source: trunk/sources/ALGLIB/ssolve.cs @ 2575

/*************************************************************************
Copyright (c) 1992-2007 The University of Tennessee.  All rights reserved.

Contributors:
    * Sergey Bochkanov (ALGLIB project). Translation from FORTRAN to
      pseudocode.

See subroutines comments for additional copyrights.

>>> SOURCE LICENSE >>>
This program is free software; you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation (www.fsf.org); either version 2 of the 
License, or (at your option) any later version.

This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
GNU General Public License for more details.

A copy of the GNU General Public License is available at
http://www.fsf.org/licensing/licenses

>>> END OF LICENSE >>>
*************************************************************************/

using System;

namespace alglib
{
    public class ssolve
    {
        /*************************************************************************
        Solving  a system  of linear equations  with a system matrix  given by its
        LDLT decomposition

        The algorithm solves systems with a square matrix only.

        Input parameters:
            A       -   LDLT decomposition of the matrix (the result of the
                        SMatrixLDLT subroutine).
            Pivots  -   row permutation table (the result of the SMatrixLDLT subroutine).
            B       -   right side of a system.
                        Array whose index ranges within [0..N-1].
            N       -   size of matrix A.
            IsUpper -   points to the triangle of matrix A in which the LDLT
                        decomposition is stored.
                        If IsUpper=True, the decomposition has the form of U*D*U',
                        matrix U is stored in the upper triangle of  matrix A  (in
                        that case, the lower triangle isn't used and isn't changed
                        by the subroutine).
                        Similarly, if IsUpper=False, the decomposition has the form
                        of L*D*L' and the lower triangle stores matrix L.

        Output parameters:
            X       -   solution of a system.
                        Array whose index ranges within [0..N-1].

        Result:
            True, if the matrix is not singular. X contains the solution.
            False, if the matrix is singular (the determinant of matrix D is equal
        to 0). In this case, X doesn't contain a solution.
        *************************************************************************/
        public static bool smatrixldltsolve(ref double[,] a,
            ref int[] pivots,
            double[] b,
            int n,
            bool isupper,
            ref double[] x)
        {
            bool result = new bool();
            int i = 0;
            int j = 0;
            int k = 0;
            int kp = 0;
            double ak = 0;
            double akm1 = 0;
            double akm1k = 0;
            double bk = 0;
            double bkm1 = 0;
            double denom = 0;
            double v = 0;
            int i_ = 0;

            b = (double[])b.Clone();

            
            //
            // Quick return if possible
            //
            result = true;
            if( n==0 )
            {
                return result;
            }
            
            //
            // Check that the diagonal matrix D is nonsingular
            //
            if( isupper )
            {
                
                //
                // Upper triangular storage: examine D from bottom to top
                //
                for(i=n-1; i>=0; i--)
                {
                    if( pivots[i]>=0 & (double)(a[i,i])==(double)(0) )
                    {
                        result = false;
                        return result;
                    }
                }
            }
            else
            {
                
                //
                // Lower triangular storage: examine D from top to bottom.
                //
                for(i=0; i<=n-1; i++)
                {
                    if( pivots[i]>=0 & (double)(a[i,i])==(double)(0) )
                    {
                        result = false;
                        return result;
                    }
                }
            }
            
            //
            // Solve Ax = b
            //
            if( isupper )
            {
                
                //
                // Solve A*X = B, where A = U*D*U'.
                //
                // First solve U*D*X = B, overwriting B with X.
                //
                // K+1 is the main loop index, decreasing from N to 1 in steps of
                // 1 or 2, depending on the size of the diagonal blocks.
                //
                k = n-1;
                while( k>=0 )
                {
                    if( pivots[k]>=0 )
                    {
                        
                        //
                        // 1 x 1 diagonal block
                        //
                        // Interchange rows K+1 and IPIV(K+1).
                        //
                        kp = pivots[k];
                        if( kp!=k )
                        {
                            v = b[k];
                            b[k] = b[kp];
                            b[kp] = v;
                        }
                        
                        //
                        // Multiply by inv(U(K+1)), where U(K+1) is the transformation
                        // stored in column K+1 of A.
                        //
                        v = b[k];
                        for(i_=0; i_<=k-1;i_++)
                        {
                            b[i_] = b[i_] - v*a[i_,k];
                        }
                        
                        //
                        // Multiply by the inverse of the diagonal block.
                        //
                        b[k] = b[k]/a[k,k];
                        k = k-1;
                    }
                    else
                    {
                        
                        //
                        // 2 x 2 diagonal block
                        //
                        // Interchange rows K+1-1 and -IPIV(K+1).
                        //
                        kp = pivots[k]+n;
                        if( kp!=k-1 )
                        {
                            v = b[k-1];
                            b[k-1] = b[kp];
                            b[kp] = v;
                        }
                        
                        //
                        // Multiply by inv(U(K+1)), where U(K+1) is the transformation
                        // stored in columns K+1-1 and K+1 of A.
                        //
                        v = b[k];
                        for(i_=0; i_<=k-2;i_++)
                        {
                            b[i_] = b[i_] - v*a[i_,k];
                        }
                        v = b[k-1];
                        for(i_=0; i_<=k-2;i_++)
                        {
                            b[i_] = b[i_] - v*a[i_,k-1];
                        }
                        
                        //
                        // Multiply by the inverse of the diagonal block.
                        //
                        akm1k = a[k-1,k];
                        akm1 = a[k-1,k-1]/akm1k;
                        ak = a[k,k]/akm1k;
                        denom = akm1*ak-1;
                        bkm1 = b[k-1]/akm1k;
                        bk = b[k]/akm1k;
                        b[k-1] = (ak*bkm1-bk)/denom;
                        b[k] = (akm1*bk-bkm1)/denom;
                        k = k-2;
                    }
                }
                
                //
                // Next solve U'*X = B, overwriting B with X.
                //
                // K+1 is the main loop index, increasing from 1 to N in steps of
                // 1 or 2, depending on the size of the diagonal blocks.
                //
                k = 0;
                while( k<=n-1 )
                {
                    if( pivots[k]>=0 )
                    {
                        
                        //
                        // 1 x 1 diagonal block
                        //
                        // Multiply by inv(U'(K+1)), where U(K+1) is the transformation
                        // stored in column K+1 of A.
                        //
                        v = 0.0;
                        for(i_=0; i_<=k-1;i_++)
                        {
                            v += b[i_]*a[i_,k];
                        }
                        b[k] = b[k]-v;
                        
                        //
                        // Interchange rows K+1 and IPIV(K+1).
                        //
                        kp = pivots[k];
                        if( kp!=k )
                        {
                            v = b[k];
                            b[k] = b[kp];
                            b[kp] = v;
                        }
                        k = k+1;
                    }
                    else
                    {
                        
                        //
                        // 2 x 2 diagonal block
                        //
                        // Multiply by inv(U'(K+1+1)), where U(K+1+1) is the transformation
                        // stored in columns K+1 and K+1+1 of A.
                        //
                        v = 0.0;
                        for(i_=0; i_<=k-1;i_++)
                        {
                            v += b[i_]*a[i_,k];
                        }
                        b[k] = b[k]-v;
                        v = 0.0;
                        for(i_=0; i_<=k-1;i_++)
                        {
                            v += b[i_]*a[i_,k+1];
                        }
                        b[k+1] = b[k+1]-v;
                        
                        //
                        // Interchange rows K+1 and -IPIV(K+1).
                        //
                        kp = pivots[k]+n;
                        if( kp!=k )
                        {
                            v = b[k];
                            b[k] = b[kp];
                            b[kp] = v;
                        }
                        k = k+2;
                    }
                }
            }
            else
            {
                
                //
                // Solve A*X = B, where A = L*D*L'.
                //
                // First solve L*D*X = B, overwriting B with X.
                //
                // K+1 is the main loop index, increasing from 1 to N in steps of
                // 1 or 2, depending on the size of the diagonal blocks.
                //
                k = 0;
                while( k<=n-1 )
                {
                    if( pivots[k]>=0 )
                    {
                        
                        //
                        // 1 x 1 diagonal block
                        //
                        // Interchange rows K+1 and IPIV(K+1).
                        //
                        kp = pivots[k];
                        if( kp!=k )
                        {
                            v = b[k];
                            b[k] = b[kp];
                            b[kp] = v;
                        }
                        
                        //
                        // Multiply by inv(L(K+1)), where L(K+1) is the transformation
                        // stored in column K+1 of A.
                        //
                        if( k+1<n )
                        {
                            v = b[k];
                            for(i_=k+1; i_<=n-1;i_++)
                            {
                                b[i_] = b[i_] - v*a[i_,k];
                            }
                        }
                        
                        //
                        // Multiply by the inverse of the diagonal block.
                        //
                        b[k] = b[k]/a[k,k];
                        k = k+1;
                    }
                    else
                    {
                        
                        //
                        // 2 x 2 diagonal block
                        //
                        // Interchange rows K+1+1 and -IPIV(K+1).
                        //
                        kp = pivots[k]+n;
                        if( kp!=k+1 )
                        {
                            v = b[k+1];
                            b[k+1] = b[kp];
                            b[kp] = v;
                        }
                        
                        //
                        // Multiply by inv(L(K+1)), where L(K+1) is the transformation
                        // stored in columns K+1 and K+1+1 of A.
                        //
                        if( k+1<n-1 )
                        {
                            v = b[k];
                            for(i_=k+2; i_<=n-1;i_++)
                            {
                                b[i_] = b[i_] - v*a[i_,k];
                            }
                            v = b[k+1];
                            for(i_=k+2; i_<=n-1;i_++)
                            {
                                b[i_] = b[i_] - v*a[i_,k+1];
                            }
                        }
                        
                        //
                        // Multiply by the inverse of the diagonal block.
                        //
                        akm1k = a[k+1,k];
                        akm1 = a[k,k]/akm1k;
                        ak = a[k+1,k+1]/akm1k;
                        denom = akm1*ak-1;
                        bkm1 = b[k]/akm1k;
                        bk = b[k+1]/akm1k;
                        b[k] = (ak*bkm1-bk)/denom;
                        b[k+1] = (akm1*bk-bkm1)/denom;
                        k = k+2;
                    }
                }
                
                //
                // Next solve L'*X = B, overwriting B with X.
                //
                // K+1 is the main loop index, decreasing from N to 1 in steps of
                // 1 or 2, depending on the size of the diagonal blocks.
                //
                k = n-1;
                while( k>=0 )
                {
                    if( pivots[k]>=0 )
                    {
                        
                        //
                        // 1 x 1 diagonal block
                        //
                        // Multiply by inv(L'(K+1)), where L(K+1) is the transformation
                        // stored in column K+1 of A.
                        //
                        if( k+1<n )
                        {
                            v = 0.0;
                            for(i_=k+1; i_<=n-1;i_++)
                            {
                                v += b[i_]*a[i_,k];
                            }
                            b[k] = b[k]-v;
                        }
                        
                        //
                        // Interchange rows K+1 and IPIV(K+1).
                        //
                        kp = pivots[k];
                        if( kp!=k )
                        {
                            v = b[k];
                            b[k] = b[kp];
                            b[kp] = v;
                        }
                        k = k-1;
                    }
                    else
                    {
                        
                        //
                        // 2 x 2 diagonal block
                        //
                        // Multiply by inv(L'(K+1-1)), where L(K+1-1) is the transformation
                        // stored in columns K+1-1 and K+1 of A.
                        //
                        if( k+1<n )
                        {
                            v = 0.0;
                            for(i_=k+1; i_<=n-1;i_++)
                            {
                                v += b[i_]*a[i_,k];
                            }
                            b[k] = b[k]-v;
                            v = 0.0;
                            for(i_=k+1; i_<=n-1;i_++)
                            {
                                v += b[i_]*a[i_,k-1];
                            }
                            b[k-1] = b[k-1]-v;
                        }
                        
                        //
                        // Interchange rows K+1 and -IPIV(K+1).
                        //
                        kp = pivots[k]+n;
                        if( kp!=k )
                        {
                            v = b[k];
                            b[k] = b[kp];
                            b[kp] = v;
                        }
                        k = k-2;
                    }
                }
            }
            x = new double[n-1+1];
            for(i_=0; i_<=n-1;i_++)
            {
                x[i_] = b[i_];
            }
            return result;
        }


        /*************************************************************************
        Solving a system of linear equations with a symmetric system matrix

        Input parameters:
            A       -   system matrix (upper or lower triangle).
                        Array whose indexes range within [0..N-1, 0..N-1].
            B       -   right side of a system.
                        Array whose index ranges within [0..N-1].
            N       -   size of matrix A.
            IsUpper -   If IsUpper = True, A contains the upper triangle,
                        otherwise A contains the lower triangle.

        Output parameters:
            X       -   solution of a system.
                        Array whose index ranges within [0..N-1].

        Result:
            True, if the matrix is not singular. X contains the solution.
            False, if the matrix is singular (the determinant of the matrix is equal
        to 0). In this case, X doesn't contain a solution.

          -- ALGLIB --
             Copyright 2005 by Bochkanov Sergey
        *************************************************************************/
        public static bool smatrixsolve(double[,] a,
            ref double[] b,
            int n,
            bool isupper,
            ref double[] x)
        {
            bool result = new bool();
            int[] pivots = new int[0];

            a = (double[,])a.Clone();

            ldlt.smatrixldlt(ref a, n, isupper, ref pivots);
            result = smatrixldltsolve(ref a, ref pivots, b, n, isupper, ref x);
            return result;
        }


        public static bool solvesystemldlt(ref double[,] a,
            ref int[] pivots,
            double[] b,
            int n,
            bool isupper,
            ref double[] x)
        {
            bool result = new bool();
            int i = 0;
            int j = 0;
            int k = 0;
            int kp = 0;
            int km1 = 0;
            int km2 = 0;
            int kp1 = 0;
            int kp2 = 0;
            double ak = 0;
            double akm1 = 0;
            double akm1k = 0;
            double bk = 0;
            double bkm1 = 0;
            double denom = 0;
            double v = 0;
            int i_ = 0;

            b = (double[])b.Clone();

            
            //
            // Quick return if possible
            //
            result = true;
            if( n==0 )
            {
                return result;
            }
            
            //
            // Check that the diagonal matrix D is nonsingular
            //
            if( isupper )
            {
                
                //
                // Upper triangular storage: examine D from bottom to top
                //
                for(i=n; i>=1; i--)
                {
                    if( pivots[i]>0 & (double)(a[i,i])==(double)(0) )
                    {
                        result = false;
                        return result;
                    }
                }
            }
            else
            {
                
                //
                // Lower triangular storage: examine D from top to bottom.
                //
                for(i=1; i<=n; i++)
                {
                    if( pivots[i]>0 & (double)(a[i,i])==(double)(0) )
                    {
                        result = false;
                        return result;
                    }
                }
            }
            
            //
            // Solve Ax = b
            //
            if( isupper )
            {
                
                //
                // Solve A*X = B, where A = U*D*U'.
                //
                // First solve U*D*X = B, overwriting B with X.
                //
                // K is the main loop index, decreasing from N to 1 in steps of
                // 1 or 2, depending on the size of the diagonal blocks.
                //
                k = n;
                while( k>=1 )
                {
                    if( pivots[k]>0 )
                    {
                        
                        //
                        // 1 x 1 diagonal block
                        //
                        // Interchange rows K and IPIV(K).
                        //
                        kp = pivots[k];
                        if( kp!=k )
                        {
                            v = b[k];
                            b[k] = b[kp];
                            b[kp] = v;
                        }
                        
                        //
                        // Multiply by inv(U(K)), where U(K) is the transformation
                        // stored in column K of A.
                        //
                        km1 = k-1;
                        v = b[k];
                        for(i_=1; i_<=km1;i_++)
                        {
                            b[i_] = b[i_] - v*a[i_,k];
                        }
                        
                        //
                        // Multiply by the inverse of the diagonal block.
                        //
                        b[k] = b[k]/a[k,k];
                        k = k-1;
                    }
                    else
                    {
                        
                        //
                        // 2 x 2 diagonal block
                        //
                        // Interchange rows K-1 and -IPIV(K).
                        //
                        kp = -pivots[k];
                        if( kp!=k-1 )
                        {
                            v = b[k-1];
                            b[k-1] = b[kp];
                            b[kp] = v;
                        }
                        
                        //
                        // Multiply by inv(U(K)), where U(K) is the transformation
                        // stored in columns K-1 and K of A.
                        //
                        km2 = k-2;
                        km1 = k-1;
                        v = b[k];
                        for(i_=1; i_<=km2;i_++)
                        {
                            b[i_] = b[i_] - v*a[i_,k];
                        }
                        v = b[k-1];
                        for(i_=1; i_<=km2;i_++)
                        {
                            b[i_] = b[i_] - v*a[i_,km1];
                        }
                        
                        //
                        // Multiply by the inverse of the diagonal block.
                        //
                        akm1k = a[k-1,k];
                        akm1 = a[k-1,k-1]/akm1k;
                        ak = a[k,k]/akm1k;
                        denom = akm1*ak-1;
                        bkm1 = b[k-1]/akm1k;
                        bk = b[k]/akm1k;
                        b[k-1] = (ak*bkm1-bk)/denom;
                        b[k] = (akm1*bk-bkm1)/denom;
                        k = k-2;
                    }
                }
                
                //
                // Next solve U'*X = B, overwriting B with X.
                //
                // K is the main loop index, increasing from 1 to N in steps of
                // 1 or 2, depending on the size of the diagonal blocks.
                //
                k = 1;
                while( k<=n )
                {
                    if( pivots[k]>0 )
                    {
                        
                        //
                        // 1 x 1 diagonal block
                        //
                        // Multiply by inv(U'(K)), where U(K) is the transformation
                        // stored in column K of A.
                        //
                        km1 = k-1;
                        v = 0.0;
                        for(i_=1; i_<=km1;i_++)
                        {
                            v += b[i_]*a[i_,k];
                        }
                        b[k] = b[k]-v;
                        
                        //
                        // Interchange rows K and IPIV(K).
                        //
                        kp = pivots[k];
                        if( kp!=k )
                        {
                            v = b[k];
                            b[k] = b[kp];
                            b[kp] = v;
                        }
                        k = k+1;
                    }
                    else
                    {
                        
                        //
                        // 2 x 2 diagonal block
                        //
                        // Multiply by inv(U'(K+1)), where U(K+1) is the transformation
                        // stored in columns K and K+1 of A.
                        //
                        km1 = k-1;
                        kp1 = k+1;
                        v = 0.0;
                        for(i_=1; i_<=km1;i_++)
                        {
                            v += b[i_]*a[i_,k];
                        }
                        b[k] = b[k]-v;
                        v = 0.0;
                        for(i_=1; i_<=km1;i_++)
                        {
                            v += b[i_]*a[i_,kp1];
                        }
                        b[k+1] = b[k+1]-v;
                        
                        //
                        // Interchange rows K and -IPIV(K).
                        //
                        kp = -pivots[k];
                        if( kp!=k )
                        {
                            v = b[k];
                            b[k] = b[kp];
                            b[kp] = v;
                        }
                        k = k+2;
                    }
                }
            }
            else
            {
                
                //
                // Solve A*X = B, where A = L*D*L'.
                //
                // First solve L*D*X = B, overwriting B with X.
                //
                // K is the main loop index, increasing from 1 to N in steps of
                // 1 or 2, depending on the size of the diagonal blocks.
                //
                k = 1;
                while( k<=n )
                {
                    if( pivots[k]>0 )
                    {
                        
                        //
                        // 1 x 1 diagonal block
                        //
                        // Interchange rows K and IPIV(K).
                        //
                        kp = pivots[k];
                        if( kp!=k )
                        {
                            v = b[k];
                            b[k] = b[kp];
                            b[kp] = v;
                        }
                        
                        //
                        // Multiply by inv(L(K)), where L(K) is the transformation
                        // stored in column K of A.
                        //
                        if( k<n )
                        {
                            kp1 = k+1;
                            v = b[k];
                            for(i_=kp1; i_<=n;i_++)
                            {
                                b[i_] = b[i_] - v*a[i_,k];
                            }
                        }
                        
                        //
                        // Multiply by the inverse of the diagonal block.
                        //
                        b[k] = b[k]/a[k,k];
                        k = k+1;
                    }
                    else
                    {
                        
                        //
                        // 2 x 2 diagonal block
                        //
                        // Interchange rows K+1 and -IPIV(K).
                        //
                        kp = -pivots[k];
                        if( kp!=k+1 )
                        {
                            v = b[k+1];
                            b[k+1] = b[kp];
                            b[kp] = v;
                        }
                        
                        //
                        // Multiply by inv(L(K)), where L(K) is the transformation
                        // stored in columns K and K+1 of A.
                        //
                        if( k<n-1 )
                        {
                            kp1 = k+1;
                            kp2 = k+2;
                            v = b[k];
                            for(i_=kp2; i_<=n;i_++)
                            {
                                b[i_] = b[i_] - v*a[i_,k];
                            }
                            v = b[k+1];
                            for(i_=kp2; i_<=n;i_++)
                            {
                                b[i_] = b[i_] - v*a[i_,kp1];
                            }
                        }
                        
                        //
                        // Multiply by the inverse of the diagonal block.
                        //
                        akm1k = a[k+1,k];
                        akm1 = a[k,k]/akm1k;
                        ak = a[k+1,k+1]/akm1k;
                        denom = akm1*ak-1;
                        bkm1 = b[k]/akm1k;
                        bk = b[k+1]/akm1k;
                        b[k] = (ak*bkm1-bk)/denom;
                        b[k+1] = (akm1*bk-bkm1)/denom;
                        k = k+2;
                    }
                }
                
                //
                // Next solve L'*X = B, overwriting B with X.
                //
                // K is the main loop index, decreasing from N to 1 in steps of
                // 1 or 2, depending on the size of the diagonal blocks.
                //
                k = n;
                while( k>=1 )
                {
                    if( pivots[k]>0 )
                    {
                        
                        //
                        // 1 x 1 diagonal block
                        //
                        // Multiply by inv(L'(K)), where L(K) is the transformation
                        // stored in column K of A.
                        //
                        if( k<n )
                        {
                            kp1 = k+1;
                            v = 0.0;
                            for(i_=kp1; i_<=n;i_++)
                            {
                                v += b[i_]*a[i_,k];
                            }
                            b[k] = b[k]-v;
                        }
                        
                        //
                        // Interchange rows K and IPIV(K).
                        //
                        kp = pivots[k];
                        if( kp!=k )
                        {
                            v = b[k];
                            b[k] = b[kp];
                            b[kp] = v;
                        }
                        k = k-1;
                    }
                    else
                    {
                        
                        //
                        // 2 x 2 diagonal block
                        //
                        // Multiply by inv(L'(K-1)), where L(K-1) is the transformation
                        // stored in columns K-1 and K of A.
                        //
                        if( k<n )
                        {
                            kp1 = k+1;
                            km1 = k-1;
                            v = 0.0;
                            for(i_=kp1; i_<=n;i_++)
                            {
                                v += b[i_]*a[i_,k];
                            }
                            b[k] = b[k]-v;
                            v = 0.0;
                            for(i_=kp1; i_<=n;i_++)
                            {
                                v += b[i_]*a[i_,km1];
                            }
                            b[k-1] = b[k-1]-v;
                        }
                        
                        //
                        // Interchange rows K and -IPIV(K).
                        //
                        kp = -pivots[k];
                        if( kp!=k )
                        {
                            v = b[k];
                            b[k] = b[kp];
                            b[kp] = v;
                        }
                        k = k-2;
                    }
                }
            }
            x = new double[n+1];
            for(i_=1; i_<=n;i_++)
            {
                x[i_] = b[i_];
            }
            return result;
        }


        public static bool solvesymmetricsystem(double[,] a,
            double[] b,
            int n,
            bool isupper,
            ref double[] x)
        {
            bool result = new bool();
            int[] pivots = new int[0];

            a = (double[,])a.Clone();
            b = (double[])b.Clone();

            ldlt.ldltdecomposition(ref a, n, isupper, ref pivots);
            result = solvesystemldlt(ref a, ref pivots, b, n, isupper, ref x);
            return result;
        }
    }
}