidth);
|
---|
| 1743 | mheappush(ref state.heap, state.heapused, state.heapwidth);
|
---|
| 1744 | state.heapused = state.heapused+1;
|
---|
| 1745 | goto lbl_14;
|
---|
| 1746 | lbl_15:
|
---|
| 1747 | result = false;
|
---|
| 1748 | return result;
|
---|
| 1749 |
|
---|
| 1750 | //
|
---|
| 1751 | // Saving state
|
---|
| 1752 | //
|
---|
| 1753 | lbl_rcomm:
|
---|
| 1754 | result = true;
|
---|
| 1755 | state.rstate.ia[0] = i;
|
---|
| 1756 | state.rstate.ia[1] = j;
|
---|
| 1757 | state.rstate.ia[2] = ns;
|
---|
| 1758 | state.rstate.ia[3] = info;
|
---|
| 1759 | state.rstate.ra[0] = c1;
|
---|
| 1760 | state.rstate.ra[1] = c2;
|
---|
| 1761 | state.rstate.ra[2] = intg;
|
---|
| 1762 | state.rstate.ra[3] = intk;
|
---|
| 1763 | state.rstate.ra[4] = inta;
|
---|
| 1764 | state.rstate.ra[5] = v;
|
---|
| 1765 | state.rstate.ra[6] = ta;
|
---|
| 1766 | state.rstate.ra[7] = tb;
|
---|
| 1767 | state.rstate.ra[8] = qeps;
|
---|
| 1768 | return result;
|
---|
| 1769 | }
|
---|
| 1770 |
|
---|
| 1771 |
|
---|
| 1772 | private static void mheappop(ref double[,] heap,
|
---|
| 1773 | int heapsize,
|
---|
| 1774 | int heapwidth)
|
---|
| 1775 | {
|
---|
| 1776 | int i = 0;
|
---|
| 1777 | int p = 0;
|
---|
| 1778 | double t = 0;
|
---|
| 1779 | int maxcp = 0;
|
---|
| 1780 |
|
---|
| 1781 | if( heapsize==1 )
|
---|
| 1782 | {
|
---|
| 1783 | return;
|
---|
| 1784 | }
|
---|
| 1785 | for(i=0; i<=heapwidth-1; i++)
|
---|
| 1786 | {
|
---|
| 1787 | t = heap[heapsize-1,i];
|
---|
| 1788 | heap[heapsize-1,i] = heap[0,i];
|
---|
| 1789 | heap[0,i] = t;
|
---|
| 1790 | }
|
---|
| 1791 | p = 0;
|
---|
| 1792 | while( 2*p+1<heapsize-1 )
|
---|
| 1793 | {
|
---|
| 1794 | maxcp = 2*p+1;
|
---|
| 1795 | if( 2*p+2<heapsize-1 )
|
---|
| 1796 | {
|
---|
| 1797 | if( (double)(heap[2*p+2,0])>(double)(heap[2*p+1,0]) )
|
---|
| 1798 | {
|
---|
| 1799 | maxcp = 2*p+2;
|
---|
| 1800 | }
|
---|
| 1801 | }
|
---|
| 1802 | if( (double)(heap[p,0])<(double)(heap[maxcp,0]) )
|
---|
| 1803 | {
|
---|
| 1804 | for(i=0; i<=heapwidth-1; i++)
|
---|
| 1805 | {
|
---|
| 1806 | t = heap[p,i];
|
---|
| 1807 | heap[p,i] = heap[maxcp,i];
|
---|
| 1808 | heap[maxcp,i] = t;
|
---|
| 1809 | }
|
---|
| 1810 | p = maxcp;
|
---|
| 1811 | }
|
---|
| 1812 | else
|
---|
| 1813 | {
|
---|
| 1814 | break;
|
---|
| 1815 | }
|
---|
| 1816 | }
|
---|
| 1817 | }
|
---|
| 1818 |
|
---|
| 1819 |
|
---|
| 1820 | private static void mheappush(ref double[,] heap,
|
---|
| 1821 | int heapsize,
|
---|
| 1822 | int heapwidth)
|
---|
| 1823 | {
|
---|
| 1824 | int i = 0;
|
---|
| 1825 | int p = 0;
|
---|
| 1826 | double t = 0;
|
---|
| 1827 | int parent = 0;
|
---|
| 1828 |
|
---|
| 1829 | if( heapsize==0 )
|
---|
| 1830 | {
|
---|
| 1831 | return;
|
---|
| 1832 | }
|
---|
| 1833 | p = heapsize;
|
---|
| 1834 | while( p!=0 )
|
---|
| 1835 | {
|
---|
| 1836 | parent = (p-1)/2;
|
---|
| 1837 | if( (double)(heap[p,0])>(double)(heap[parent,0]) )
|
---|
| 1838 | {
|
---|
| 1839 | for(i=0; i<=heapwidth-1; i++)
|
---|
| 1840 | {
|
---|
| 1841 | t = heap[p,i];
|
---|
| 1842 | heap[p,i] = heap[parent,i];
|
---|
| 1843 | heap[parent,i] = t;
|
---|
| 1844 | }
|
---|
| 1845 | p = parent;
|
---|
| 1846 | }
|
---|
| 1847 | else
|
---|
| 1848 | {
|
---|
| 1849 | break;
|
---|
| 1850 | }
|
---|
| 1851 | }
|
---|
| 1852 | }
|
---|
| 1853 |
|
---|
| 1854 |
|
---|
| 1855 | private static void mheapresize(ref double[,] heap,
|
---|
| 1856 | ref int heapsize,
|
---|
| 1857 | int newheapsize,
|
---|
| 1858 | int heapwidth)
|
---|
| 1859 | {
|
---|
| 1860 | double[,] tmp = new double[0,0];
|
---|
| 1861 | int i = 0;
|
---|
| 1862 | int i_ = 0;
|
---|
| 1863 |
|
---|
| 1864 | tmp = new double[heapsize, heapwidth];
|
---|
| 1865 | for(i=0; i<=heapsize-1; i++)
|
---|
| 1866 | {
|
---|
| 1867 | for(i_=0; i_<=heapwidth-1;i_++)
|
---|
| 1868 | {
|
---|
| 1869 | tmp[i,i_] = heap[i,i_];
|
---|
| 1870 | }
|
---|
| 1871 | }
|
---|
| 1872 | heap = new double[newheapsize, heapwidth];
|
---|
| 1873 | for(i=0; i<=heapsize-1; i++)
|
---|
| 1874 | {
|
---|
| 1875 | for(i_=0; i_<=heapwidth-1;i_++)
|
---|
| 1876 | {
|
---|
| 1877 | heap[i,i_] = tmp[i,i_];
|
---|
| 1878 | }
|
---|
| 1879 | }
|
---|
| 1880 | heapsize = newheapsize;
|
---|
| 1881 | }
|
---|
| 1882 |
|
---|
| 1883 |
|
---|
| 1884 | }
|
---|
| 1885 | public class gq
|
---|
| 1886 | {
|
---|
| 1887 | /*************************************************************************
|
---|
| 1888 | Computation of nodes and weights for a Gauss quadrature formula
|
---|
| 1889 |
|
---|
| 1890 | The algorithm generates the N-point Gauss quadrature formula with weight
|
---|
| 1891 | function given by coefficients alpha and beta of a recurrence relation
|
---|
| 1892 | which generates a system of orthogonal polynomials:
|
---|
| 1893 |
|
---|
| 1894 | P-1(x) = 0
|
---|
| 1895 | P0(x) = 1
|
---|
| 1896 | Pn+1(x) = (x-alpha(n))*Pn(x) - beta(n)*Pn-1(x)
|
---|
| 1897 |
|
---|
| 1898 | and zeroth moment Mu0
|
---|
| 1899 |
|
---|
| 1900 | Mu0 = integral(W(x)dx,a,b)
|
---|
| 1901 |
|
---|
| 1902 | INPUT PARAMETERS:
|
---|
| 1903 | Alpha array[0..N-1], alpha coefficients
|
---|
| 1904 | Beta array[0..N-1], beta coefficients
|
---|
| 1905 | Zero-indexed element is not used and may be arbitrary.
|
---|
| 1906 | Beta[I]>0.
|
---|
| 1907 | Mu0 zeroth moment of the weight function.
|
---|
| 1908 | N number of nodes of the quadrature formula, N>=1
|
---|
| 1909 |
|
---|
| 1910 | OUTPUT PARAMETERS:
|
---|
| 1911 | Info - error code:
|
---|
| 1912 | * -3 internal eigenproblem solver hasn't converged
|
---|
| 1913 | * -2 Beta[i]<=0
|
---|
| 1914 | * -1 incorrect N was passed
|
---|
| 1915 | * 1 OK
|
---|
| 1916 | X - array[0..N-1] - array of quadrature nodes,
|
---|
| 1917 | in ascending order.
|
---|
| 1918 | W - array[0..N-1] - array of quadrature weights.
|
---|
| 1919 |
|
---|
| 1920 | -- ALGLIB --
|
---|
| 1921 | Copyright 2005-2009 by Bochkanov Sergey
|
---|
| 1922 | *************************************************************************/
|
---|
| 1923 | public static void gqgeneraterec(double[] alpha,
|
---|
| 1924 | double[] beta,
|
---|
| 1925 | double mu0,
|
---|
| 1926 | int n,
|
---|
| 1927 | ref int info,
|
---|
| 1928 | ref double[] x,
|
---|
| 1929 | ref double[] w)
|
---|
| 1930 | {
|
---|
| 1931 | int i = 0;
|
---|
| 1932 | double[] d = new double[0];
|
---|
| 1933 | double[] e = new double[0];
|
---|
| 1934 | double[,] z = new double[0,0];
|
---|
| 1935 |
|
---|
| 1936 | info = 0;
|
---|
| 1937 | x = new double[0];
|
---|
| 1938 | w = new double[0];
|
---|
| 1939 |
|
---|
| 1940 | if( n<1 )
|
---|
| 1941 | {
|
---|
| 1942 | info = -1;
|
---|
| 1943 | return;
|
---|
| 1944 | }
|
---|
| 1945 | info = 1;
|
---|
| 1946 |
|
---|
| 1947 | //
|
---|
| 1948 | // Initialize
|
---|
| 1949 | //
|
---|
| 1950 | d = new double[n];
|
---|
| 1951 | e = new double[n];
|
---|
| 1952 | for(i=1; i<=n-1; i++)
|
---|
| 1953 | {
|
---|
| 1954 | d[i-1] = alpha[i-1];
|
---|
| 1955 | if( (double)(beta[i])<=(double)(0) )
|
---|
| 1956 | {
|
---|
| 1957 | info = -2;
|
---|
| 1958 | return;
|
---|
| 1959 | }
|
---|
| 1960 | e[i-1] = Math.Sqrt(beta[i]);
|
---|
| 1961 | }
|
---|
| 1962 | d[n-1] = alpha[n-1];
|
---|
| 1963 |
|
---|
| 1964 | //
|
---|
| 1965 | // EVD
|
---|
| 1966 | //
|
---|
| 1967 | if( !evd.smatrixtdevd(ref d, e, n, 3, ref z) )
|
---|
| 1968 | {
|
---|
| 1969 | info = -3;
|
---|
| 1970 | return;
|
---|
| 1971 | }
|
---|
| 1972 |
|
---|
| 1973 | //
|
---|
| 1974 | // Generate
|
---|
| 1975 | //
|
---|
| 1976 | x = new double[n];
|
---|
| 1977 | w = new double[n];
|
---|
| 1978 | for(i=1; i<=n; i++)
|
---|
| 1979 | {
|
---|
| 1980 | x[i-1] = d[i-1];
|
---|
| 1981 | w[i-1] = mu0*math.sqr(z[0,i-1]);
|
---|
| 1982 | }
|
---|
| 1983 | }
|
---|
| 1984 |
|
---|
| 1985 |
|
---|
| 1986 | /*************************************************************************
|
---|
| 1987 | Computation of nodes and weights for a Gauss-Lobatto quadrature formula
|
---|
| 1988 |
|
---|
| 1989 | The algorithm generates the N-point Gauss-Lobatto quadrature formula with
|
---|
| 1990 | weight function given by coefficients alpha and beta of a recurrence which
|
---|
| 1991 | generates a system of orthogonal polynomials.
|
---|
| 1992 |
|
---|
| 1993 | P-1(x) = 0
|
---|
| 1994 | P0(x) = 1
|
---|
| 1995 | Pn+1(x) = (x-alpha(n))*Pn(x) - beta(n)*Pn-1(x)
|
---|
| 1996 |
|
---|
| 1997 | and zeroth moment Mu0
|
---|
| 1998 |
|
---|
| 1999 | Mu0 = integral(W(x)dx,a,b)
|
---|
| 2000 |
|
---|
| 2001 | INPUT PARAMETERS:
|
---|
| 2002 | Alpha array[0..N-2], alpha coefficients
|
---|
| 2003 | Beta array[0..N-2], beta coefficients.
|
---|
| 2004 | Zero-indexed element is not used, may be arbitrary.
|
---|
| 2005 | Beta[I]>0
|
---|
| 2006 | Mu0 zeroth moment of the weighting function.
|
---|
| 2007 | A left boundary of the integration interval.
|
---|
| 2008 | B right boundary of the integration interval.
|
---|
| 2009 | N number of nodes of the quadrature formula, N>=3
|
---|
| 2010 | (including the left and right boundary nodes).
|
---|
| 2011 |
|
---|
| 2012 | OUTPUT PARAMETERS:
|
---|
| 2013 | Info - error code:
|
---|
| 2014 | * -3 internal eigenproblem solver hasn't converged
|
---|
| 2015 | * -2 Beta[i]<=0
|
---|
| 2016 | * -1 incorrect N was passed
|
---|
| 2017 | * 1 OK
|
---|
| 2018 | X - array[0..N-1] - array of quadrature nodes,
|
---|
| 2019 | in ascending order.
|
---|
| 2020 | W - array[0..N-1] - array of quadrature weights.
|
---|
| 2021 |
|
---|
| 2022 | -- ALGLIB --
|
---|
| 2023 | Copyright 2005-2009 by Bochkanov Sergey
|
---|
| 2024 | *************************************************************************/
|
---|
| 2025 | public static void gqgenerategausslobattorec(double[] alpha,
|
---|
| 2026 | double[] beta,
|
---|
| 2027 | double mu0,
|
---|
| 2028 | double a,
|
---|
| 2029 | double b,
|
---|
| 2030 | int n,
|
---|
| 2031 | ref int info,
|
---|
| 2032 | ref double[] x,
|
---|
| 2033 | ref double[] w)
|
---|
| 2034 | {
|
---|
| 2035 | int i = 0;
|
---|
| 2036 | double[] d = new double[0];
|
---|
| 2037 | double[] e = new double[0];
|
---|
| 2038 | double[,] z = new double[0,0];
|
---|
| 2039 | double pim1a = 0;
|
---|
| 2040 | double pia = 0;
|
---|
| 2041 | double pim1b = 0;
|
---|
| 2042 | double pib = 0;
|
---|
| 2043 | double t = 0;
|
---|
| 2044 | double a11 = 0;
|
---|
| 2045 | double a12 = 0;
|
---|
| 2046 | double a21 = 0;
|
---|
| 2047 | double a22 = 0;
|
---|
| 2048 | double b1 = 0;
|
---|
| 2049 | double b2 = 0;
|
---|
| 2050 | double alph = 0;
|
---|
| 2051 | double bet = 0;
|
---|
| 2052 |
|
---|
| 2053 | alpha = (double[])alpha.Clone();
|
---|
| 2054 | beta = (double[])beta.Clone();
|
---|
| 2055 | info = 0;
|
---|
| 2056 | x = new double[0];
|
---|
| 2057 | w = new double[0];
|
---|
| 2058 |
|
---|
| 2059 | if( n<=2 )
|
---|
| 2060 | {
|
---|
| 2061 | info = -1;
|
---|
| 2062 | return;
|
---|
| 2063 | }
|
---|
| 2064 | info = 1;
|
---|
| 2065 |
|
---|
| 2066 | //
|
---|
| 2067 | // Initialize, D[1:N+1], E[1:N]
|
---|
| 2068 | //
|
---|
| 2069 | n = n-2;
|
---|
| 2070 | d = new double[n+2];
|
---|
| 2071 | e = new double[n+1];
|
---|
| 2072 | for(i=1; i<=n+1; i++)
|
---|
| 2073 | {
|
---|
| 2074 | d[i-1] = alpha[i-1];
|
---|
| 2075 | }
|
---|
| 2076 | for(i=1; i<=n; i++)
|
---|
| 2077 | {
|
---|
| 2078 | if( (double)(beta[i])<=(double)(0) )
|
---|
| 2079 | {
|
---|
| 2080 | info = -2;
|
---|
| 2081 | return;
|
---|
| 2082 | }
|
---|
| 2083 | e[i-1] = Math.Sqrt(beta[i]);
|
---|
| 2084 | }
|
---|
| 2085 |
|
---|
| 2086 | //
|
---|
| 2087 | // Caclulate Pn(a), Pn+1(a), Pn(b), Pn+1(b)
|
---|
| 2088 | //
|
---|
| 2089 | beta[0] = 0;
|
---|
| 2090 | pim1a = 0;
|
---|
| 2091 | pia = 1;
|
---|
| 2092 | pim1b = 0;
|
---|
| 2093 | pib = 1;
|
---|
| 2094 | for(i=1; i<=n+1; i++)
|
---|
| 2095 | {
|
---|
| 2096 |
|
---|
| 2097 | //
|
---|
| 2098 | // Pi(a)
|
---|
| 2099 | //
|
---|
| 2100 | t = (a-alpha[i-1])*pia-beta[i-1]*pim1a;
|
---|
| 2101 | pim1a = pia;
|
---|
| 2102 | pia = t;
|
---|
| 2103 |
|
---|
| 2104 | //
|
---|
| 2105 | // Pi(b)
|
---|
| 2106 | //
|
---|
| 2107 | t = (b-alpha[i-1])*pib-beta[i-1]*pim1b;
|
---|
| 2108 | pim1b = pib;
|
---|
| 2109 | pib = t;
|
---|
| 2110 | }
|
---|
| 2111 |
|
---|
| 2112 | //
|
---|
| 2113 | // Calculate alpha'(n+1), beta'(n+1)
|
---|
| 2114 | //
|
---|
| 2115 | a11 = pia;
|
---|
| 2116 | a12 = pim1a;
|
---|
| 2117 | a21 = pib;
|
---|
| 2118 | a22 = pim1b;
|
---|
| 2119 | b1 = a*pia;
|
---|
| 2120 | b2 = b*pib;
|
---|
| 2121 | if( (double)(Math.Abs(a11))>(double)(Math.Abs(a21)) )
|
---|
| 2122 | {
|
---|
| 2123 | a22 = a22-a12*a21/a11;
|
---|
| 2124 | b2 = b2-b1*a21/a11;
|
---|
| 2125 | bet = b2/a22;
|
---|
| 2126 | alph = (b1-bet*a12)/a11;
|
---|
| 2127 | }
|
---|
| 2128 | else
|
---|
| 2129 | {
|
---|
| 2130 | a12 = a12-a22*a11/a21;
|
---|
| 2131 | b1 = b1-b2*a11/a21;
|
---|
| 2132 | bet = b1/a12;
|
---|
| 2133 | alph = (b2-bet*a22)/a21;
|
---|
| 2134 | }
|
---|
| 2135 | if( (double)(bet)<(double)(0) )
|
---|
| 2136 | {
|
---|
| 2137 | info = -3;
|
---|
| 2138 | return;
|
---|
| 2139 | }
|
---|
| 2140 | d[n+1] = alph;
|
---|
| 2141 | e[n] = Math.Sqrt(bet);
|
---|
| 2142 |
|
---|
| 2143 | //
|
---|
| 2144 | // EVD
|
---|
| 2145 | //
|
---|
| 2146 | if( !evd.smatrixtdevd(ref d, e, n+2, 3, ref z) )
|
---|
| 2147 | {
|
---|
| 2148 | info = -3;
|
---|
| 2149 | return;
|
---|
| 2150 | }
|
---|
| 2151 |
|
---|
| 2152 | //
|
---|
| 2153 | // Generate
|
---|
| 2154 | //
|
---|
| 2155 | x = new double[n+2];
|
---|
| 2156 | w = new double[n+2];
|
---|
| 2157 | for(i=1; i<=n+2; i++)
|
---|
| 2158 | {
|
---|
| 2159 | x[i-1] = d[i-1];
|
---|
| 2160 | w[i-1] = mu0*math.sqr(z[0,i-1]);
|
---|
| 2161 | }
|
---|
| 2162 | }
|
---|
| 2163 |
|
---|
| 2164 |
|
---|
| 2165 | /*************************************************************************
|
---|
| 2166 | Computation of nodes and weights for a Gauss-Radau quadrature formula
|
---|
| 2167 |
|
---|
| 2168 | The algorithm generates the N-point Gauss-Radau quadrature formula with
|
---|
| 2169 | weight function given by the coefficients alpha and beta of a recurrence
|
---|
| 2170 | which generates a system of orthogonal polynomials.
|
---|
| 2171 |
|
---|
| 2172 | P-1(x) = 0
|
---|
| 2173 | P0(x) = 1
|
---|
| 2174 | Pn+1(x) = (x-alpha(n))*Pn(x) - beta(n)*Pn-1(x)
|
---|
| 2175 |
|
---|
| 2176 | and zeroth moment Mu0
|
---|
| 2177 |
|
---|
| 2178 | Mu0 = integral(W(x)dx,a,b)
|
---|
| 2179 |
|
---|
| 2180 | INPUT PARAMETERS:
|
---|
| 2181 | Alpha array[0..N-2], alpha coefficients.
|
---|
| 2182 | Beta array[0..N-1], beta coefficients
|
---|
| 2183 | Zero-indexed element is not used.
|
---|
| 2184 | Beta[I]>0
|
---|
| 2185 | Mu0 zeroth moment of the weighting function.
|
---|
| 2186 | A left boundary of the integration interval.
|
---|
| 2187 | N number of nodes of the quadrature formula, N>=2
|
---|
| 2188 | (including the left boundary node).
|
---|
| 2189 |
|
---|
| 2190 | OUTPUT PARAMETERS:
|
---|
| 2191 | Info - error code:
|
---|
| 2192 | * -3 internal eigenproblem solver hasn't converged
|
---|
| 2193 | * -2 Beta[i]<=0
|
---|
| 2194 | * -1 incorrect N was passed
|
---|
| 2195 | * 1 OK
|
---|
| 2196 | X - array[0..N-1] - array of quadrature nodes,
|
---|
| 2197 | in ascending order.
|
---|
| 2198 | W - array[0..N-1] - array of quadrature weights.
|
---|
| 2199 |
|
---|
| 2200 |
|
---|
| 2201 | -- ALGLIB --
|
---|
| 2202 | Copyright 2005-2009 by Bochkanov Sergey
|
---|
| 2203 | *************************************************************************/
|
---|
| 2204 | public static void gqgenerategaussradaurec(double[] alpha,
|
---|
| 2205 | double[] beta,
|
---|
| 2206 | double mu0,
|
---|
| 2207 | double a,
|
---|
| 2208 | int n,
|
---|
| 2209 | ref int info,
|
---|
| 2210 | ref double[] x,
|
---|
| 2211 | ref double[] w)
|
---|
| 2212 | {
|
---|
| 2213 | int i = 0;
|
---|
| 2214 | double[] d = new double[0];
|
---|
| 2215 | double[] e = new double[0];
|
---|
| 2216 | double[,] z = new double[0,0];
|
---|
| 2217 | double polim1 = 0;
|
---|
| 2218 | double poli = 0;
|
---|
| 2219 | double t = 0;
|
---|
| 2220 |
|
---|
| 2221 | alpha = (double[])alpha.Clone();
|
---|
| 2222 | beta = (double[])beta.Clone();
|
---|
| 2223 | info = 0;
|
---|
| 2224 | x = new double[0];
|
---|
| 2225 | w = new double[0];
|
---|
| 2226 |
|
---|
| 2227 | if( n<2 )
|
---|
| 2228 | {
|
---|
| 2229 | info = -1;
|
---|
| 2230 | return;
|
---|
| 2231 | }
|
---|
| 2232 | info = 1;
|
---|
| 2233 |
|
---|
| 2234 | //
|
---|
| 2235 | // Initialize, D[1:N], E[1:N]
|
---|
| 2236 | //
|
---|
| 2237 | n = n-1;
|
---|
| 2238 | d = new double[n+1];
|
---|
| 2239 | e = new double[n];
|
---|
| 2240 | for(i=1; i<=n; i++)
|
---|
| 2241 | {
|
---|
| 2242 | d[i-1] = alpha[i-1];
|
---|
| 2243 | if( (double)(beta[i])<=(double)(0) )
|
---|
| 2244 | {
|
---|
| 2245 | info = -2;
|
---|
| 2246 | return;
|
---|
| 2247 | }
|
---|
| 2248 | e[i-1] = Math.Sqrt(beta[i]);
|
---|
| 2249 | }
|
---|
| 2250 |
|
---|
| 2251 | //
|
---|
| 2252 | // Caclulate Pn(a), Pn-1(a), and D[N+1]
|
---|
| 2253 | //
|
---|
| 2254 | beta[0] = 0;
|
---|
| 2255 | polim1 = 0;
|
---|
| 2256 | poli = 1;
|
---|
| 2257 | for(i=1; i<=n; i++)
|
---|
| 2258 | {
|
---|
| 2259 | t = (a-alpha[i-1])*poli-beta[i-1]*polim1;
|
---|
| 2260 | polim1 = poli;
|
---|
| 2261 | poli = t;
|
---|
| 2262 | }
|
---|
| 2263 | d[n] = a-beta[n]*polim1/poli;
|
---|
| 2264 |
|
---|
| 2265 | //
|
---|
| 2266 | // EVD
|
---|
| 2267 | //
|
---|
| 2268 | if( !evd.smatrixtdevd(ref d, e, n+1, 3, ref z) )
|
---|
| 2269 | {
|
---|
| 2270 | info = -3;
|
---|
| 2271 | return;
|
---|
| 2272 | }
|
---|
| 2273 |
|
---|
| 2274 | //
|
---|
| 2275 | // Generate
|
---|
| 2276 | //
|
---|
| 2277 | x = new double[n+1];
|
---|
| 2278 | w = new double[n+1];
|
---|
| 2279 | for(i=1; i<=n+1; i++)
|
---|
| 2280 | {
|
---|
| 2281 | x[i-1] = d[i-1];
|
---|
| 2282 | w[i-1] = mu0*math.sqr(z[0,i-1]);
|
---|
| 2283 | }
|
---|
| 2284 | }
|
---|
| 2285 |
|
---|
| 2286 |
|
---|
| 2287 | /*************************************************************************
|
---|
| 2288 | Returns nodes/weights for Gauss-Legendre quadrature on [-1,1] with N
|
---|
| 2289 | nodes.
|
---|
| 2290 |
|
---|
| 2291 | INPUT PARAMETERS:
|
---|
| 2292 | N - number of nodes, >=1
|
---|
| 2293 |
|
---|
| 2294 | OUTPUT PARAMETERS:
|
---|
| 2295 | Info - error code:
|
---|
| 2296 | * -4 an error was detected when calculating
|
---|
| 2297 | weights/nodes. N is too large to obtain
|
---|
| 2298 | weights/nodes with high enough accuracy.
|
---|
| 2299 | Try to use multiple precision version.
|
---|
| 2300 | * -3 internal eigenproblem solver hasn't converged
|
---|
| 2301 | * -1 incorrect N was passed
|
---|
| 2302 | * +1 OK
|
---|
| 2303 | X - array[0..N-1] - array of quadrature nodes,
|
---|
| 2304 | in ascending order.
|
---|
| 2305 | W - array[0..N-1] - array of quadrature weights.
|
---|
| 2306 |
|
---|
| 2307 |
|
---|
| 2308 | -- ALGLIB --
|
---|
| 2309 | Copyright 12.05.2009 by Bochkanov Sergey
|
---|
| 2310 | *************************************************************************/
|
---|
| 2311 | public static void gqgenerategausslegendre(int n,
|
---|
| 2312 | ref int info,
|
---|
| 2313 | ref double[] x,
|
---|
| 2314 | ref double[] w)
|
---|
| 2315 | {
|
---|
| 2316 | double[] alpha = new double[0];
|
---|
| 2317 | double[] beta = new double[0];
|
---|
| 2318 | int i = 0;
|
---|
| 2319 |
|
---|
| 2320 | info = 0;
|
---|
| 2321 | x = new double[0];
|
---|
| 2322 | w = new double[0];
|
---|
| 2323 |
|
---|
| 2324 | if( n<1 )
|
---|
| 2325 | {
|
---|
| 2326 | info = -1;
|
---|
| 2327 | return;
|
---|
| 2328 | }
|
---|
| 2329 | alpha = new double[n];
|
---|
| 2330 | beta = new double[n];
|
---|
| 2331 | for(i=0; i<=n-1; i++)
|
---|
| 2332 | {
|
---|
| 2333 | alpha[i] = 0;
|
---|
| 2334 | }
|
---|
| 2335 | beta[0] = 2;
|
---|
| 2336 | for(i=1; i<=n-1; i++)
|
---|
| 2337 | {
|
---|
| 2338 | beta[i] = 1/(4-1/math.sqr(i));
|
---|
| 2339 | }
|
---|
| 2340 | gqgeneraterec(alpha, beta, beta[0], n, ref info, ref x, ref w);
|
---|
| 2341 |
|
---|
| 2342 | //
|
---|
| 2343 | // test basic properties to detect errors
|
---|
| 2344 | //
|
---|
| 2345 | if( info>0 )
|
---|
| 2346 | {
|
---|
| 2347 | if( (double)(x[0])<(double)(-1) | (double)(x[n-1])>(double)(1) )
|
---|
| 2348 | {
|
---|
| 2349 | info = -4;
|
---|
| 2350 | }
|
---|
| 2351 | for(i=0; i<=n-2; i++)
|
---|
| 2352 | {
|
---|
| 2353 | if( (double)(x[i])>=(double)(x[i+1]) )
|
---|
| 2354 | {
|
---|
| 2355 | info = -4;
|
---|
| 2356 | }
|
---|
| 2357 | }
|
---|
| 2358 | }
|
---|
| 2359 | }
|
---|
| 2360 |
|
---|
| 2361 |
|
---|
| 2362 | /*************************************************************************
|
---|
| 2363 | Returns nodes/weights for Gauss-Jacobi quadrature on [-1,1] with weight
|
---|
| 2364 | function W(x)=Power(1-x,Alpha)*Power(1+x,Beta).
|
---|
| 2365 |
|
---|
| 2366 | INPUT PARAMETERS:
|
---|
| 2367 | N - number of nodes, >=1
|
---|
| 2368 | Alpha - power-law coefficient, Alpha>-1
|
---|
| 2369 | Beta - power-law coefficient, Beta>-1
|
---|
| 2370 |
|
---|
| 2371 | OUTPUT PARAMETERS:
|
---|
| 2372 | Info - error code:
|
---|
| 2373 | * -4 an error was detected when calculating
|
---|
| 2374 | weights/nodes. Alpha or Beta are too close
|
---|
| 2375 | to -1 to obtain weights/nodes with high enough
|
---|
| 2376 | accuracy, or, may be, N is too large. Try to
|
---|
| 2377 | use multiple precision version.
|
---|
| 2378 | * -3 internal eigenproblem solver hasn't converged
|
---|
| 2379 | * -1 incorrect N/Alpha/Beta was passed
|
---|
| 2380 | * +1 OK
|
---|
| 2381 | X - array[0..N-1] - array of quadrature nodes,
|
---|
| 2382 | in ascending order.
|
---|
| 2383 | W - array[0..N-1] - array of quadrature weights.
|
---|
| 2384 |
|
---|
| 2385 |
|
---|
| 2386 | -- ALGLIB --
|
---|
| 2387 | Copyright 12.05.2009 by Bochkanov Sergey
|
---|
| 2388 | *************************************************************************/
|
---|
| 2389 | public static void gqgenerategaussjacobi(int n,
|
---|
| 2390 | double alpha,
|
---|
| 2391 | double beta,
|
---|
| 2392 | ref int info,
|
---|
| 2393 | ref double[] x,
|
---|
| 2394 | ref double[] w)
|
---|
| 2395 | {
|
---|
| 2396 | double[] a = new double[0];
|
---|
| 2397 | double[] b = new double[0];
|
---|
| 2398 | double alpha2 = 0;
|
---|
| 2399 | double beta2 = 0;
|
---|
| 2400 | double apb = 0;
|
---|
| 2401 | double t = 0;
|
---|
| 2402 | int i = 0;
|
---|
| 2403 | double s = 0;
|
---|
| 2404 |
|
---|
| 2405 | info = 0;
|
---|
| 2406 | x = new double[0];
|
---|
| 2407 | w = new double[0];
|
---|
| 2408 |
|
---|
| 2409 | if( (n<1 | (double)(alpha)<=(double)(-1)) | (double)(beta)<=(double)(-1) )
|
---|
| 2410 | {
|
---|
| 2411 | info = -1;
|
---|
| 2412 | return;
|
---|
| 2413 | }
|
---|
| 2414 | a = new double[n];
|
---|
| 2415 | b = new double[n];
|
---|
| 2416 | apb = alpha+beta;
|
---|
| 2417 | a[0] = (beta-alpha)/(apb+2);
|
---|
| 2418 | t = (apb+1)*Math.Log(2)+gammafunc.lngamma(alpha+1, ref s)+gammafunc.lngamma(beta+1, ref s)-gammafunc.lngamma(apb+2, ref s);
|
---|
| 2419 | if( (double)(t)>(double)(Math.Log(math.maxrealnumber)) )
|
---|
| 2420 | {
|
---|
| 2421 | info = -4;
|
---|
| 2422 | return;
|
---|
| 2423 | }
|
---|
| 2424 | b[0] = Math.Exp(t);
|
---|
| 2425 | if( n>1 )
|
---|
| 2426 | {
|
---|
| 2427 | alpha2 = math.sqr(alpha);
|
---|
| 2428 | beta2 = math.sqr(beta);
|
---|
| 2429 | a[1] = (beta2-alpha2)/((apb+2)*(apb+4));
|
---|
| 2430 | b[1] = 4*(alpha+1)*(beta+1)/((apb+3)*math.sqr(apb+2));
|
---|
| 2431 | for(i=2; i<=n-1; i++)
|
---|
| 2432 | {
|
---|
| 2433 | a[i] = 0.25*(beta2-alpha2)/(i*i*(1+0.5*apb/i)*(1+0.5*(apb+2)/i));
|
---|
| 2434 | b[i] = 0.25*(1+alpha/i)*(1+beta/i)*(1+apb/i)/((1+0.5*(apb+1)/i)*(1+0.5*(apb-1)/i)*math.sqr(1+0.5*apb/i));
|
---|
| 2435 | }
|
---|
| 2436 | }
|
---|
| 2437 | gqgeneraterec(a, b, b[0], n, ref info, ref x, ref w);
|
---|
| 2438 |
|
---|
| 2439 | //
|
---|
| 2440 | // test basic properties to detect errors
|
---|
| 2441 | //
|
---|
| 2442 | if( info>0 )
|
---|
| 2443 | {
|
---|
| 2444 | if( (double)(x[0])<(double)(-1) | (double)(x[n-1])>(double)(1) )
|
---|
| 2445 | {
|
---|
| 2446 | info = -4;
|
---|
| 2447 | }
|
---|
| 2448 | for(i=0; i<=n-2; i++)
|
---|
| 2449 | {
|
---|
| 2450 | if( (double)(x[i])>=(double)(x[i+1]) )
|
---|
| 2451 | {
|
---|
| 2452 | info = -4;
|
---|
| 2453 | }
|
---|
| 2454 | }
|
---|
| 2455 | }
|
---|
| 2456 | }
|
---|
| 2457 |
|
---|
| 2458 |
|
---|
| 2459 | /*************************************************************************
|
---|
| 2460 | Returns nodes/weights for Gauss-Laguerre quadrature on [0,+inf) with
|
---|
| 2461 | weight function W(x)=Power(x,Alpha)*Exp(-x)
|
---|
| 2462 |
|
---|
| 2463 | INPUT PARAMETERS:
|
---|
| 2464 | N - number of nodes, >=1
|
---|
| 2465 | Alpha - power-law coefficient, Alpha>-1
|
---|
| 2466 |
|
---|
| 2467 | OUTPUT PARAMETERS:
|
---|
| 2468 | Info - error code:
|
---|
| 2469 | * -4 an error was detected when calculating
|
---|
| 2470 | weights/nodes. Alpha is too close to -1 to
|
---|
| 2471 | obtain weights/nodes with high enough accuracy
|
---|
| 2472 | or, may be, N is too large. Try to use
|
---|
| 2473 | multiple precision version.
|
---|
| 2474 | * -3 internal eigenproblem solver hasn't converged
|
---|
| 2475 | * -1 incorrect N/Alpha was passed
|
---|
| 2476 | * +1 OK
|
---|
| 2477 | X - array[0..N-1] - array of quadrature nodes,
|
---|
| 2478 | in ascending order.
|
---|
| 2479 | W - array[0..N-1] - array of quadrature weights.
|
---|
| 2480 |
|
---|
| 2481 |
|
---|
| 2482 | -- ALGLIB --
|
---|
| 2483 | Copyright 12.05.2009 by Bochkanov Sergey
|
---|
| 2484 | *************************************************************************/
|
---|
| 2485 | public static void gqgenerategausslaguerre(int n,
|
---|
| 2486 | double alpha,
|
---|
| 2487 | ref int info,
|
---|
| 2488 | ref double[] x,
|
---|
| 2489 | ref double[] w)
|
---|
| 2490 | {
|
---|
| 2491 | double[] a = new double[0];
|
---|
| 2492 | double[] b = new double[0];
|
---|
| 2493 | double t = 0;
|
---|
| 2494 | int i = 0;
|
---|
| 2495 | double s = 0;
|
---|
| 2496 |
|
---|
| 2497 | info = 0;
|
---|
| 2498 | x = new double[0];
|
---|
| 2499 | w = new double[0];
|
---|
| 2500 |
|
---|
| 2501 | if( n<1 | (double)(alpha)<=(double)(-1) )
|
---|
| 2502 | {
|
---|
| 2503 | info = -1;
|
---|
| 2504 | return;
|
---|
| 2505 | }
|
---|
| 2506 | a = new double[n];
|
---|
| 2507 | b = new double[n];
|
---|
| 2508 | a[0] = alpha+1;
|
---|
| 2509 | t = gammafunc.lngamma(alpha+1, ref s);
|
---|
| 2510 | if( (double)(t)>=(double)(Math.Log(math.maxrealnumber)) )
|
---|
| 2511 | {
|
---|
| 2512 | info = -4;
|
---|
| 2513 | return;
|
---|
| 2514 | }
|
---|
| 2515 | b[0] = Math.Exp(t);
|
---|
| 2516 | if( n>1 )
|
---|
| 2517 | {
|
---|
| 2518 | for(i=1; i<=n-1; i++)
|
---|
| 2519 | {
|
---|
| 2520 | a[i] = 2*i+alpha+1;
|
---|
| 2521 | b[i] = i*(i+alpha);
|
---|
| 2522 | }
|
---|
| 2523 | }
|
---|
| 2524 | gqgeneraterec(a, b, b[0], n, ref info, ref x, ref w);
|
---|
| 2525 |
|
---|
| 2526 | //
|
---|
| 2527 | // test basic properties to detect errors
|
---|
| 2528 | //
|
---|
| 2529 | if( info>0 )
|
---|
| 2530 | {
|
---|
| 2531 | if( (double)(x[0])<(double)(0) )
|
---|
| 2532 | {
|
---|
| 2533 | info = -4;
|
---|
| 2534 | }
|
---|
| 2535 | for(i=0; i<=n-2; i++)
|
---|
| 2536 | {
|
---|
| 2537 | if( (double)(x[i])>=(double)(x[i+1]) )
|
---|
| 2538 | {
|
---|
| 2539 | info = -4;
|
---|
| 2540 | }
|
---|
| 2541 | }
|
---|
| 2542 | }
|
---|
| 2543 | }
|
---|
| 2544 |
|
---|
| 2545 |
|
---|
| 2546 | /*************************************************************************
|
---|
| 2547 | Returns nodes/weights for Gauss-Hermite quadrature on (-inf,+inf) with
|
---|
| 2548 | weight function W(x)=Exp(-x*x)
|
---|
| 2549 |
|
---|
| 2550 | INPUT PARAMETERS:
|
---|
| 2551 | N - number of nodes, >=1
|
---|
| 2552 |
|
---|
| 2553 | OUTPUT PARAMETERS:
|
---|
| 2554 | Info - error code:
|
---|
| 2555 | * -4 an error was detected when calculating
|
---|
| 2556 | weights/nodes. May be, N is too large. Try to
|
---|
| 2557 | use multiple precision version.
|
---|
| 2558 | * -3 internal eigenproblem solver hasn't converged
|
---|
| 2559 | * -1 incorrect N/Alpha was passed
|
---|
| 2560 | * +1 OK
|
---|
| 2561 | X - array[0..N-1] - array of quadrature nodes,
|
---|
| 2562 | in ascending order.
|
---|
| 2563 | W - array[0..N-1] - array of quadrature weights.
|
---|
| 2564 |
|
---|
| 2565 |
|
---|
| 2566 | -- ALGLIB --
|
---|
| 2567 | Copyright 12.05.2009 by Bochkanov Sergey
|
---|
| 2568 | *************************************************************************/
|
---|
| 2569 | public static void gqgenerategausshermite(int n,
|
---|
| 2570 | ref int info,
|
---|
| 2571 | ref double[] x,
|
---|
| 2572 | ref double[] w)
|
---|
| 2573 | {
|
---|
| 2574 | double[] a = new double[0];
|
---|
| 2575 | double[] b = new double[0];
|
---|
| 2576 | int i = 0;
|
---|
| 2577 |
|
---|
| 2578 | info = 0;
|
---|
| 2579 | x = new double[0];
|
---|
| 2580 | w = new double[0];
|
---|
| 2581 |
|
---|
| 2582 | if( n<1 )
|
---|
| 2583 | {
|
---|
| 2584 | info = -1;
|
---|
| 2585 | return;
|
---|
| 2586 | }
|
---|
| 2587 | a = new double[n];
|
---|
| 2588 | b = new double[n];
|
---|
| 2589 | for(i=0; i<=n-1; i++)
|
---|
| 2590 | {
|
---|
| 2591 | a[i] = 0;
|
---|
| 2592 | }
|
---|
| 2593 | b[0] = Math.Sqrt(4*Math.Atan(1));
|
---|
| 2594 | if( n>1 )
|
---|
| 2595 | {
|
---|
| 2596 | for(i=1; i<=n-1; i++)
|
---|
| 2597 | {
|
---|
| 2598 | b[i] = 0.5*i;
|
---|
| 2599 | }
|
---|
| 2600 | }
|
---|
| 2601 | gqgeneraterec(a, b, b[0], n, ref info, ref x, ref w);
|
---|
| 2602 |
|
---|
| 2603 | //
|
---|
| 2604 | // test basic properties to detect errors
|
---|
| 2605 | //
|
---|
| 2606 | if( info>0 )
|
---|
| 2607 | {
|
---|
| 2608 | for(i=0; i<=n-2; i++)
|
---|
| 2609 | {
|
---|
| 2610 | if( (double)(x[i])>=(double)(x[i+1]) )
|
---|
| 2611 | {
|
---|
| 2612 | info = -4;
|
---|
| 2613 | }
|
---|
| 2614 | }
|
---|
| 2615 | }
|
---|
| 2616 | }
|
---|
| 2617 |
|
---|
| 2618 |
|
---|
| 2619 | }
|
---|
| 2620 | public class gkq
|
---|
| 2621 | {
|
---|
| 2622 | /*************************************************************************
|
---|
| 2623 | Computation of nodes and weights of a Gauss-Kronrod quadrature formula
|
---|
| 2624 |
|
---|
| 2625 | The algorithm generates the N-point Gauss-Kronrod quadrature formula with
|
---|
| 2626 | weight function given by coefficients alpha and beta of a recurrence
|
---|
| 2627 | relation which generates a system of orthogonal polynomials:
|
---|
| 2628 |
|
---|
| 2629 | P-1(x) = 0
|
---|
| 2630 | P0(x) = 1
|
---|
| 2631 | Pn+1(x) = (x-alpha(n))*Pn(x) - beta(n)*Pn-1(x)
|
---|
| 2632 |
|
---|
| 2633 | and zero moment Mu0
|
---|
| 2634 |
|
---|
| 2635 | Mu0 = integral(W(x)dx,a,b)
|
---|
| 2636 |
|
---|
| 2637 |
|
---|
| 2638 | INPUT PARAMETERS:
|
---|
| 2639 | Alpha alpha coefficients, array[0..floor(3*K/2)].
|
---|
| 2640 | Beta beta coefficients, array[0..ceil(3*K/2)].
|
---|
| 2641 | Beta[0] is not used and may be arbitrary.
|
---|
| 2642 | Beta[I]>0.
|
---|
| 2643 | Mu0 zeroth moment of the weight function.
|
---|
| 2644 | N number of nodes of the Gauss-Kronrod quadrature formula,
|
---|
| 2645 | N >= 3,
|
---|
| 2646 | N = 2*K+1.
|
---|
| 2647 |
|
---|
| 2648 | OUTPUT PARAMETERS:
|
---|
| 2649 | Info - error code:
|
---|
| 2650 | * -5 no real and positive Gauss-Kronrod formula can
|
---|
| 2651 | be created for such a weight function with a
|
---|
| 2652 | given number of nodes.
|
---|
| 2653 | * -4 N is too large, task may be ill conditioned -
|
---|
| 2654 | x[i]=x[i+1] found.
|
---|
| 2655 | * -3 internal eigenproblem solver hasn't converged
|
---|
| 2656 | * -2 Beta[i]<=0
|
---|
| 2657 | * -1 incorrect N was passed
|
---|
| 2658 | * +1 OK
|
---|
| 2659 | X - array[0..N-1] - array of quadrature nodes,
|
---|
| 2660 | in ascending order.
|
---|
| 2661 | WKronrod - array[0..N-1] - Kronrod weights
|
---|
| 2662 | WGauss - array[0..N-1] - Gauss weights (interleaved with zeros
|
---|
| 2663 | corresponding to extended Kronrod nodes).
|
---|
| 2664 |
|
---|
| 2665 | -- ALGLIB --
|
---|
| 2666 | Copyright 08.05.2009 by Bochkanov Sergey
|
---|
| 2667 | *************************************************************************/
|
---|
| 2668 | public static void gkqgeneraterec(double[] alpha,
|
---|
| 2669 | double[] beta,
|
---|
| 2670 | double mu0,
|
---|
| 2671 | int n,
|
---|
| 2672 | ref int info,
|
---|
| 2673 | ref double[] x,
|
---|
| 2674 | ref double[] wkronrod,
|
---|
| 2675 | ref double[] wgauss)
|
---|
| 2676 | {
|
---|
| 2677 | double[] ta = new double[0];
|
---|
| 2678 | int i = 0;
|
---|
| 2679 | int j = 0;
|
---|
| 2680 | double[] t = new double[0];
|
---|
| 2681 | double[] s = new double[0];
|
---|
| 2682 | int wlen = 0;
|
---|
| 2683 | int woffs = 0;
|
---|
| 2684 | double u = 0;
|
---|
| 2685 | int m = 0;
|
---|
| 2686 | int l = 0;
|
---|
| 2687 | int k = 0;
|
---|
| 2688 | double[] xgtmp = new double[0];
|
---|
| 2689 | double[] wgtmp = new double[0];
|
---|
| 2690 | int i_ = 0;
|
---|
| 2691 |
|
---|
| 2692 | alpha = (double[])alpha.Clone();
|
---|
| 2693 | beta = (double[])beta.Clone();
|
---|
| 2694 | info = 0;
|
---|
| 2695 | x = new double[0];
|
---|
| 2696 | wkronrod = new double[0];
|
---|
| 2697 | wgauss = new double[0];
|
---|
| 2698 |
|
---|
| 2699 | if( n%2!=1 | n<3 )
|
---|
| 2700 | {
|
---|
| 2701 | info = -1;
|
---|
| 2702 | return;
|
---|
| 2703 | }
|
---|
| 2704 | for(i=0; i<=(int)Math.Ceiling((double)(3*(n/2))/(double)2); i++)
|
---|
| 2705 | {
|
---|
| 2706 | if( (double)(beta[i])<=(double)(0) )
|
---|
| 2707 | {
|
---|
| 2708 | info = -2;
|
---|
| 2709 | return;
|
---|
| 2710 | }
|
---|
| 2711 | }
|
---|
| 2712 | info = 1;
|
---|
| 2713 |
|
---|
| 2714 | //
|
---|
| 2715 | // from external conventions about N/Beta/Mu0 to internal
|
---|
| 2716 | //
|
---|
| 2717 | n = n/2;
|
---|
| 2718 | beta[0] = mu0;
|
---|
| 2719 |
|
---|
| 2720 | //
|
---|
| 2721 | // Calculate Gauss nodes/weights, save them for later processing
|
---|
| 2722 | //
|
---|
| 2723 | gq.gqgeneraterec(alpha, beta, mu0, n, ref info, ref xgtmp, ref wgtmp);
|
---|
| 2724 | if( info<0 )
|
---|
| 2725 | {
|
---|
| 2726 | return;
|
---|
| 2727 | }
|
---|
| 2728 |
|
---|
| 2729 | //
|
---|
| 2730 | // Resize:
|
---|
| 2731 | // * A from 0..floor(3*n/2) to 0..2*n
|
---|
| 2732 | // * B from 0..ceil(3*n/2) to 0..2*n
|
---|
| 2733 | //
|
---|
| 2734 | ta = new double[(int)Math.Floor((double)(3*n)/(double)2)+1];
|
---|
| 2735 | for(i_=0; i_<=(int)Math.Floor((double)(3*n)/(double)2);i_++)
|
---|
| 2736 | {
|
---|
| 2737 | ta[i_] = alpha[i_];
|
---|
| 2738 | }
|
---|
| 2739 | alpha = new double[2*n+1];
|
---|
| 2740 | for(i_=0; i_<=(int)Math.Floor((double)(3*n)/(double)2);i_++)
|
---|
| 2741 | {
|
---|
| 2742 | alpha[i_] = ta[i_];
|
---|
| 2743 | }
|
---|
| 2744 | for(i=(int)Math.Floor((double)(3*n)/(double)2)+1; i<=2*n; i++)
|
---|
| 2745 | {
|
---|
| 2746 | alpha[i] = 0;
|
---|
| 2747 | }
|
---|
| 2748 | ta = new double[(int)Math.Ceiling((double)(3*n)/(double)2)+1];
|
---|
| 2749 | for(i_=0; i_<=(int)Math.Ceiling((double)(3*n)/(double)2);i_++)
|
---|
| 2750 | {
|
---|
| 2751 | ta[i_] = beta[i_];
|
---|
| 2752 | }
|
---|
| 2753 | beta = new double[2*n+1];
|
---|
| 2754 | for(i_=0; i_<=(int)Math.Ceiling((double)(3*n)/(double)2);i_++)
|
---|
| 2755 | {
|
---|
| 2756 | beta[i_] = ta[i_];
|
---|
| 2757 | }
|
---|
| 2758 | for(i=(int)Math.Ceiling((double)(3*n)/(double)2)+1; i<=2*n; i++)
|
---|
| 2759 | {
|
---|
| 2760 | beta[i] = 0;
|
---|
| 2761 | }
|
---|
| 2762 |
|
---|
| 2763 | //
|
---|
| 2764 | // Initialize T, S
|
---|
| 2765 | //
|
---|
| 2766 | wlen = 2+n/2;
|
---|
| 2767 | t = new double[wlen];
|
---|
| 2768 | s = new double[wlen];
|
---|
| 2769 | ta = new double[wlen];
|
---|
| 2770 | woffs = 1;
|
---|
| 2771 | for(i=0; i<=wlen-1; i++)
|
---|
| 2772 | {
|
---|
| 2773 | t[i] = 0;
|
---|
| 2774 | s[i] = 0;
|
---|
| 2775 | }
|
---|
| 2776 |
|
---|
| 2777 | //
|
---|
| 2778 | // Algorithm from Dirk P. Laurie, "Calculation of Gauss-Kronrod quadrature rules", 1997.
|
---|
| 2779 | //
|
---|
| 2780 | t[woffs+0] = beta[n+1];
|
---|
| 2781 | for(m=0; m<=n-2; m++)
|
---|
| 2782 | {
|
---|
| 2783 | u = 0;
|
---|
| 2784 | for(k=(m+1)/2; k>=0; k--)
|
---|
| 2785 | {
|
---|
| 2786 | l = m-k;
|
---|
| 2787 | u = u+(alpha[k+n+1]-alpha[l])*t[woffs+k]+beta[k+n+1]*s[woffs+k-1]-beta[l]*s[woffs+k];
|
---|
| 2788 | s[woffs+k] = u;
|
---|
| 2789 | }
|
---|
| 2790 | for(i_=0; i_<=wlen-1;i_++)
|
---|
| 2791 | {
|
---|
| 2792 | ta[i_] = t[i_];
|
---|
| 2793 | }
|
---|
| 2794 | for(i_=0; i_<=wlen-1;i_++)
|
---|
| 2795 | {
|
---|
| 2796 | t[i_] = s[i_];
|
---|
| 2797 | }
|
---|
| 2798 | for(i_=0; i_<=wlen-1;i_++)
|
---|
| 2799 | {
|
---|
| 2800 | s[i_] = ta[i_];
|
---|
| 2801 | }
|
---|
| 2802 | }
|
---|
| 2803 | for(j=n/2; j>=0; j--)
|
---|
| 2804 | {
|
---|
| 2805 | s[woffs+j] = s[woffs+j-1];
|
---|
| 2806 | }
|
---|
| 2807 | for(m=n-1; m<=2*n-3; m++)
|
---|
| 2808 | {
|
---|
| 2809 | u = 0;
|
---|
| 2810 | for(k=m+1-n; k<=(m-1)/2; k++)
|
---|
| 2811 | {
|
---|
| 2812 | l = m-k;
|
---|
| 2813 | j = n-1-l;
|
---|
| 2814 | u = u-(alpha[k+n+1]-alpha[l])*t[woffs+j]-beta[k+n+1]*s[woffs+j]+beta[l]*s[woffs+j+1];
|
---|
| 2815 | s[woffs+j] = u;
|
---|
| 2816 | }
|
---|
| 2817 | if( m%2==0 )
|
---|
| 2818 | {
|
---|
| 2819 | k = m/2;
|
---|
| 2820 | alpha[k+n+1] = alpha[k]+(s[woffs+j]-beta[k+n+1]*s[woffs+j+1])/t[woffs+j+1];
|
---|
| 2821 | }
|
---|
| 2822 | else
|
---|
| 2823 | {
|
---|
| 2824 | k = (m+1)/2;
|
---|
| 2825 | beta[k+n+1] = s[woffs+j]/s[woffs+j+1];
|
---|
| 2826 | }
|
---|
| 2827 | for(i_=0; i_<=wlen-1;i_++)
|
---|
| 2828 | {
|
---|
| 2829 | ta[i_] = t[i_];
|
---|
| 2830 | }
|
---|
| 2831 | for(i_=0; i_<=wlen-1;i_++)
|
---|
| 2832 | {
|
---|
| 2833 | t[i_] = s[i_];
|
---|
| 2834 | }
|
---|
| 2835 | for(i_=0; i_<=wlen-1;i_++)
|
---|
| 2836 | {
|
---|
| 2837 | s[i_] = ta[i_];
|
---|
| 2838 | }
|
---|
| 2839 | }
|
---|
| 2840 | alpha[2*n] = alpha[n-1]-beta[2*n]*s[woffs+0]/t[woffs+0];
|
---|
| 2841 |
|
---|
| 2842 | //
|
---|
| 2843 | // calculation of Kronrod nodes and weights, unpacking of Gauss weights
|
---|
| 2844 | //
|
---|
| 2845 | gq.gqgeneraterec(alpha, beta, mu0, 2*n+1, ref info, ref x, ref wkronrod);
|
---|
| 2846 | if( info==-2 )
|
---|
| 2847 | {
|
---|
| 2848 | info = -5;
|
---|
| 2849 | }
|
---|
| 2850 | if( info<0 )
|
---|
| 2851 | {
|
---|
| 2852 | return;
|
---|
| 2853 | }
|
---|
| 2854 | for(i=0; i<=2*n-1; i++)
|
---|
| 2855 | {
|
---|
| 2856 | if( (double)(x[i])>=(double)(x[i+1]) )
|
---|
| 2857 | {
|
---|
| 2858 | info = -4;
|
---|
| 2859 | }
|
---|
| 2860 | }
|
---|
| 2861 | if( info<0 )
|
---|
| 2862 | {
|
---|
| 2863 | return;
|
---|
| 2864 | }
|
---|
| 2865 | wgauss = new double[2*n+1];
|
---|
| 2866 | for(i=0; i<=2*n; i++)
|
---|
| 2867 | {
|
---|
| 2868 | wgauss[i] = 0;
|
---|
| 2869 | }
|
---|
| 2870 | for(i=0; i<=n-1; i++)
|
---|
| 2871 | {
|
---|
| 2872 | wgauss[2*i+1] = wgtmp[i];
|
---|
| 2873 | }
|
---|
| 2874 | }
|
---|
| 2875 |
|
---|
| 2876 |
|
---|
| 2877 | /*************************************************************************
|
---|
| 2878 | Returns Gauss and Gauss-Kronrod nodes/weights for Gauss-Legendre
|
---|
| 2879 | quadrature with N points.
|
---|
| 2880 |
|
---|
| 2881 | GKQLegendreCalc (calculation) or GKQLegendreTbl (precomputed table) is
|
---|
| 2882 | used depending on machine precision and number of nodes.
|
---|
| 2883 |
|
---|
| 2884 | INPUT PARAMETERS:
|
---|
| 2885 | N - number of Kronrod nodes, must be odd number, >=3.
|
---|
| 2886 |
|
---|
| 2887 | OUTPUT PARAMETERS:
|
---|
| 2888 | Info - error code:
|
---|
| 2889 | * -4 an error was detected when calculating
|
---|
| 2890 | weights/nodes. N is too large to obtain
|
---|
| 2891 | weights/nodes with high enough accuracy.
|
---|
| 2892 | Try to use multiple precision version.
|
---|
| 2893 | * -3 internal eigenproblem solver hasn't converged
|
---|
| 2894 | * -1 incorrect N was passed
|
---|
| 2895 | * +1 OK
|
---|
| 2896 | X - array[0..N-1] - array of quadrature nodes, ordered in
|
---|
| 2897 | ascending order.
|
---|
| 2898 | WKronrod - array[0..N-1] - Kronrod weights
|
---|
| 2899 | WGauss - array[0..N-1] - Gauss weights (interleaved with zeros
|
---|
| 2900 | corresponding to extended Kronrod nodes).
|
---|
| 2901 |
|
---|
| 2902 |
|
---|
| 2903 | -- ALGLIB --
|
---|
| 2904 | Copyright 12.05.2009 by Bochkanov Sergey
|
---|
| 2905 | *************************************************************************/
|
---|
| 2906 | public static void gkqgenerategausslegendre(int n,
|
---|
| 2907 | ref int info,
|
---|
| 2908 | ref double[] x,
|
---|
| 2909 | ref double[] wkronrod,
|
---|
| 2910 | ref double[] wgauss)
|
---|
| 2911 | {
|
---|
| 2912 | double eps = 0;
|
---|
| 2913 |
|
---|
| 2914 | info = 0;
|
---|
| 2915 | x = new double[0];
|
---|
| 2916 | wkronrod = new double[0];
|
---|
| 2917 | wgauss = new double[0];
|
---|
| 2918 |
|
---|
| 2919 | if( (double)(math.machineepsilon)>(double)(1.0E-32) & (((((n==15 | n==21) | n==31) | n==41) | n==51) | n==61) )
|
---|
| 2920 | {
|
---|
| 2921 | info = 1;
|
---|
| 2922 | gkqlegendretbl(n, ref x, ref wkronrod, ref wgauss, ref eps);
|
---|
| 2923 | }
|
---|
| 2924 | else
|
---|
| 2925 | {
|
---|
| 2926 | gkqlegendrecalc(n, ref info, ref x, ref wkronrod, ref wgauss);
|
---|
| 2927 | }
|
---|
| 2928 | }
|
---|
| 2929 |
|
---|
| 2930 |
|
---|
| 2931 | /*************************************************************************
|
---|
| 2932 | Returns Gauss and Gauss-Kronrod nodes/weights for Gauss-Jacobi
|
---|
| 2933 | quadrature on [-1,1] with weight function
|
---|
| 2934 |
|
---|
| 2935 | W(x)=Power(1-x,Alpha)*Power(1+x,Beta).
|
---|
| 2936 |
|
---|
| 2937 | INPUT PARAMETERS:
|
---|
| 2938 | N - number of Kronrod nodes, must be odd number, >=3.
|
---|
| 2939 | Alpha - power-law coefficient, Alpha>-1
|
---|
| 2940 | Beta - power-law coefficient, Beta>-1
|
---|
| 2941 |
|
---|
| 2942 | OUTPUT PARAMETERS:
|
---|
| 2943 | Info - error code:
|
---|
| 2944 | * -5 no real and positive Gauss-Kronrod formula can
|
---|
| 2945 | be created for such a weight function with a
|
---|
| 2946 | given number of nodes.
|
---|
| 2947 | * -4 an error was detected when calculating
|
---|
| 2948 | weights/nodes. Alpha or Beta are too close
|
---|
| 2949 | to -1 to obtain weights/nodes with high enough
|
---|
| 2950 | accuracy, or, may be, N is too large. Try to
|
---|
| 2951 | use multiple precision version.
|
---|
| 2952 | * -3 internal eigenproblem solver hasn't converged
|
---|
| 2953 | * -1 incorrect N was passed
|
---|
| 2954 | * +1 OK
|
---|
| 2955 | * +2 OK, but quadrature rule have exterior nodes,
|
---|
| 2956 | x[0]<-1 or x[n-1]>+1
|
---|
| 2957 | X - array[0..N-1] - array of quadrature nodes, ordered in
|
---|
| 2958 | ascending order.
|
---|
| 2959 | WKronrod - array[0..N-1] - Kronrod weights
|
---|
| 2960 | WGauss - array[0..N-1] - Gauss weights (interleaved with zeros
|
---|
| 2961 | corresponding to extended Kronrod nodes).
|
---|
| 2962 |
|
---|
| 2963 |
|
---|
| 2964 | -- ALGLIB --
|
---|
| 2965 | Copyright 12.05.2009 by Bochkanov Sergey
|
---|
| 2966 | *************************************************************************/
|
---|
| 2967 | public static void gkqgenerategaussjacobi(int n,
|
---|
| 2968 | double alpha,
|
---|
| 2969 | double beta,
|
---|
| 2970 | ref int info,
|
---|
| 2971 | ref double[] x,
|
---|
| 2972 | ref double[] wkronrod,
|
---|
| 2973 | ref double[] wgauss)
|
---|
| 2974 | {
|
---|
| 2975 | int clen = 0;
|
---|
| 2976 | double[] a = new double[0];
|
---|
| 2977 | double[] b = new double[0];
|
---|
| 2978 | double alpha2 = 0;
|
---|
| 2979 | double beta2 = 0;
|
---|
| 2980 | double apb = 0;
|
---|
| 2981 | double t = 0;
|
---|
| 2982 | int i = 0;
|
---|
| 2983 | double s = 0;
|
---|
| 2984 |
|
---|
| 2985 | info = 0;
|
---|
| 2986 | x = new double[0];
|
---|
| 2987 | wkronrod = new double[0];
|
---|
| 2988 | wgauss = new double[0];
|
---|
| 2989 |
|
---|
| 2990 | if( n%2!=1 | n<3 )
|
---|
| 2991 | {
|
---|
| 2992 | info = -1;
|
---|
| 2993 | return;
|
---|
| 2994 | }
|
---|
| 2995 | if( (double)(alpha)<=(double)(-1) | (double)(beta)<=(double)(-1) )
|
---|
| 2996 | {
|
---|
| 2997 | info = -1;
|
---|
| 2998 | return;
|
---|
| 2999 | }
|
---|
| 3000 | clen = (int)Math.Ceiling((double)(3*(n/2))/(double)2)+1;
|
---|
| 3001 | a = new double[clen];
|
---|
| 3002 | b = new double[clen];
|
---|
| 3003 | for(i=0; i<=clen-1; i++)
|
---|
| 3004 | {
|
---|
| 3005 | a[i] = 0;
|
---|
| 3006 | }
|
---|
| 3007 | apb = alpha+beta;
|
---|
| 3008 | a[0] = (beta-alpha)/(apb+2);
|
---|
| 3009 | t = (apb+1)*Math.Log(2)+gammafunc.lngamma(alpha+1, ref s)+gammafunc.lngamma(beta+1, ref s)-gammafunc.lngamma(apb+2, ref s);
|
---|
| 3010 | if( (double)(t)>(double)(Math.Log(math.maxrealnumber)) )
|
---|
| 3011 | {
|
---|
| 3012 | info = -4;
|
---|
| 3013 | return;
|
---|
| 3014 | }
|
---|
| 3015 | b[0] = Math.Exp(t);
|
---|
| 3016 | if( clen>1 )
|
---|
| 3017 | {
|
---|
| 3018 | alpha2 = math.sqr(alpha);
|
---|
| 3019 | beta2 = math.sqr(beta);
|
---|
| 3020 | a[1] = (beta2-alpha2)/((apb+2)*(apb+4));
|
---|
| 3021 | b[1] = 4*(alpha+1)*(beta+1)/((apb+3)*math.sqr(apb+2));
|
---|
| 3022 | for(i=2; i<=clen-1; i++)
|
---|
| 3023 | {
|
---|
| 3024 | a[i] = 0.25*(beta2-alpha2)/(i*i*(1+0.5*apb/i)*(1+0.5*(apb+2)/i));
|
---|
| 3025 | b[i] = 0.25*(1+alpha/i)*(1+beta/i)*(1+apb/i)/((1+0.5*(apb+1)/i)*(1+0.5*(apb-1)/i)*math.sqr(1+0.5*apb/i));
|
---|
| 3026 | }
|
---|
| 3027 | }
|
---|
| 3028 | gkqgeneraterec(a, b, b[0], n, ref info, ref x, ref wkronrod, ref wgauss);
|
---|
| 3029 |
|
---|
| 3030 | //
|
---|
| 3031 | // test basic properties to detect errors
|
---|
| 3032 | //
|
---|
| 3033 | if( info>0 )
|
---|
| 3034 | {
|
---|
| 3035 | if( (double)(x[0])<(double)(-1) | (double)(x[n-1])>(double)(1) )
|
---|
| 3036 | {
|
---|
| 3037 | info = 2;
|
---|
| 3038 | }
|
---|
| 3039 | for(i=0; i<=n-2; i++)
|
---|
| 3040 | {
|
---|
| 3041 | if( (double)(x[i])>=(double)(x[i+1]) )
|
---|
| 3042 | {
|
---|
| 3043 | info = -4;
|
---|
| 3044 | }
|
---|
| 3045 | }
|
---|
| 3046 | }
|
---|
| 3047 | }
|
---|
| 3048 |
|
---|
| 3049 |
|
---|
| 3050 | /*************************************************************************
|
---|
| 3051 | Returns Gauss and Gauss-Kronrod nodes for quadrature with N points.
|
---|
| 3052 |
|
---|
| 3053 | Reduction to tridiagonal eigenproblem is used.
|
---|
| 3054 |
|
---|
| 3055 | INPUT PARAMETERS:
|
---|
| 3056 | N - number of Kronrod nodes, must be odd number, >=3.
|
---|
| 3057 |
|
---|
| 3058 | OUTPUT PARAMETERS:
|
---|
| 3059 | Info - error code:
|
---|
| 3060 | * -4 an error was detected when calculating
|
---|
| 3061 | weights/nodes. N is too large to obtain
|
---|
| 3062 | weights/nodes with high enough accuracy.
|
---|
| 3063 | Try to use multiple precision version.
|
---|
| 3064 | * -3 internal eigenproblem solver hasn't converged
|
---|
| 3065 | * -1 incorrect N was passed
|
---|
| 3066 | * +1 OK
|
---|
| 3067 | X - array[0..N-1] - array of quadrature nodes, ordered in
|
---|
| 3068 | ascending order.
|
---|
| 3069 | WKronrod - array[0..N-1] - Kronrod weights
|
---|
| 3070 | WGauss - array[0..N-1] - Gauss weights (interleaved with zeros
|
---|
| 3071 | corresponding to extended Kronrod nodes).
|
---|
| 3072 |
|
---|
| 3073 | -- ALGLIB --
|
---|
| 3074 | Copyright 12.05.2009 by Bochkanov Sergey
|
---|
| 3075 | *************************************************************************/
|
---|
| 3076 | public static void gkqlegendrecalc(int n,
|
---|
| 3077 | ref int info,
|
---|
| 3078 | ref double[] x,
|
---|
| 3079 | ref double[] wkronrod,
|
---|
| 3080 | ref double[] wgauss)
|
---|
| 3081 | {
|
---|
| 3082 | double[] alpha = new double[0];
|
---|
| 3083 | double[] beta = new double[0];
|
---|
| 3084 | int alen = 0;
|
---|
| 3085 | int blen = 0;
|
---|
| 3086 | double mu0 = 0;
|
---|
| 3087 | int k = 0;
|
---|
| 3088 | int i = 0;
|
---|
| 3089 |
|
---|
| 3090 | info = 0;
|
---|
| 3091 | x = new double[0];
|
---|
| 3092 | wkronrod = new double[0];
|
---|
| 3093 | wgauss = new double[0];
|
---|
| 3094 |
|
---|
| 3095 | if( n%2!=1 | n<3 )
|
---|
| 3096 | {
|
---|
| 3097 | info = -1;
|
---|
| 3098 | return;
|
---|
| 3099 | }
|
---|
| 3100 | mu0 = 2;
|
---|
| 3101 | alen = (int)Math.Floor((double)(3*(n/2))/(double)2)+1;
|
---|
| 3102 | blen = (int)Math.Ceiling((double)(3*(n/2))/(double)2)+1;
|
---|
| 3103 | alpha = new double[alen];
|
---|
| 3104 | beta = new double[blen];
|
---|
| 3105 | for(k=0; k<=alen-1; k++)
|
---|
| 3106 | {
|
---|
| 3107 | alpha[k] = 0;
|
---|
| 3108 | }
|
---|
| 3109 | beta[0] = 2;
|
---|
| 3110 | for(k=1; k<=blen-1; k++)
|
---|
| 3111 | {
|
---|
| 3112 | beta[k] = 1/(4-1/math.sqr(k));
|
---|
| 3113 | }
|
---|
| 3114 | gkqgeneraterec(alpha, beta, mu0, n, ref info, ref x, ref wkronrod, ref wgauss);
|
---|
| 3115 |
|
---|
| 3116 | //
|
---|
| 3117 | // test basic properties to detect errors
|
---|
| 3118 | //
|
---|
| 3119 | if( info>0 )
|
---|
| 3120 | {
|
---|
| 3121 | if( (double)(x[0])<(double)(-1) | (double)(x[n-1])>(double)(1) )
|
---|
| 3122 | {
|
---|
| 3123 | info = -4;
|
---|
| 3124 | }
|
---|
| 3125 | for(i=0; i<=n-2; i++)
|
---|
| 3126 | {
|
---|
| 3127 | if( (double)(x[i])>=(double)(x[i+1]) )
|
---|
| 3128 | {
|
---|
| 3129 | info = -4;
|
---|
| 3130 | }
|
---|
| 3131 | }
|
---|
| 3132 | }
|
---|
| 3133 | }
|
---|
| 3134 |
|
---|
| 3135 |
|
---|
| 3136 | /*************************************************************************
|
---|
| 3137 | Returns Gauss and Gauss-Kronrod nodes for quadrature with N points using
|
---|
| 3138 | pre-calculated table. Nodes/weights were computed with accuracy up to
|
---|
| 3139 | 1.0E-32 (if MPFR version of ALGLIB is used). In standard double precision
|
---|
| 3140 | accuracy reduces to something about 2.0E-16 (depending on your compiler's
|
---|
| 3141 | handling of long floating point constants).
|
---|
| 3142 |
|
---|
| 3143 | INPUT PARAMETERS:
|
---|
| 3144 | N - number of Kronrod nodes.
|
---|
| 3145 | N can be 15, 21, 31, 41, 51, 61.
|
---|
| 3146 |
|
---|
| 3147 | OUTPUT PARAMETERS:
|
---|
| 3148 | X - array[0..N-1] - array of quadrature nodes, ordered in
|
---|
| 3149 | ascending order.
|
---|
| 3150 | WKronrod - array[0..N-1] - Kronrod weights
|
---|
| 3151 | WGauss - array[0..N-1] - Gauss weights (interleaved with zeros
|
---|
| 3152 | corresponding to extended Kronrod nodes).
|
---|
| 3153 |
|
---|
| 3154 |
|
---|
| 3155 | -- ALGLIB --
|
---|
| 3156 | Copyright 12.05.2009 by Bochkanov Sergey
|
---|
| 3157 | *************************************************************************/
|
---|
| 3158 | public static void gkqlegendretbl(int n,
|
---|
| 3159 | ref double[] x,
|
---|
| 3160 | ref double[] wkronrod,
|
---|
| 3161 | ref double[] wgauss,
|
---|
| 3162 | ref double eps)
|
---|
| 3163 | {
|
---|
| 3164 | int i = 0;
|
---|
| 3165 | int ng = 0;
|
---|
| 3166 | int[] p1 = new int[0];
|
---|
| 3167 | int[] p2 = new int[0];
|
---|
| 3168 | double tmp = 0;
|
---|
| 3169 |
|
---|
| 3170 | x = new double[0];
|
---|
| 3171 | wkronrod = new double[0];
|
---|
| 3172 | wgauss = new double[0];
|
---|
| 3173 | eps = 0;
|
---|
| 3174 |
|
---|
| 3175 |
|
---|
| 3176 | //
|
---|
| 3177 | // these initializers are not really necessary,
|
---|
| 3178 | // but without them compiler complains about uninitialized locals
|
---|
| 3179 | //
|
---|
| 3180 | ng = 0;
|
---|
| 3181 |
|
---|
| 3182 | //
|
---|
| 3183 | // Process
|
---|
| 3184 | //
|
---|
| 3185 | ap.assert(((((n==15 | n==21) | n==31) | n==41) | n==51) | n==61, "GKQNodesTbl: incorrect N!");
|
---|
| 3186 | x = new double[n];
|
---|
| 3187 | wkronrod = new double[n];
|
---|
| 3188 | wgauss = new double[n];
|
---|
| 3189 | for(i=0; i<=n-1; i++)
|
---|
| 3190 | {
|
---|
| 3191 | x[i] = 0;
|
---|
| 3192 | wkronrod[i] = 0;
|
---|
| 3193 | wgauss[i] = 0;
|
---|
| 3194 | }
|
---|
| 3195 | eps = Math.Max(math.machineepsilon, 1.0E-32);
|
---|
| 3196 | if( n==15 )
|
---|
| 3197 | {
|
---|
| 3198 | ng = 4;
|
---|
| 3199 | wgauss[0] = 0.129484966168869693270611432679082;
|
---|
| 3200 | wgauss[1] = 0.279705391489276667901467771423780;
|
---|
| 3201 | wgauss[2] = 0.381830050505118944950369775488975;
|
---|
| 3202 | wgauss[3] = 0.417959183673469387755102040816327;
|
---|
| 3203 | x[0] = 0.991455371120812639206854697526329;
|
---|
| 3204 | x[1] = 0.949107912342758524526189684047851;
|
---|
| 3205 | x[2] = 0.864864423359769072789712788640926;
|
---|
| 3206 | x[3] = 0.741531185599394439863864773280788;
|
---|
| 3207 | x[4] = 0.586087235467691130294144838258730;
|
---|
| 3208 | x[5] = 0.405845151377397166906606412076961;
|
---|
| 3209 | x[6] = 0.207784955007898467600689403773245;
|
---|
| 3210 | x[7] = 0.000000000000000000000000000000000;
|
---|
| 3211 | wkronrod[0] = 0.022935322010529224963732008058970;
|
---|
| 3212 | wkronrod[1] = 0.063092092629978553290700663189204;
|
---|
| 3213 | wkronrod[2] = 0.104790010322250183839876322541518;
|
---|
| 3214 | wkronrod[3] = 0.140653259715525918745189590510238;
|
---|
| 3215 | wkronrod[4] = 0.169004726639267902826583426598550;
|
---|
| 3216 | wkronrod[5] = 0.190350578064785409913256402421014;
|
---|
| 3217 | wkronrod[6] = 0.204432940075298892414161999234649;
|
---|
| 3218 | wkronrod[7] = 0.209482141084727828012999174891714;
|
---|
| 3219 | }
|
---|
| 3220 | if( n==21 )
|
---|
| 3221 | {
|
---|
| 3222 | ng = 5;
|
---|
| 3223 | wgauss[0] = 0.066671344308688137593568809893332;
|
---|
| 3224 | wgauss[1] = 0.149451349150580593145776339657697;
|
---|
| 3225 | wgauss[2] = 0.219086362515982043995534934228163;
|
---|
| 3226 | wgauss[3] = 0.269266719309996355091226921569469;
|
---|
| 3227 | wgauss[4] = 0.295524224714752870173892994651338;
|
---|
| 3228 | x[0] = 0.995657163025808080735527280689003;
|
---|
| 3229 | x[1] = 0.973906528517171720077964012084452;
|
---|
| 3230 | x[2] = 0.930157491355708226001207180059508;
|
---|
| 3231 | x[3] = 0.865063366688984510732096688423493;
|
---|
| 3232 | x[4] = 0.780817726586416897063717578345042;
|
---|
| 3233 | x[5] = 0.679409568299024406234327365114874;
|
---|
| 3234 | x[6] = 0.562757134668604683339000099272694;
|
---|
| 3235 | x[7] = 0.433395394129247190799265943165784;
|
---|
| 3236 | x[8] = 0.294392862701460198131126603103866;
|
---|
| 3237 | x[9] = 0.148874338981631210884826001129720;
|
---|
| 3238 | x[10] = 0.000000000000000000000000000000000;
|
---|
| 3239 | wkronrod[0] = 0.011694638867371874278064396062192;
|
---|
| 3240 | wkronrod[1] = 0.032558162307964727478818972459390;
|
---|
| 3241 | wkronrod[2] = 0.054755896574351996031381300244580;
|
---|
| 3242 | wkronrod[3] = 0.075039674810919952767043140916190;
|
---|
| 3243 | wkronrod[4] = 0.093125454583697605535065465083366;
|
---|
| 3244 | wkronrod[5] = 0.109387158802297641899210590325805;
|
---|
| 3245 | wkronrod[6] = 0.123491976262065851077958109831074;
|
---|
| 3246 | wkronrod[7] = 0.134709217311473325928054001771707;
|
---|
| 3247 | wkronrod[8] = 0.142775938577060080797094273138717;
|
---|
| 3248 | wkronrod[9] = 0.147739104901338491374841515972068;
|
---|
| 3249 | wkronrod[10] = 0.149445554002916905664936468389821;
|
---|
| 3250 | }
|
---|
| 3251 | if( n==31 )
|
---|
| 3252 | {
|
---|
| 3253 | ng = 8;
|
---|
| 3254 | wgauss[0] = 0.030753241996117268354628393577204;
|
---|
| 3255 | wgauss[1] = 0.070366047488108124709267416450667;
|
---|
| 3256 | wgauss[2] = 0.107159220467171935011869546685869;
|
---|
| 3257 | wgauss[3] = 0.139570677926154314447804794511028;
|
---|
| 3258 | wgauss[4] = 0.166269205816993933553200860481209;
|
---|
| 3259 | wgauss[5] = 0.186161000015562211026800561866423;
|
---|
| 3260 | wgauss[6] = 0.198431485327111576456118326443839;
|
---|
| 3261 | wgauss[7] = 0.202578241925561272880620199967519;
|
---|
| 3262 | x[0] = 0.998002298693397060285172840152271;
|
---|
| 3263 | x[1] = 0.987992518020485428489565718586613;
|
---|
| 3264 | x[2] = 0.967739075679139134257347978784337;
|
---|
| 3265 | x[3] = 0.937273392400705904307758947710209;
|
---|
| 3266 | x[4] = 0.897264532344081900882509656454496;
|
---|
| 3267 | x[5] = 0.848206583410427216200648320774217;
|
---|
| 3268 | x[6] = 0.790418501442465932967649294817947;
|
---|
| 3269 | x[7] = 0.724417731360170047416186054613938;
|
---|
| 3270 | x[8] = 0.650996741297416970533735895313275;
|
---|
| 3271 | x[9] = 0.570972172608538847537226737253911;
|
---|
| 3272 | x[10] = 0.485081863640239680693655740232351;
|
---|
| 3273 | x[11] = 0.394151347077563369897207370981045;
|
---|
| 3274 | x[12] = 0.299180007153168812166780024266389;
|
---|
| 3275 | x[13] = 0.201194093997434522300628303394596;
|
---|
| 3276 | x[14] = 0.101142066918717499027074231447392;
|
---|
| 3277 | x[15] = 0.000000000000000000000000000000000;
|
---|
| 3278 | wkronrod[0] = 0.005377479872923348987792051430128;
|
---|
| 3279 | wkronrod[1] = 0.015007947329316122538374763075807;
|
---|
| 3280 | wkronrod[2] = 0.025460847326715320186874001019653;
|
---|
| 3281 | wkronrod[3] = 0.035346360791375846222037948478360;
|
---|
| 3282 | wkronrod[4] = 0.044589751324764876608227299373280;
|
---|
| 3283 | wkronrod[5] = 0.053481524690928087265343147239430;
|
---|
| 3284 | wkronrod[6] = 0.062009567800670640285139230960803;
|
---|
| 3285 | wkronrod[7] = 0.069854121318728258709520077099147;
|
---|
| 3286 | wkronrod[8] = 0.076849680757720378894432777482659;
|
---|
| 3287 | wkronrod[9] = 0.083080502823133021038289247286104;
|
---|
| 3288 | wkronrod[10] = 0.088564443056211770647275443693774;
|
---|
| 3289 | wkronrod[11] = 0.093126598170825321225486872747346;
|
---|
| 3290 | wkronrod[12] = 0.096642726983623678505179907627589;
|
---|
| 3291 | wkronrod[13] = 0.099173598721791959332393173484603;
|
---|
| 3292 | wkronrod[14] = 0.100769845523875595044946662617570;
|
---|
| 3293 | wkronrod[15] = 0.101330007014791549017374792767493;
|
---|
| 3294 | }
|
---|
| 3295 | if( n==41 )
|
---|
| 3296 | {
|
---|
| 3297 | ng = 10;
|
---|
| 3298 | wgauss[0] = 0.017614007139152118311861962351853;
|
---|
| 3299 | wgauss[1] = 0.040601429800386941331039952274932;
|
---|
| 3300 | wgauss[2] = 0.062672048334109063569506535187042;
|
---|
| 3301 | wgauss[3] = 0.083276741576704748724758143222046;
|
---|
| 3302 | wgauss[4] = 0.101930119817240435036750135480350;
|
---|
| 3303 | wgauss[5] = 0.118194531961518417312377377711382;
|
---|
| 3304 | wgauss[6] = 0.131688638449176626898494499748163;
|
---|
| 3305 | wgauss[7] = 0.142096109318382051329298325067165;
|
---|
| 3306 | wgauss[8] = 0.149172986472603746787828737001969;
|
---|
| 3307 | wgauss[9] = 0.152753387130725850698084331955098;
|
---|
| 3308 | x[0] = 0.998859031588277663838315576545863;
|
---|
| 3309 | x[1] = 0.993128599185094924786122388471320;
|
---|
| 3310 | x[2] = 0.981507877450250259193342994720217;
|
---|
| 3311 | x[3] = 0.963971927277913791267666131197277;
|
---|
| 3312 | x[4] = 0.940822633831754753519982722212443;
|
---|
| 3313 | x[5] = 0.912234428251325905867752441203298;
|
---|
| 3314 | x[6] = 0.878276811252281976077442995113078;
|
---|
| 3315 | x[7] = 0.839116971822218823394529061701521;
|
---|
| 3316 | x[8] = 0.795041428837551198350638833272788;
|
---|
| 3317 | x[9] = 0.746331906460150792614305070355642;
|
---|
| 3318 | x[10] = 0.693237656334751384805490711845932;
|
---|
| 3319 | x[11] = 0.636053680726515025452836696226286;
|
---|
| 3320 | x[12] = 0.575140446819710315342946036586425;
|
---|
| 3321 | x[13] = 0.510867001950827098004364050955251;
|
---|
| 3322 | x[14] = 0.443593175238725103199992213492640;
|
---|
| 3323 | x[15] = 0.373706088715419560672548177024927;
|
---|
| 3324 | x[16] = 0.301627868114913004320555356858592;
|
---|
| 3325 | x[17] = 0.227785851141645078080496195368575;
|
---|
| 3326 | x[18] = 0.152605465240922675505220241022678;
|
---|
| 3327 | x[19] = 0.076526521133497333754640409398838;
|
---|
| 3328 | x[20] = 0.000000000000000000000000000000000;
|
---|
| 3329 | wkronrod[0] = 0.003073583718520531501218293246031;
|
---|
| 3330 | wkronrod[1] = 0.008600269855642942198661787950102;
|
---|
| 3331 | wkronrod[2] = 0.014626169256971252983787960308868;
|
---|
| 3332 | wkronrod[3] = 0.020388373461266523598010231432755;
|
---|
| 3333 | wkronrod[4] = 0.025882133604951158834505067096153;
|
---|
| 3334 | wkronrod[5] = 0.031287306777032798958543119323801;
|
---|
| 3335 | wkronrod[6] = 0.036600169758200798030557240707211;
|
---|
| 3336 | wkronrod[7] = 0.041668873327973686263788305936895;
|
---|
| 3337 | wkronrod[8] = 0.046434821867497674720231880926108;
|
---|
| 3338 | wkronrod[9] = 0.050944573923728691932707670050345;
|
---|
| 3339 | wkronrod[10] = 0.055195105348285994744832372419777;
|
---|
| 3340 | wkronrod[11] = 0.059111400880639572374967220648594;
|
---|
| 3341 | wkronrod[12] = 0.062653237554781168025870122174255;
|
---|
| 3342 | wkronrod[13] = 0.065834597133618422111563556969398;
|
---|
| 3343 | wkronrod[14] = 0.068648672928521619345623411885368;
|
---|
| 3344 | wkronrod[15] = 0.071054423553444068305790361723210;
|
---|
| 3345 | wkronrod[16] = 0.073030690332786667495189417658913;
|
---|
| 3346 | wkronrod[17] = 0.074582875400499188986581418362488;
|
---|
| 3347 | wkronrod[18] = 0.075704497684556674659542775376617;
|
---|
| 3348 | wkronrod[19] = 0.076377867672080736705502835038061;
|
---|
| 3349 | wkronrod[20] = 0.076600711917999656445049901530102;
|
---|
| 3350 | }
|
---|
| 3351 | if( n==51 )
|
---|
| 3352 | {
|
---|
| 3353 | ng = 13;
|
---|
| 3354 | wgauss[0] = 0.011393798501026287947902964113235;
|
---|
| 3355 | wgauss[1] = 0.026354986615032137261901815295299;
|
---|
| 3356 | wgauss[2] = 0.040939156701306312655623487711646;
|
---|
| 3357 | wgauss[3] = 0.054904695975835191925936891540473;
|
---|
| 3358 | wgauss[4] = 0.068038333812356917207187185656708;
|
---|
| 3359 | wgauss[5] = 0.080140700335001018013234959669111;
|
---|
| 3360 | wgauss[6] = 0.091028261982963649811497220702892;
|
---|
| 3361 | wgauss[7] = 0.100535949067050644202206890392686;
|
---|
| 3362 | wgauss[8] = 0.108519624474263653116093957050117;
|
---|
| 3363 | wgauss[9] = 0.114858259145711648339325545869556;
|
---|
| 3364 | wgauss[10] = 0.119455763535784772228178126512901;
|
---|
| 3365 | wgauss[11] = 0.122242442990310041688959518945852;
|
---|
| 3366 | wgauss[12] = 0.123176053726715451203902873079050;
|
---|
| 3367 | x[0] = 0.999262104992609834193457486540341;
|
---|
| 3368 | x[1] = 0.995556969790498097908784946893902;
|
---|
| 3369 | x[2] = 0.988035794534077247637331014577406;
|
---|
| 3370 | x[3] = 0.976663921459517511498315386479594;
|
---|
| 3371 | x[4] = 0.961614986425842512418130033660167;
|
---|
| 3372 | x[5] = 0.942974571228974339414011169658471;
|
---|
| 3373 | x[6] = 0.920747115281701561746346084546331;
|
---|
| 3374 | x[7] = 0.894991997878275368851042006782805;
|
---|
| 3375 | x[8] = 0.865847065293275595448996969588340;
|
---|
| 3376 | x[9] = 0.833442628760834001421021108693570;
|
---|
| 3377 | x[10] = 0.797873797998500059410410904994307;
|
---|
| 3378 | x[11] = 0.759259263037357630577282865204361;
|
---|
| 3379 | x[12] = 0.717766406813084388186654079773298;
|
---|
| 3380 | x[13] = 0.673566368473468364485120633247622;
|
---|
| 3381 | x[14] = 0.626810099010317412788122681624518;
|
---|
| 3382 | x[15] = 0.577662930241222967723689841612654;
|
---|
| 3383 | x[16] = 0.526325284334719182599623778158010;
|
---|
| 3384 | x[17] = 0.473002731445714960522182115009192;
|
---|
| 3385 | x[18] = 0.417885382193037748851814394594572;
|
---|
| 3386 | x[19] = 0.361172305809387837735821730127641;
|
---|
| 3387 | x[20] = 0.303089538931107830167478909980339;
|
---|
| 3388 | x[21] = 0.243866883720988432045190362797452;
|
---|
| 3389 | x[22] = 0.183718939421048892015969888759528;
|
---|
| 3390 | x[23] = 0.122864692610710396387359818808037;
|
---|
| 3391 | x[24] = 0.061544483005685078886546392366797;
|
---|
| 3392 | x[25] = 0.000000000000000000000000000000000;
|
---|
| 3393 | wkronrod[0] = 0.001987383892330315926507851882843;
|
---|
| 3394 | wkronrod[1] = 0.005561932135356713758040236901066;
|
---|
| 3395 | wkronrod[2] = 0.009473973386174151607207710523655;
|
---|
| 3396 | wkronrod[3] = 0.013236229195571674813656405846976;
|
---|
| 3397 | wkronrod[4] = 0.016847817709128298231516667536336;
|
---|
| 3398 | wkronrod[5] = 0.020435371145882835456568292235939;
|
---|
| 3399 | wkronrod[6] = 0.024009945606953216220092489164881;
|
---|
| 3400 | wkronrod[7] = 0.027475317587851737802948455517811;
|
---|
| 3401 | wkronrod[8] = 0.030792300167387488891109020215229;
|
---|
| 3402 | wkronrod[9] = 0.034002130274329337836748795229551;
|
---|
| 3403 | wkronrod[10] = 0.037116271483415543560330625367620;
|
---|
| 3404 | wkronrod[11] = 0.040083825504032382074839284467076;
|
---|
| 3405 | wkronrod[12] = 0.042872845020170049476895792439495;
|
---|
| 3406 | wkronrod[13] = 0.045502913049921788909870584752660;
|
---|
| 3407 | wkronrod[14] = 0.047982537138836713906392255756915;
|
---|
| 3408 | wkronrod[15] = 0.050277679080715671963325259433440;
|
---|
| 3409 | wkronrod[16] = 0.052362885806407475864366712137873;
|
---|
| 3410 | wkronrod[17] = 0.054251129888545490144543370459876;
|
---|
| 3411 | wkronrod[18] = 0.055950811220412317308240686382747;
|
---|
| 3412 | wkronrod[19] = 0.057437116361567832853582693939506;
|
---|
| 3413 | wkronrod[20] = 0.058689680022394207961974175856788;
|
---|
| 3414 | wkronrod[21] = 0.059720340324174059979099291932562;
|
---|
| 3415 | wkronrod[22] = 0.060539455376045862945360267517565;
|
---|
| 3416 | wkronrod[23] = 0.061128509717053048305859030416293;
|
---|
| 3417 | wkronrod[24] = 0.061471189871425316661544131965264;
|
---|
| 3418 | wkronrod[25] = 0.061580818067832935078759824240055;
|
---|
| 3419 | }
|
---|
| 3420 | if( n==61 )
|
---|
| 3421 | {
|
---|
| 3422 | ng = 15;
|
---|
| 3423 | wgauss[0] = 0.007968192496166605615465883474674;
|
---|
| 3424 | wgauss[1] = 0.018466468311090959142302131912047;
|
---|
| 3425 | wgauss[2] = 0.028784707883323369349719179611292;
|
---|
| 3426 | wgauss[3] = 0.038799192569627049596801936446348;
|
---|
| 3427 | wgauss[4] = 0.048402672830594052902938140422808;
|
---|
| 3428 | wgauss[5] = 0.057493156217619066481721689402056;
|
---|
| 3429 | wgauss[6] = 0.065974229882180495128128515115962;
|
---|
| 3430 | wgauss[7] = 0.073755974737705206268243850022191;
|
---|
| 3431 | wgauss[8] = 0.080755895229420215354694938460530;
|
---|
| 3432 | wgauss[9] = 0.086899787201082979802387530715126;
|
---|
| 3433 | wgauss[10] = 0.092122522237786128717632707087619;
|
---|
| 3434 | wgauss[11] = 0.096368737174644259639468626351810;
|
---|
| 3435 | wgauss[12] = 0.099593420586795267062780282103569;
|
---|
| 3436 | wgauss[13] = 0.101762389748405504596428952168554;
|
---|
| 3437 | wgauss[14] = 0.102852652893558840341285636705415;
|
---|
| 3438 | x[0] = 0.999484410050490637571325895705811;
|
---|
| 3439 | x[1] = 0.996893484074649540271630050918695;
|
---|
| 3440 | x[2] = 0.991630996870404594858628366109486;
|
---|
| 3441 | x[3] = 0.983668123279747209970032581605663;
|
---|
| 3442 | x[4] = 0.973116322501126268374693868423707;
|
---|
| 3443 | x[5] = 0.960021864968307512216871025581798;
|
---|
| 3444 | x[6] = 0.944374444748559979415831324037439;
|
---|
| 3445 | x[7] = 0.926200047429274325879324277080474;
|
---|
| 3446 | x[8] = 0.905573307699907798546522558925958;
|
---|
| 3447 | x[9] = 0.882560535792052681543116462530226;
|
---|
| 3448 | x[10] = 0.857205233546061098958658510658944;
|
---|
| 3449 | x[11] = 0.829565762382768397442898119732502;
|
---|
| 3450 | x[12] = 0.799727835821839083013668942322683;
|
---|
| 3451 | x[13] = 0.767777432104826194917977340974503;
|
---|
| 3452 | x[14] = 0.733790062453226804726171131369528;
|
---|
| 3453 | x[15] = 0.697850494793315796932292388026640;
|
---|
| 3454 | x[16] = 0.660061064126626961370053668149271;
|
---|
| 3455 | x[17] = 0.620526182989242861140477556431189;
|
---|
| 3456 | x[18] = 0.579345235826361691756024932172540;
|
---|
| 3457 | x[19] = 0.536624148142019899264169793311073;
|
---|
| 3458 | x[20] = 0.492480467861778574993693061207709;
|
---|
| 3459 | x[21] = 0.447033769538089176780609900322854;
|
---|
| 3460 | x[22] = 0.400401254830394392535476211542661;
|
---|
| 3461 | x[23] = 0.352704725530878113471037207089374;
|
---|
| 3462 | x[24] = 0.304073202273625077372677107199257;
|
---|
| 3463 | x[25] = 0.254636926167889846439805129817805;
|
---|
| 3464 | x[26] = 0.204525116682309891438957671002025;
|
---|
| 3465 | x[27] = 0.153869913608583546963794672743256;
|
---|
| 3466 | x[28] = 0.102806937966737030147096751318001;
|
---|
| 3467 | x[29] = 0.051471842555317695833025213166723;
|
---|
| 3468 | x[30] = 0.000000000000000000000000000000000;
|
---|
| 3469 | wkronrod[0] = 0.001389013698677007624551591226760;
|
---|
| 3470 | wkronrod[1] = 0.003890461127099884051267201844516;
|
---|
| 3471 | wkronrod[2] = 0.006630703915931292173319826369750;
|
---|
| 3472 | wkronrod[3] = 0.009273279659517763428441146892024;
|
---|
| 3473 | wkronrod[4] = 0.011823015253496341742232898853251;
|
---|
| 3474 | wkronrod[5] = 0.014369729507045804812451432443580;
|
---|
| 3475 | wkronrod[6] = 0.016920889189053272627572289420322;
|
---|
| 3476 | wkronrod[7] = 0.019414141193942381173408951050128;
|
---|
| 3477 | wkronrod[8] = 0.021828035821609192297167485738339;
|
---|
| 3478 | wkronrod[9] = 0.024191162078080601365686370725232;
|
---|
| 3479 | wkronrod[10] = 0.026509954882333101610601709335075;
|
---|
| 3480 | wkronrod[11] = 0.028754048765041292843978785354334;
|
---|
| 3481 | wkronrod[12] = 0.030907257562387762472884252943092;
|
---|
| 3482 | wkronrod[13] = 0.032981447057483726031814191016854;
|
---|
| 3483 | wkronrod[14] = 0.034979338028060024137499670731468;
|
---|
| 3484 | wkronrod[15] = 0.036882364651821229223911065617136;
|
---|
| 3485 | wkronrod[16] = 0.038678945624727592950348651532281;
|
---|
| 3486 | wkronrod[17] = 0.040374538951535959111995279752468;
|
---|
| 3487 | wkronrod[18] = 0.041969810215164246147147541285970;
|
---|
| 3488 | wkronrod[19] = 0.043452539701356069316831728117073;
|
---|
| 3489 | wkronrod[20] = 0.044814800133162663192355551616723;
|
---|
| 3490 | wkronrod[21] = 0.046059238271006988116271735559374;
|
---|
| 3491 | wkronrod[22] = 0.047185546569299153945261478181099;
|
---|
| 3492 | wkronrod[23] = 0.048185861757087129140779492298305;
|
---|
| 3493 | wkronrod[24] = 0.049055434555029778887528165367238;
|
---|
| 3494 | wkronrod[25] = 0.049795683427074206357811569379942;
|
---|
| 3495 | wkronrod[26] = 0.050405921402782346840893085653585;
|
---|
| 3496 | wkronrod[27] = 0.050881795898749606492297473049805;
|
---|
| 3497 | wkronrod[28] = 0.051221547849258772170656282604944;
|
---|
| 3498 | wkronrod[29] = 0.051426128537459025933862879215781;
|
---|
| 3499 | wkronrod[30] = 0.051494729429451567558340433647099;
|
---|
| 3500 | }
|
---|
| 3501 |
|
---|
| 3502 | //
|
---|
| 3503 | // copy nodes
|
---|
| 3504 | //
|
---|
| 3505 | for(i=n-1; i>=n/2; i--)
|
---|
| 3506 | {
|
---|
| 3507 | x[i] = -x[n-1-i];
|
---|
| 3508 | }
|
---|
| 3509 |
|
---|
| 3510 | //
|
---|
| 3511 | // copy Kronrod weights
|
---|
| 3512 | //
|
---|
| 3513 | for(i=n-1; i>=n/2; i--)
|
---|
| 3514 | {
|
---|
| 3515 | wkronrod[i] = wkronrod[n-1-i];
|
---|
| 3516 | }
|
---|
| 3517 |
|
---|
| 3518 | //
|
---|
| 3519 | // copy Gauss weights
|
---|
| 3520 | //
|
---|
| 3521 | for(i=ng-1; i>=0; i--)
|
---|
| 3522 | {
|
---|
| 3523 | wgauss[n-2-2*i] = wgauss[i];
|
---|
| 3524 | wgauss[1+2*i] = wgauss[i];
|
---|
| 3525 | }
|
---|
| 3526 | for(i=0; i<=n/2; i++)
|
---|
| 3527 | {
|
---|
| 3528 | wgauss[2*i] = 0;
|
---|
| 3529 | }
|
---|
| 3530 |
|
---|
| 3531 | //
|
---|
| 3532 | // reorder
|
---|
| 3533 | //
|
---|
| 3534 | tsort.tagsort(ref x, n, ref p1, ref p2);
|
---|
| 3535 | for(i=0; i<=n-1; i++)
|
---|
| 3536 | {
|
---|
| 3537 | tmp = wkronrod[i];
|
---|
| 3538 | wkronrod[i] = wkronrod[p2[i]];
|
---|
| 3539 | wkronrod[p2[i]] = tmp;
|
---|
| 3540 | tmp = wgauss[i];
|
---|
| 3541 | wgauss[i] = wgauss[p2[i]];
|
---|
| 3542 | wgauss[p2[i]] = tmp;
|
---|
| 3543 | }
|
---|
| 3544 | }
|
---|
| 3545 |
|
---|
| 3546 |
|
---|
| 3547 | }
|
---|
| 3548 | }
|
---|
| 3549 |
|
---|