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source: branches/HeuristicLab.Problems.GaussianProcessTuning/ILNumerics.2.14.4735.573/Misc/ILComplex.cs @ 10355

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Last change on this file since 10355 was 9102, checked in by gkronber, 12 years ago
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1	///
2	/// This file is part of ILNumerics Community Edition.
3	///
4	/// ILNumerics Community Edition - high performance computing for applications.
5	/// Copyright (C) 2006 - 2012 Haymo Kutschbach, http://ilnumerics.net
6	///
7	/// ILNumerics Community Edition is free software: you can redistribute it and/or modify
8	/// it under the terms of the GNU General Public License version 3 as published by
9	/// the Free Software Foundation.
10	///
11	/// ILNumerics Community Edition is distributed in the hope that it will be useful,
12	/// but WITHOUT ANY WARRANTY; without even the implied warranty of
13	/// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
14	/// GNU General Public License for more details.
15	///
16	/// You should have received a copy of the GNU General Public License
17	/// along with ILNumerics Community Edition. See the file License.txt in the root
18	/// of your distribution package. If not, see <http://www.gnu.org/licenses/>.
19	///
20	/// In addition this software uses the following components and/or licenses:
21	///
22	/// =================================================================================
23	/// The Open Toolkit Library License
24	///
25	/// Copyright (c) 2006 - 2009 the Open Toolkit library.
26	///
27	/// Permission is hereby granted, free of charge, to any person obtaining a copy
28	/// of this software and associated documentation files (the "Software"), to deal
29	/// in the Software without restriction, including without limitation the rights to
30	/// use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of
31	/// the Software, and to permit persons to whom the Software is furnished to do
32	/// so, subject to the following conditions:
33	///
34	/// The above copyright notice and this permission notice shall be included in all
35	/// copies or substantial portions of the Software.
36	///
37	/// =================================================================================
38	///
39
40	#pragma warning disable 162
41	using System;
42	using System.Collections.Generic;
43	using System.Text;
44	using System.Runtime.InteropServices;
45
46
47
48	namespace ILNumerics {
49	/// <summary>
50	/// Floating point complex value data type of double precision
51	/// </summary>
52	/// <remarks>This class extends the system value types for real numbers to complex double
53	/// values. Besides the publicly available members 'real' and 'imag' it provides all the
54	/// basis functionality the floating point system.double brings (abs, log, sqrt, tan etc.),
55	/// as well as it overrides the basic unary and binary operators for all common system value
56	/// types including rarely used types (e.g. UInt16). This includes the basic numerical operations
57	/// like '+','-','/','*' and the relational operators: '==','>','>=' etc. Also there are some
58	/// explicit and some implicit casting operators from / to complex value into system
59	/// value types.</remarks>
60	[Serializable]
61	[StructLayout(LayoutKind.Sequential)]
62	public struct complex : IEquatable<complex> {
63	/// <summary>
64	/// Real part of this complex number
65	/// </summary>
66	public double real;
67	/// <summary>
68	/// Imaginary part of this complex number
69	/// </summary>
70	public double imag;
71	/// <summary>
72	/// Imaginary unit
73	/// </summary>
74	public static readonly complex i = new complex(0.0f,1.0f);
75
76	/// <summary>
77	/// Constructor creating a new complex value
78	/// </summary>
79	/// <param name="real">Real part</param>
80	/// <param name="imag">Imaginary part</param>
81	public complex(double real, double imag) {
82	this.real = real;
83	this.imag = imag;
84	}
85	/// <summary>
86	/// Complex conjugate
87	/// </summary>
88	public complex conj {
89	get{
90	return new complex(real,-imag);
91	}
92	}
93
94	/// <summary>
95	/// Positive infinity for real and imag part of complex value
96	/// </summary>
97	public static complex INF {
98	get {
99	return new complex(
100	double.PositiveInfinity,
101	double.PositiveInfinity
102	);
103	}
104	}
105
106	/// <summary>
107	/// New complex, real and imaginary parts are zero
108	/// </summary>
109	public static complex Zero {
110	get {
111	return new complex(0.0,0.0);
112	}
113	}
114
115	/// <summary>
116	/// Complex quantity, marked as being "not a number"
117	/// </summary>
118	public static complex NaN {
119	get {
120	return new complex(double.NaN,double.NaN);
121	}
122	}
123
124	/// <summary>
125	/// Are obj's real and imaginary part identical to the real and imaginary parts of this fcomplex
126	/// </summary>
127	/// <param name="obj">fcomplex object to determine the equality for</param>
128	/// <returns>true if obj is of fcomplex type and its real and imag part has the same
129	/// values as the real and imaginary part of this array.</returns>
130	public override bool Equals(object obj) {
131	if (obj is complex && ((complex)obj) == this)
132	return true;
133	return false;
134	}
135	/// <summary>
136	/// Check if a complex number equals this complex number
137	/// </summary>
138	/// <param name="other">other complex number</param>
139	/// <returns>true if both, real and imaginary parts of both complex number are (binary) equal, false otherwise</returns>
140	public bool Equals(complex other) {
141	return real.Equals(other.real) && imag.Equals(other.imag);
142	}
143	/// <summary>
144	/// Hash code of this comples
145	/// </summary>
146	/// <returns>Hash code of this complex</returns>
147	public override int GetHashCode() {
148	return 77101 * real.GetHashCode() + imag.GetHashCode();
149	}
150
151
152
153	/// <summary>
154	/// Add two complex numbers
155	/// </summary>
156	/// <param name="A">First summand</param>
157	/// <param name="B">Second summand</param>
158	/// <returns>result</returns>
159	public static complex operator +( complex A, complex B) {
160	complex ret;
161	ret.real = (double) (A.real + B.real );
162	ret.imag = (double) (A.imag + B.imag );
163	return ret;
164	}
165	/// <summary>
166	/// Subtract two complex values
167	/// </summary>
168	/// <param name="A">Minuend</param>
169	/// <param name="B">Subtrahend</param>
170	/// <returns>result</returns>
171	public static complex operator -( complex A, complex B) {
172	complex ret;
173	ret.real = (double) (A.real - B.real );
174	ret.imag = (double) (A.imag - B.imag );
175	return ret;
176	}
177	/// <summary>
178	/// Multiply two complex values
179	/// </summary>
180	/// <param name="A">First factor</param>
181	/// <param name="B">Second factor</param>
182	/// <returns>result</returns>
183	public static complex operator *( complex A, complex B) {
184	complex ret;
185	ret.real = (double) ((A.real * B.real ) - (A.imag * B.imag ));
186	ret.imag = (double) ((A.real * B.imag ) + (A.imag * B.real ));
187	return ret;
188	}
189	/// <summary>
190	/// Divide two numbers
191	/// </summary>
192	/// <param name="A">Divident</param>
193	/// <param name="B">Divisor</param>
194	/// <returns>Result</returns>
195	/// <remarks><para>Unless the operator must handle special inputs (Inf or 0 values),
196	/// the algorithm described in [1] is used for division. This is considered to be
197	/// more robust against floating point overflow than the naive approach of simple
198	/// cartesian division.</para>
199	/// <para>References: [1]: Smith, R.L., Algorithm 116: Complex division. Commun.ACM 5,8 (1962),435 <br />
200	/// [2]: Stewart, G.W., A note on complex division, ACM trans.on math software, Vol.11, N.3 (1985)</para></remarks>
201	public static complex operator /( complex A, complex B) {
202	if (B.imag == 0) return A / B.real;
203	return A * (1 / B);
204	if (IsNaN(A) \|\| complex .IsNaN(B)) return NaN;
205	//if ( complex .IsInfinity(B)) return NaN;
206	//if (A.real == 0 && A.imag == 0) return ( complex )0;
207	complex ret;
208	if (B.real == 0) {
209	ret.imag = (double) -(A.real / B.imag);
210	ret.real = (double) (A.imag / B.imag);
211	return ret;
212	}
213	// this would be the naive approach. But it come with to little robustness against overflow
214	//double norm2 = B.real * B.real + B.imag * B.imag;
215	//if (norm2 == 0) return INF; // this may be removed, since division by 0 results in inf anyway ?
216	//ret.real = (double) (((A.real * B.real ) + (A.imag * B.imag )) / norm2);
217	//ret.imag = (double) (((A.imag * B.real ) - (A.real * B.imag )) / norm2);
218
219	// this algorithm is taken from [1]. The one described in [2] was not taken. Tests
220	// did not show any advantage when using double precision floating point arithmetic.
221	double tmp1, tmp2;
222	if (Math.Abs(B.real) >= Math.Abs(B.imag)) {
223	tmp1 = (double) (B.imag * (1/B.real));
224	tmp2 = (double) (B.real + B.imag*tmp1);
225	ret.real = (double) (A.real + A.imag*tmp1)/tmp2;
226	ret.imag = (double) (A.imag - A.real*tmp1)/tmp2;
227	} else {
228	tmp1 = (double) (B.real * (1/B.imag));
229	tmp2 = (double) (B.imag + B.real*tmp1);
230	ret.real = (double) (A.imag + A.real*tmp1)/tmp2;
231	ret.imag = - (double) (A.real - A.imag*tmp1)/tmp2;
232	}
233	return ret;
234	}
235	/// <summary>
236	/// Equality comparison for complex numbers
237	/// </summary>
238	/// <param name="A">Left side</param>
239	/// <param name="B">Right side</param>
240	/// <returns>true, if real and imaginary part are identical</returns>
241	public static bool operator ==( complex A, complex B) {
242	return (A.imag == B.imag ) && (A.real == B.real );
243	}
244	/// <summary>
245	/// Unequality comparison for complex numbers
246	/// </summary>
247	/// <param name="A">Left side</param>
248	/// <param name="B">Right side</param>
249	/// <returns>true if real and imaginary parts of A and B are not equal, false otherwise</returns>
250	public static bool operator !=( complex A, complex B) {
251	return (A.imag != B.imag ) \|\| (A.real != B.real );
252	}
253	/// <summary>
254	/// Greater than comparison for complex numbers
255	/// </summary>
256	/// <param name="A">Left side</param>
257	/// <param name="B">Right side</param>
258	/// <returns>true if real part of A is greater than real part of B, false otherwise</returns>
259	/// <remarks>Only the real parts are compared!</remarks>
260	public static bool operator > ( complex A, complex B) {
261	return (A.real > B.real );
262	}
263	/// <summary>
264	/// Lower than comparison for complex numbers
265	/// </summary>
266	/// <param name="A">Left side</param>
267	/// <param name="B">Right side</param>
268	/// <returns>true if real part of A is lower then real part of B, false otherwise</returns>
269	/// <remarks>Only the real parts are compared!</remarks>
270	public static bool operator < ( complex A, complex B) {
271	return (A.real < B.real );
272	}
273	/// <summary>
274	/// Greater than or equal to comparison for complex numbers
275	/// </summary>
276	/// <param name="A">Left side</param>
277	/// <param name="B">Right side</param>
278	/// <returns>true if real part of A is greater than real part of B, false otherwise</returns>
279	/// <remarks>Only the real parts are compared!</remarks>
280	public static bool operator >=( complex A, complex B) {
281	return (A.real >= B.real );
282	}
283	/// <summary>
284	/// Lower than or equal to comparison for complex numbers
285	/// </summary>
286	/// <param name="A">Left side</param>
287	/// <param name="B">Right side</param>
288	/// <returns>true if real part of A is lower then real part of B, false otherwise</returns>
289	/// <remarks>Only the real parts are compared!</remarks>
290	public static bool operator <=( complex A, complex B) {
291	return (A.real <= B.real );
292	}
293
294	#region HYCALPER AUTO GENERATED CODE
295
296
297	/// <summary>
298	/// Add two complex numbers
299	/// </summary>
300	/// <param name="A">First summand</param>
301	/// <param name="B">Second summand</param>
302	/// <returns>result</returns>
303	public static complex operator +( complex A, fcomplex B) {
304	complex ret;
305	ret.real = (double) (A.real + B.real );
306	ret.imag = (double) (A.imag + B.imag );
307	return ret;
308	}
309	/// <summary>
310	/// Subtract two complex values
311	/// </summary>
312	/// <param name="A">Minuend</param>
313	/// <param name="B">Subtrahend</param>
314	/// <returns>result</returns>
315	public static complex operator -( complex A, fcomplex B) {
316	complex ret;
317	ret.real = (double) (A.real - B.real );
318	ret.imag = (double) (A.imag - B.imag );
319	return ret;
320	}
321	/// <summary>
322	/// Multiply two complex values
323	/// </summary>
324	/// <param name="A">First factor</param>
325	/// <param name="B">Second factor</param>
326	/// <returns>result</returns>
327	public static complex operator *( complex A, fcomplex B) {
328	complex ret;
329	ret.real = (double) ((A.real * B.real ) - (A.imag * B.imag ));
330	ret.imag = (double) ((A.real * B.imag ) + (A.imag * B.real ));
331	return ret;
332	}
333	/// <summary>
334	/// Divide two numbers
335	/// </summary>
336	/// <param name="A">Divident</param>
337	/// <param name="B">Divisor</param>
338	/// <returns>Result</returns>
339	/// <remarks><para>Unless the operator must handle special inputs (Inf or 0 values),
340	/// the algorithm described in [1] is used for division. This is considered to be
341	/// more robust against floating point overflow than the naive approach of simple
342	/// cartesian division.</para>
343	/// <para>References: [1]: Smith, R.L., Algorithm 116: Complex division. Commun.ACM 5,8 (1962),435 <br />
344	/// [2]: Stewart, G.W., A note on complex division, ACM trans.on math software, Vol.11, N.3 (1985)</para></remarks>
345	public static complex operator /( complex A, fcomplex B) {
346	if (B.imag == 0) return A / B.real;
347	return A * (1 / B);
348	if (IsNaN(A) \|\| fcomplex .IsNaN(B)) return NaN;
349	//if ( fcomplex .IsInfinity(B)) return NaN;
350	//if (A.real == 0 && A.imag == 0) return ( complex )0;
351	complex ret;
352	if (B.real == 0) {
353	ret.imag = (double) -(A.real / B.imag);
354	ret.real = (double) (A.imag / B.imag);
355	return ret;
356	}
357	// this would be the naive approach. But it come with to little robustness against overflow
358	//double norm2 = B.real * B.real + B.imag * B.imag;
359	//if (norm2 == 0) return INF; // this may be removed, since division by 0 results in inf anyway ?
360	//ret.real = (double) (((A.real * B.real ) + (A.imag * B.imag )) / norm2);
361	//ret.imag = (double) (((A.imag * B.real ) - (A.real * B.imag )) / norm2);
362
363	// this algorithm is taken from [1]. The one described in [2] was not taken. Tests
364	// did not show any advantage when using double precision floating point arithmetic.
365	double tmp1, tmp2;
366	if (Math.Abs(B.real) >= Math.Abs(B.imag)) {
367	tmp1 = (double) (B.imag * (1/B.real));
368	tmp2 = (double) (B.real + B.imag*tmp1);
369	ret.real = (double) (A.real + A.imag*tmp1)/tmp2;
370	ret.imag = (double) (A.imag - A.real*tmp1)/tmp2;
371	} else {
372	tmp1 = (double) (B.real * (1/B.imag));
373	tmp2 = (double) (B.imag + B.real*tmp1);
374	ret.real = (double) (A.imag + A.real*tmp1)/tmp2;
375	ret.imag = - (double) (A.real - A.imag*tmp1)/tmp2;
376	}
377	return ret;
378	}
379	/// <summary>
380	/// Equality comparison for complex numbers
381	/// </summary>
382	/// <param name="A">Left side</param>
383	/// <param name="B">Right side</param>
384	/// <returns>true, if real and imaginary part are identical</returns>
385	public static bool operator ==( complex A, fcomplex B) {
386	return (A.imag == B.imag ) && (A.real == B.real );
387	}
388	/// <summary>
389	/// Unequality comparison for complex numbers
390	/// </summary>
391	/// <param name="A">Left side</param>
392	/// <param name="B">Right side</param>
393	/// <returns>true if real and imaginary parts of A and B are not equal, false otherwise</returns>
394	public static bool operator !=( complex A, fcomplex B) {
395	return (A.imag != B.imag ) \|\| (A.real != B.real );
396	}
397	/// <summary>
398	/// Greater than comparison for complex numbers
399	/// </summary>
400	/// <param name="A">Left side</param>
401	/// <param name="B">Right side</param>
402	/// <returns>true if real part of A is greater than real part of B, false otherwise</returns>
403	/// <remarks>Only the real parts are compared!</remarks>
404	public static bool operator > ( complex A, fcomplex B) {
405	return (A.real > B.real );
406	}
407	/// <summary>
408	/// Lower than comparison for complex numbers
409	/// </summary>
410	/// <param name="A">Left side</param>
411	/// <param name="B">Right side</param>
412	/// <returns>true if real part of A is lower then real part of B, false otherwise</returns>
413	/// <remarks>Only the real parts are compared!</remarks>
414	public static bool operator < ( complex A, fcomplex B) {
415	return (A.real < B.real );
416	}
417	/// <summary>
418	/// Greater than or equal to comparison for complex numbers
419	/// </summary>
420	/// <param name="A">Left side</param>
421	/// <param name="B">Right side</param>
422	/// <returns>true if real part of A is greater than real part of B, false otherwise</returns>
423	/// <remarks>Only the real parts are compared!</remarks>
424	public static bool operator >=( complex A, fcomplex B) {
425	return (A.real >= B.real );
426	}
427	/// <summary>
428	/// Lower than or equal to comparison for complex numbers
429	/// </summary>
430	/// <param name="A">Left side</param>
431	/// <param name="B">Right side</param>
432	/// <returns>true if real part of A is lower then real part of B, false otherwise</returns>
433	/// <remarks>Only the real parts are compared!</remarks>
434	public static bool operator <=( complex A, fcomplex B) {
435	return (A.real <= B.real );
436	}
437
438	#endregion HYCALPER AUTO GENERATED CODE
439
440
441
442	/// <summary>
443	/// Add two complex numbers
444	/// </summary>
445	/// <param name="A">First summand</param>
446	/// <param name="B">Second summand</param>
447	/// <returns>Result</returns>
448	public static complex operator +( complex A, double B) {
449	complex ret;
450	ret.real = (double) (A.real + B);
451	ret.imag = (double) A.imag;
452	return ret;
453	}
454	/// <summary>
455	/// Subtract two values
456	/// </summary>
457	/// <param name="A">Minuend</param>
458	/// <param name="B">Subtrahend</param>
459	/// <returns>result</returns>
460	public static complex operator -( complex A, double B) {
461	complex ret;
462	ret.real = (double) (A.real - B);
463	ret.imag = (double) A.imag;
464	return ret;
465	}
466	/// <summary>
467	/// Multiply two values
468	/// </summary>
469	/// <param name="A">First factor</param>
470	/// <param name="B">Second factor</param>
471	/// <returns>result</returns>
472	public static complex operator *( complex A, double B) {
473	complex ret;
474	ret.real = (double) (A.real * B);
475	ret.imag = (double) (A.imag * B);
476	return ret;
477	}
478	/// <summary>
479	/// Divide two numbers
480	/// </summary>
481	/// <param name="A">Divident</param>
482	/// <param name="B">Divisor</param>
483	/// <returns>result</returns>
484	public static complex operator /( complex A, double B) {
485	if (IsNaN(A)) return NaN;
486
487	if (double.IsNaN(B)) return NaN;
488	if (A.real == 0 && A.imag == 0) {
489	if (B == 0) return NaN;
490	return ( complex )0;
491	} else {
492
493	if (double .IsInfinity(B))
494	{
495	if (IsInfinity(A)) {
496	return NaN;
497	} else {
498	return ( complex )0;
499	}
500	}
501	}
502	complex ret;
503	if (B == 0) return INF ;
504	ret.real = (double) (A.real / B);
505	ret.imag = (double) (A.imag / B);
506	return ret;
507	}
508	/// <summary>
509	/// Equality comparison for complex numbers
510	/// </summary>
511	/// <param name="A">Left side</param>
512	/// <param name="B">Right side</param>
513	/// <returns>result</returns>
514	/// <remarks>Real inputs are converted to a complex number and the result is compared to the complex input.</remarks>
515	public static bool operator ==( complex A, double B) {
516	return (A.real == B && A.imag == 0.0);
517	}
518	/// <summary>
519	/// Unequality comparison for complex numbers
520	/// </summary>
521	/// <param name="A">Left side</param>
522	/// <param name="B">Right side</param>
523	/// <returns>true if real and imaginary parts of A and B are not equal, false otherwise</returns>
524	/// <remarks>Real inputs are converted to a complex number and the result is compared to the complex input.</remarks>
525	public static bool operator !=( complex A, double B) {
526	return (A.imag != 0.0) \|\| (A.real != B);
527	}
528	/// <summary>
529	/// Freater than comparison for complex numbers
530	/// </summary>
531	/// <param name="A">Left side</param>
532	/// <param name="B">Right side</param>
533	/// <returns>true if real part of A is greater than real part of B, false otherwise</returns>
534	/// <remarks>Only the real parts are compared!</remarks>
535	public static bool operator > ( complex A, double B) {
536	return (A.real > B);
537	}
538	/// <summary>
539	/// Lower than comparison for complex numbers
540	/// </summary>
541	/// <param name="A">Left side</param>
542	/// <param name="B">Right side</param>
543	/// <returns>true if real part of A is lower then real part of B, false otherwise</returns>
544	/// <remarks>Only the real parts are compared!</remarks>
545	public static bool operator <( complex A, double B) {
546	return (A.real < B);
547	}
548	/// <summary>
549	/// Greater than or equal to comparison for complex numbers
550	/// </summary>
551	/// <param name="A">Left side</param>
552	/// <param name="B">Right side</param>
553	/// <returns>true if real part of A is greater than real part of B, false otherwise</returns>
554	/// <remarks>Only the real parts are compared!</remarks>
555	public static bool operator >=( complex A, double B) {
556	return (A.real >= B);
557	}
558	/// <summary>
559	/// Lower than or equal to comparison for complex numbers
560	/// </summary>
561	/// <param name="A">Left side</param>
562	/// <param name="B">Right side</param>
563	/// <returns>true if real part of A is lower then real part of B, false otherwise</returns>
564	/// <remarks>Only the real parts are compared!</remarks>
565	public static bool operator <=( complex A, double B) {
566	return (A.real <= B);
567	}
568
569	#region HYCALPER AUTO GENERATED CODE
570
571
572	/// <summary>
573	/// Add two complex numbers
574	/// </summary>
575	/// <param name="A">First summand</param>
576	/// <param name="B">Second summand</param>
577	/// <returns>Result</returns>
578	public static complex operator +( complex A, Int64 B) {
579	complex ret;
580	ret.real = (double) (A.real + B);
581	ret.imag = (double) A.imag;
582	return ret;
583	}
584	/// <summary>
585	/// Subtract two values
586	/// </summary>
587	/// <param name="A">Minuend</param>
588	/// <param name="B">Subtrahend</param>
589	/// <returns>result</returns>
590	public static complex operator -( complex A, Int64 B) {
591	complex ret;
592	ret.real = (double) (A.real - B);
593	ret.imag = (double) A.imag;
594	return ret;
595	}
596	/// <summary>
597	/// Multiply two values
598	/// </summary>
599	/// <param name="A">First factor</param>
600	/// <param name="B">Second factor</param>
601	/// <returns>result</returns>
602	public static complex operator *( complex A, Int64 B) {
603	complex ret;
604	ret.real = (double) (A.real * B);
605	ret.imag = (double) (A.imag * B);
606	return ret;
607	}
608	/// <summary>
609	/// Divide two numbers
610	/// </summary>
611	/// <param name="A">Divident</param>
612	/// <param name="B">Divisor</param>
613	/// <returns>result</returns>
614	public static complex operator /( complex A, Int64 B) {
615	if (IsNaN(A)) return NaN;
616
617	if (A.real == 0 && A.imag == 0) {
618	if (B == 0) return NaN;
619	return ( complex )0;
620	} else {
621	if (false)
622	{
623	if (IsInfinity(A)) {
624	return NaN;
625	} else {
626	return ( complex )0;
627	}
628	}
629	}
630	complex ret;
631	if (B == 0) return INF ;
632	ret.real = (double) (A.real / B);
633	ret.imag = (double) (A.imag / B);
634	return ret;
635	}
636	/// <summary>
637	/// Equality comparison for complex numbers
638	/// </summary>
639	/// <param name="A">Left side</param>
640	/// <param name="B">Right side</param>
641	/// <returns>result</returns>
642	/// <remarks>Real inputs are converted to a complex number and the result is compared to the complex input.</remarks>
643	public static bool operator ==( complex A, Int64 B) {
644	return (A.real == B && A.imag == 0.0);
645	}
646	/// <summary>
647	/// Unequality comparison for complex numbers
648	/// </summary>
649	/// <param name="A">Left side</param>
650	/// <param name="B">Right side</param>
651	/// <returns>true if real and imaginary parts of A and B are not equal, false otherwise</returns>
652	/// <remarks>Real inputs are converted to a complex number and the result is compared to the complex input.</remarks>
653	public static bool operator !=( complex A, Int64 B) {
654	return (A.imag != 0.0) \|\| (A.real != B);
655	}
656	/// <summary>
657	/// Freater than comparison for complex numbers
658	/// </summary>
659	/// <param name="A">Left side</param>
660	/// <param name="B">Right side</param>
661	/// <returns>true if real part of A is greater than real part of B, false otherwise</returns>
662	/// <remarks>Only the real parts are compared!</remarks>
663	public static bool operator > ( complex A, Int64 B) {
664	return (A.real > B);
665	}
666	/// <summary>
667	/// Lower than comparison for complex numbers
668	/// </summary>
669	/// <param name="A">Left side</param>
670	/// <param name="B">Right side</param>
671	/// <returns>true if real part of A is lower then real part of B, false otherwise</returns>
672	/// <remarks>Only the real parts are compared!</remarks>
673	public static bool operator <( complex A, Int64 B) {
674	return (A.real < B);
675	}
676	/// <summary>
677	/// Greater than or equal to comparison for complex numbers
678	/// </summary>
679	/// <param name="A">Left side</param>
680	/// <param name="B">Right side</param>
681	/// <returns>true if real part of A is greater than real part of B, false otherwise</returns>
682	/// <remarks>Only the real parts are compared!</remarks>
683	public static bool operator >=( complex A, Int64 B) {
684	return (A.real >= B);
685	}
686	/// <summary>
687	/// Lower than or equal to comparison for complex numbers
688	/// </summary>
689	/// <param name="A">Left side</param>
690	/// <param name="B">Right side</param>
691	/// <returns>true if real part of A is lower then real part of B, false otherwise</returns>
692	/// <remarks>Only the real parts are compared!</remarks>
693	public static bool operator <=( complex A, Int64 B) {
694	return (A.real <= B);
695	}
696
697	/// <summary>
698	/// Add two complex numbers
699	/// </summary>
700	/// <param name="A">First summand</param>
701	/// <param name="B">Second summand</param>
702	/// <returns>Result</returns>
703	public static complex operator +( complex A, Int32 B) {
704	complex ret;
705	ret.real = (double) (A.real + B);
706	ret.imag = (double) A.imag;
707	return ret;
708	}
709	/// <summary>
710	/// Subtract two values
711	/// </summary>
712	/// <param name="A">Minuend</param>
713	/// <param name="B">Subtrahend</param>
714	/// <returns>result</returns>
715	public static complex operator -( complex A, Int32 B) {
716	complex ret;
717	ret.real = (double) (A.real - B);
718	ret.imag = (double) A.imag;
719	return ret;
720	}
721	/// <summary>
722	/// Multiply two values
723	/// </summary>
724	/// <param name="A">First factor</param>
725	/// <param name="B">Second factor</param>
726	/// <returns>result</returns>
727	public static complex operator *( complex A, Int32 B) {
728	complex ret;
729	ret.real = (double) (A.real * B);
730	ret.imag = (double) (A.imag * B);
731	return ret;
732	}
733	/// <summary>
734	/// Divide two numbers
735	/// </summary>
736	/// <param name="A">Divident</param>
737	/// <param name="B">Divisor</param>
738	/// <returns>result</returns>
739	public static complex operator /( complex A, Int32 B) {
740	if (IsNaN(A)) return NaN;
741
742	if (A.real == 0 && A.imag == 0) {
743	if (B == 0) return NaN;
744	return ( complex )0;
745	} else {
746	if (false)
747	{
748	if (IsInfinity(A)) {
749	return NaN;
750	} else {
751	return ( complex )0;
752	}
753	}
754	}
755	complex ret;
756	if (B == 0) return INF ;
757	ret.real = (double) (A.real / B);
758	ret.imag = (double) (A.imag / B);
759	return ret;
760	}
761	/// <summary>
762	/// Equality comparison for complex numbers
763	/// </summary>
764	/// <param name="A">Left side</param>
765	/// <param name="B">Right side</param>
766	/// <returns>result</returns>
767	/// <remarks>Real inputs are converted to a complex number and the result is compared to the complex input.</remarks>
768	public static bool operator ==( complex A, Int32 B) {
769	return (A.real == B && A.imag == 0.0);
770	}
771	/// <summary>
772	/// Unequality comparison for complex numbers
773	/// </summary>
774	/// <param name="A">Left side</param>
775	/// <param name="B">Right side</param>
776	/// <returns>true if real and imaginary parts of A and B are not equal, false otherwise</returns>
777	/// <remarks>Real inputs are converted to a complex number and the result is compared to the complex input.</remarks>
778	public static bool operator !=( complex A, Int32 B) {
779	return (A.imag != 0.0) \|\| (A.real != B);
780	}
781	/// <summary>
782	/// Freater than comparison for complex numbers
783	/// </summary>
784	/// <param name="A">Left side</param>
785	/// <param name="B">Right side</param>
786	/// <returns>true if real part of A is greater than real part of B, false otherwise</returns>
787	/// <remarks>Only the real parts are compared!</remarks>
788	public static bool operator > ( complex A, Int32 B) {
789	return (A.real > B);
790	}
791	/// <summary>
792	/// Lower than comparison for complex numbers
793	/// </summary>
794	/// <param name="A">Left side</param>
795	/// <param name="B">Right side</param>
796	/// <returns>true if real part of A is lower then real part of B, false otherwise</returns>
797	/// <remarks>Only the real parts are compared!</remarks>
798	public static bool operator <( complex A, Int32 B) {
799	return (A.real < B);
800	}
801	/// <summary>
802	/// Greater than or equal to comparison for complex numbers
803	/// </summary>
804	/// <param name="A">Left side</param>
805	/// <param name="B">Right side</param>
806	/// <returns>true if real part of A is greater than real part of B, false otherwise</returns>
807	/// <remarks>Only the real parts are compared!</remarks>
808	public static bool operator >=( complex A, Int32 B) {
809	return (A.real >= B);
810	}
811	/// <summary>
812	/// Lower than or equal to comparison for complex numbers
813	/// </summary>
814	/// <param name="A">Left side</param>
815	/// <param name="B">Right side</param>
816	/// <returns>true if real part of A is lower then real part of B, false otherwise</returns>
817	/// <remarks>Only the real parts are compared!</remarks>
818	public static bool operator <=( complex A, Int32 B) {
819	return (A.real <= B);
820	}
821
822	/// <summary>
823	/// Add two complex numbers
824	/// </summary>
825	/// <param name="A">First summand</param>
826	/// <param name="B">Second summand</param>
827	/// <returns>Result</returns>
828	public static complex operator +( complex A, float B) {
829	complex ret;
830	ret.real = (double) (A.real + B);
831	ret.imag = (double) A.imag;
832	return ret;
833	}
834	/// <summary>
835	/// Subtract two values
836	/// </summary>
837	/// <param name="A">Minuend</param>
838	/// <param name="B">Subtrahend</param>
839	/// <returns>result</returns>
840	public static complex operator -( complex A, float B) {
841	complex ret;
842	ret.real = (double) (A.real - B);
843	ret.imag = (double) A.imag;
844	return ret;
845	}
846	/// <summary>
847	/// Multiply two values
848	/// </summary>
849	/// <param name="A">First factor</param>
850	/// <param name="B">Second factor</param>
851	/// <returns>result</returns>
852	public static complex operator *( complex A, float B) {
853	complex ret;
854	ret.real = (double) (A.real * B);
855	ret.imag = (double) (A.imag * B);
856	return ret;
857	}
858	/// <summary>
859	/// Divide two numbers
860	/// </summary>
861	/// <param name="A">Divident</param>
862	/// <param name="B">Divisor</param>
863	/// <returns>result</returns>
864	public static complex operator /( complex A, float B) {
865	if (IsNaN(A)) return NaN;
866	if (float.IsNaN(B)) return NaN;
867	if (A.real == 0 && A.imag == 0) {
868	if (B == 0) return NaN;
869	return ( complex )0;
870	} else {
871	if (float.IsInfinity(B))
872	{
873	if (IsInfinity(A)) {
874	return NaN;
875	} else {
876	return ( complex )0;
877	}
878	}
879	}
880	complex ret;
881	if (B == 0) return INF ;
882	ret.real = (double) (A.real / B);
883	ret.imag = (double) (A.imag / B);
884	return ret;
885	}
886	/// <summary>
887	/// Equality comparison for complex numbers
888	/// </summary>
889	/// <param name="A">Left side</param>
890	/// <param name="B">Right side</param>
891	/// <returns>result</returns>
892	/// <remarks>Real inputs are converted to a complex number and the result is compared to the complex input.</remarks>
893	public static bool operator ==( complex A, float B) {
894	return (A.real == B && A.imag == 0.0);
895	}
896	/// <summary>
897	/// Unequality comparison for complex numbers
898	/// </summary>
899	/// <param name="A">Left side</param>
900	/// <param name="B">Right side</param>
901	/// <returns>true if real and imaginary parts of A and B are not equal, false otherwise</returns>
902	/// <remarks>Real inputs are converted to a complex number and the result is compared to the complex input.</remarks>
903	public static bool operator !=( complex A, float B) {
904	return (A.imag != 0.0) \|\| (A.real != B);
905	}
906	/// <summary>
907	/// Freater than comparison for complex numbers
908	/// </summary>
909	/// <param name="A">Left side</param>
910	/// <param name="B">Right side</param>
911	/// <returns>true if real part of A is greater than real part of B, false otherwise</returns>
912	/// <remarks>Only the real parts are compared!</remarks>
913	public static bool operator > ( complex A, float B) {
914	return (A.real > B);
915	}
916	/// <summary>
917	/// Lower than comparison for complex numbers
918	/// </summary>
919	/// <param name="A">Left side</param>
920	/// <param name="B">Right side</param>
921	/// <returns>true if real part of A is lower then real part of B, false otherwise</returns>
922	/// <remarks>Only the real parts are compared!</remarks>
923	public static bool operator <( complex A, float B) {
924	return (A.real < B);
925	}
926	/// <summary>
927	/// Greater than or equal to comparison for complex numbers
928	/// </summary>
929	/// <param name="A">Left side</param>
930	/// <param name="B">Right side</param>
931	/// <returns>true if real part of A is greater than real part of B, false otherwise</returns>
932	/// <remarks>Only the real parts are compared!</remarks>
933	public static bool operator >=( complex A, float B) {
934	return (A.real >= B);
935	}
936	/// <summary>
937	/// Lower than or equal to comparison for complex numbers
938	/// </summary>
939	/// <param name="A">Left side</param>
940	/// <param name="B">Right side</param>
941	/// <returns>true if real part of A is lower then real part of B, false otherwise</returns>
942	/// <remarks>Only the real parts are compared!</remarks>
943	public static bool operator <=( complex A, float B) {
944	return (A.real <= B);
945	}
946
947	/// <summary>
948	/// Add two complex numbers
949	/// </summary>
950	/// <param name="A">First summand</param>
951	/// <param name="B">Second summand</param>
952	/// <returns>Result</returns>
953	public static complex operator +( complex A, byte B) {
954	complex ret;
955	ret.real = (double) (A.real + B);
956	ret.imag = (double) A.imag;
957	return ret;
958	}
959	/// <summary>
960	/// Subtract two values
961	/// </summary>
962	/// <param name="A">Minuend</param>
963	/// <param name="B">Subtrahend</param>
964	/// <returns>result</returns>
965	public static complex operator -( complex A, byte B) {
966	complex ret;
967	ret.real = (double) (A.real - B);
968	ret.imag = (double) A.imag;
969	return ret;
970	}
971	/// <summary>
972	/// Multiply two values
973	/// </summary>
974	/// <param name="A">First factor</param>
975	/// <param name="B">Second factor</param>
976	/// <returns>result</returns>
977	public static complex operator *( complex A, byte B) {
978	complex ret;
979	ret.real = (double) (A.real * B);
980	ret.imag = (double) (A.imag * B);
981	return ret;
982	}
983	/// <summary>
984	/// Divide two numbers
985	/// </summary>
986	/// <param name="A">Divident</param>
987	/// <param name="B">Divisor</param>
988	/// <returns>result</returns>
989	public static complex operator /( complex A, byte B) {
990	if (IsNaN(A)) return NaN;
991
992	if (A.real == 0 && A.imag == 0) {
993	if (B == 0) return NaN;
994	return ( complex )0;
995	} else {
996	if (false)
997	{
998	if (IsInfinity(A)) {
999	return NaN;
1000	} else {
1001	return ( complex )0;
1002	}
1003	}
1004	}
1005	complex ret;
1006	if (B == 0) return INF ;
1007	ret.real = (double) (A.real / B);
1008	ret.imag = (double) (A.imag / B);
1009	return ret;
1010	}
1011	/// <summary>
1012	/// Equality comparison for complex numbers
1013	/// </summary>
1014	/// <param name="A">Left side</param>
1015	/// <param name="B">Right side</param>
1016	/// <returns>result</returns>
1017	/// <remarks>Real inputs are converted to a complex number and the result is compared to the complex input.</remarks>
1018	public static bool operator ==( complex A, byte B) {
1019	return (A.real == B && A.imag == 0.0);
1020	}
1021	/// <summary>
1022	/// Unequality comparison for complex numbers
1023	/// </summary>
1024	/// <param name="A">Left side</param>
1025	/// <param name="B">Right side</param>
1026	/// <returns>true if real and imaginary parts of A and B are not equal, false otherwise</returns>
1027	/// <remarks>Real inputs are converted to a complex number and the result is compared to the complex input.</remarks>
1028	public static bool operator !=( complex A, byte B) {
1029	return (A.imag != 0.0) \|\| (A.real != B);
1030	}
1031	/// <summary>
1032	/// Freater than comparison for complex numbers
1033	/// </summary>
1034	/// <param name="A">Left side</param>
1035	/// <param name="B">Right side</param>
1036	/// <returns>true if real part of A is greater than real part of B, false otherwise</returns>
1037	/// <remarks>Only the real parts are compared!</remarks>
1038	public static bool operator > ( complex A, byte B) {
1039	return (A.real > B);
1040	}
1041	/// <summary>
1042	/// Lower than comparison for complex numbers
1043	/// </summary>
1044	/// <param name="A">Left side</param>
1045	/// <param name="B">Right side</param>
1046	/// <returns>true if real part of A is lower then real part of B, false otherwise</returns>
1047	/// <remarks>Only the real parts are compared!</remarks>
1048	public static bool operator <( complex A, byte B) {
1049	return (A.real < B);
1050	}
1051	/// <summary>
1052	/// Greater than or equal to comparison for complex numbers
1053	/// </summary>
1054	/// <param name="A">Left side</param>
1055	/// <param name="B">Right side</param>
1056	/// <returns>true if real part of A is greater than real part of B, false otherwise</returns>
1057	/// <remarks>Only the real parts are compared!</remarks>
1058	public static bool operator >=( complex A, byte B) {
1059	return (A.real >= B);
1060	}
1061	/// <summary>
1062	/// Lower than or equal to comparison for complex numbers
1063	/// </summary>
1064	/// <param name="A">Left side</param>
1065	/// <param name="B">Right side</param>
1066	/// <returns>true if real part of A is lower then real part of B, false otherwise</returns>
1067	/// <remarks>Only the real parts are compared!</remarks>
1068	public static bool operator <=( complex A, byte B) {
1069	return (A.real <= B);
1070	}
1071
1072	#endregion HYCALPER AUTO GENERATED CODE
1073
1074
1075
1076	/// <summary>
1077	/// Add two complex values
1078	/// </summary>
1079	/// <param name="A">First summand</param>
1080	/// <param name="B">Second summand</param>
1081	/// <returns>Result</returns>
1082	public static complex operator +( double A, complex B) {
1083	complex ret;
1084	ret.real = (double) (A + B.real);
1085	ret.imag = (double) B.imag;
1086	return ret;
1087	}
1088	/// <summary>
1089	/// Subtract two values
1090	/// </summary>
1091	/// <param name="A">Minuend</param>
1092	/// <param name="B">Subtrahend</param>
1093	/// <returns>Result</returns>
1094	public static complex operator -( double A, complex B) {
1095	complex ret;
1096	ret.real = (double) (A - B.real);
1097	ret.imag = - (double) B.imag;
1098	return ret;
1099	}
1100	/// <summary>
1101	/// Multiply two values
1102	/// </summary>
1103	/// <param name="A">First factor</param>
1104	/// <param name="B">Second factor</param>
1105	/// <returns>Result</returns>
1106	public static complex operator *( double A, complex B) {
1107	complex ret;
1108	ret.real = (double) (A * B.real);
1109	ret.imag = (double) (A * B.imag);
1110	return ret;
1111	}
1112	/// <summary>
1113	/// Divide two values
1114	/// </summary>
1115	/// <param name="A">Divident</param>
1116	/// <param name="B">Divisor</param>
1117	/// <returns>Result</returns>
1118	public static complex operator /( double A, complex B) {
1119	complex ret;
1120	if (A == 0) {
1121	if (IsInfinity(B)) return NaN;
1122	} else {
1123	if (IsInfinity(B)) return ( complex )0;
1124	}
1125	if (B.real == 0 && B.imag == 0) {
1126	return INF;
1127	}
1128	// this algorithm is taken from [1]. The one described in [2] was not taken. Tests
1129	// did not show any advantage when using double precision floating point arithmetic.
1130	double tmp;
1131	if (Math.Abs(B.real) >= Math.Abs(B.imag)) {
1132	tmp = (double) (B.imag * (1/B.real));
1133	ret.imag = (double) (B.real + B.imag*tmp);
1134	ret.real = (double) A/ret.imag;
1135	ret.imag = - (double) (A*tmp)/ret.imag;
1136	} else {
1137	tmp = (double) (B.real * (1/B.imag));
1138	ret.imag = (double) (B.imag + B.real*tmp);
1139	ret.real = (double) (A*tmp)/ret.imag;
1140	ret.imag = - (double) A/ret.imag;
1141	}
1142	return ret;
1143	}
1144	/// <summary>
1145	/// Equality comparison for complex numbers
1146	/// </summary>
1147	/// <param name="A">Left side</param>
1148	/// <param name="B">Right side</param>
1149	/// <returns>Result</returns>
1150	/// <remarks>Real inputs are converted to a complex number and the result is compared to the complex input.</remarks>
1151	public static bool operator ==( double A, complex B) {
1152	return (B.real == A && B.imag == 0.0);
1153	}
1154	/// <summary>
1155	/// Unequality comparison for complex numbers
1156	/// </summary>
1157	/// <param name="A">Left side</param>
1158	/// <param name="B">Right side</param>
1159	/// <returns>true if real and imaginary parts of A and B are not equal, false otherwise</returns>
1160	/// <remarks>Real inputs are converted to a complex number and the result is compared to the complex input.</remarks>
1161	public static bool operator !=( double A, complex B) {
1162	return (B.imag != 0.0) \|\| (B.real != A);
1163	}
1164	/// <summary>
1165	/// Greater than comparison for complex numbers
1166	/// </summary>
1167	/// <param name="A">Left side</param>
1168	/// <param name="B">Right side</param>
1169	/// <returns>true if real part of A is greater than real part of B, false otherwise</returns>
1170	/// <remarks>Only the real parts are compared!</remarks>
1171	public static bool operator > ( double A, complex B) {
1172	return (A > B.real);
1173	}
1174	/// <summary>
1175	/// Lower than comparison for complex numbers
1176	/// </summary>
1177	/// <param name="A">Left side</param>
1178	/// <param name="B">Right side</param>
1179	/// <returns>true if real part of A is lower then real part of B, false otherwise</returns>
1180	/// <remarks>Only the real parts are compared!</remarks>
1181	public static bool operator < ( double A, complex B) {
1182	return (A < B.real);
1183	}
1184	/// <summary>
1185	/// Greater than or equal to comparison for complex numbers
1186	/// </summary>
1187	/// <param name="A">Left side</param>
1188	/// <param name="B">Right side</param>
1189	/// <returns>true if real part of A is greater than real part of B, false otherwise</returns>
1190	/// <remarks>Only the real parts are compared!</remarks>
1191	public static bool operator >=( double A, complex B) {
1192	return (A >= B.real);
1193	}
1194	/// <summary>
1195	/// Lower than or equal to comparison for complex numbers
1196	/// </summary>
1197	/// <param name="A">Left side</param>
1198	/// <param name="B">Right side</param>
1199	/// <returns>true if real part of A is lower then real part of B, false otherwise</returns>
1200	/// <remarks>Only the real parts are compared!</remarks>
1201	public static bool operator <=( double A, complex B) {
1202	return (A <= B.real);
1203	}
1204
1205	#region HYCALPER AUTO GENERATED CODE
1206
1207
1208	/// <summary>
1209	/// Add two complex values
1210	/// </summary>
1211	/// <param name="A">First summand</param>
1212	/// <param name="B">Second summand</param>
1213	/// <returns>Result</returns>
1214	public static complex operator +( Int64 A, complex B) {
1215	complex ret;
1216	ret.real = (double) (A + B.real);
1217	ret.imag = (double) B.imag;
1218	return ret;
1219	}
1220	/// <summary>
1221	/// Subtract two values
1222	/// </summary>
1223	/// <param name="A">Minuend</param>
1224	/// <param name="B">Subtrahend</param>
1225	/// <returns>Result</returns>
1226	public static complex operator -( Int64 A, complex B) {
1227	complex ret;
1228	ret.real = (double) (A - B.real);
1229	ret.imag = - (double) B.imag;
1230	return ret;
1231	}
1232	/// <summary>
1233	/// Multiply two values
1234	/// </summary>
1235	/// <param name="A">First factor</param>
1236	/// <param name="B">Second factor</param>
1237	/// <returns>Result</returns>
1238	public static complex operator *( Int64 A, complex B) {
1239	complex ret;
1240	ret.real = (double) (A * B.real);
1241	ret.imag = (double) (A * B.imag);
1242	return ret;
1243	}
1244	/// <summary>
1245	/// Divide two values
1246	/// </summary>
1247	/// <param name="A">Divident</param>
1248	/// <param name="B">Divisor</param>
1249	/// <returns>Result</returns>
1250	public static complex operator /( Int64 A, complex B) {
1251	complex ret;
1252	if (A == 0) {
1253	if (IsInfinity(B)) return NaN;
1254	} else {
1255	if (IsInfinity(B)) return ( complex )0;
1256	}
1257	if (B.real == 0 && B.imag == 0) {
1258	return INF;
1259	}
1260	// this algorithm is taken from [1]. The one described in [2] was not taken. Tests
1261	// did not show any advantage when using double precision floating point arithmetic.
1262	double tmp;
1263	if (Math.Abs(B.real) >= Math.Abs(B.imag)) {
1264	tmp = (double) (B.imag * (1/B.real));
1265	ret.imag = (double) (B.real + B.imag*tmp);
1266	ret.real = (double) A/ret.imag;
1267	ret.imag = - (double) (A*tmp)/ret.imag;
1268	} else {
1269	tmp = (double) (B.real * (1/B.imag));
1270	ret.imag = (double) (B.imag + B.real*tmp);
1271	ret.real = (double) (A*tmp)/ret.imag;
1272	ret.imag = - (double) A/ret.imag;
1273	}
1274	return ret;
1275	}
1276	/// <summary>
1277	/// Equality comparison for complex numbers
1278	/// </summary>
1279	/// <param name="A">Left side</param>
1280	/// <param name="B">Right side</param>
1281	/// <returns>Result</returns>
1282	/// <remarks>Real inputs are converted to a complex number and the result is compared to the complex input.</remarks>
1283	public static bool operator ==( Int64 A, complex B) {
1284	return (B.real == A && B.imag == 0.0);
1285	}
1286	/// <summary>
1287	/// Unequality comparison for complex numbers
1288	/// </summary>
1289	/// <param name="A">Left side</param>
1290	/// <param name="B">Right side</param>
1291	/// <returns>true if real and imaginary parts of A and B are not equal, false otherwise</returns>
1292	/// <remarks>Real inputs are converted to a complex number and the result is compared to the complex input.</remarks>
1293	public static bool operator !=( Int64 A, complex B) {
1294	return (B.imag != 0.0) \|\| (B.real != A);
1295	}
1296	/// <summary>
1297	/// Greater than comparison for complex numbers
1298	/// </summary>
1299	/// <param name="A">Left side</param>
1300	/// <param name="B">Right side</param>
1301	/// <returns>true if real part of A is greater than real part of B, false otherwise</returns>
1302	/// <remarks>Only the real parts are compared!</remarks>
1303	public static bool operator > ( Int64 A, complex B) {
1304	return (A > B.real);
1305	}
1306	/// <summary>
1307	/// Lower than comparison for complex numbers
1308	/// </summary>
1309	/// <param name="A">Left side</param>
1310	/// <param name="B">Right side</param>
1311	/// <returns>true if real part of A is lower then real part of B, false otherwise</returns>
1312	/// <remarks>Only the real parts are compared!</remarks>
1313	public static bool operator < ( Int64 A, complex B) {
1314	return (A < B.real);
1315	}
1316	/// <summary>
1317	/// Greater than or equal to comparison for complex numbers
1318	/// </summary>
1319	/// <param name="A">Left side</param>
1320	/// <param name="B">Right side</param>
1321	/// <returns>true if real part of A is greater than real part of B, false otherwise</returns>
1322	/// <remarks>Only the real parts are compared!</remarks>
1323	public static bool operator >=( Int64 A, complex B) {
1324	return (A >= B.real);
1325	}
1326	/// <summary>
1327	/// Lower than or equal to comparison for complex numbers
1328	/// </summary>
1329	/// <param name="A">Left side</param>
1330	/// <param name="B">Right side</param>
1331	/// <returns>true if real part of A is lower then real part of B, false otherwise</returns>
1332	/// <remarks>Only the real parts are compared!</remarks>
1333	public static bool operator <=( Int64 A, complex B) {
1334	return (A <= B.real);
1335	}
1336
1337	/// <summary>
1338	/// Add two complex values
1339	/// </summary>
1340	/// <param name="A">First summand</param>
1341	/// <param name="B">Second summand</param>
1342	/// <returns>Result</returns>
1343	public static complex operator +( Int32 A, complex B) {
1344	complex ret;
1345	ret.real = (double) (A + B.real);
1346	ret.imag = (double) B.imag;
1347	return ret;
1348	}
1349	/// <summary>
1350	/// Subtract two values
1351	/// </summary>
1352	/// <param name="A">Minuend</param>
1353	/// <param name="B">Subtrahend</param>
1354	/// <returns>Result</returns>
1355	public static complex operator -( Int32 A, complex B) {
1356	complex ret;
1357	ret.real = (double) (A - B.real);
1358	ret.imag = - (double) B.imag;
1359	return ret;
1360	}
1361	/// <summary>
1362	/// Multiply two values
1363	/// </summary>
1364	/// <param name="A">First factor</param>
1365	/// <param name="B">Second factor</param>
1366	/// <returns>Result</returns>
1367	public static complex operator *( Int32 A, complex B) {
1368	complex ret;
1369	ret.real = (double) (A * B.real);
1370	ret.imag = (double) (A * B.imag);
1371	return ret;
1372	}
1373	/// <summary>
1374	/// Divide two values
1375	/// </summary>
1376	/// <param name="A">Divident</param>
1377	/// <param name="B">Divisor</param>
1378	/// <returns>Result</returns>
1379	public static complex operator /( Int32 A, complex B) {
1380	complex ret;
1381	if (A == 0) {
1382	if (IsInfinity(B)) return NaN;
1383	} else {
1384	if (IsInfinity(B)) return ( complex )0;
1385	}
1386	if (B.real == 0 && B.imag == 0) {
1387	return INF;
1388	}
1389	// this algorithm is taken from [1]. The one described in [2] was not taken. Tests
1390	// did not show any advantage when using double precision floating point arithmetic.
1391	double tmp;
1392	if (Math.Abs(B.real) >= Math.Abs(B.imag)) {
1393	tmp = (double) (B.imag * (1/B.real));
1394	ret.imag = (double) (B.real + B.imag*tmp);
1395	ret.real = (double) A/ret.imag;
1396	ret.imag = - (double) (A*tmp)/ret.imag;
1397	} else {
1398	tmp = (double) (B.real * (1/B.imag));
1399	ret.imag = (double) (B.imag + B.real*tmp);
1400	ret.real = (double) (A*tmp)/ret.imag;
1401	ret.imag = - (double) A/ret.imag;
1402	}
1403	return ret;
1404	}
1405	/// <summary>
1406	/// Equality comparison for complex numbers
1407	/// </summary>
1408	/// <param name="A">Left side</param>
1409	/// <param name="B">Right side</param>
1410	/// <returns>Result</returns>
1411	/// <remarks>Real inputs are converted to a complex number and the result is compared to the complex input.</remarks>
1412	public static bool operator ==( Int32 A, complex B) {
1413	return (B.real == A && B.imag == 0.0);
1414	}
1415	/// <summary>
1416	/// Unequality comparison for complex numbers
1417	/// </summary>
1418	/// <param name="A">Left side</param>
1419	/// <param name="B">Right side</param>
1420	/// <returns>true if real and imaginary parts of A and B are not equal, false otherwise</returns>
1421	/// <remarks>Real inputs are converted to a complex number and the result is compared to the complex input.</remarks>
1422	public static bool operator !=( Int32 A, complex B) {
1423	return (B.imag != 0.0) \|\| (B.real != A);
1424	}
1425	/// <summary>
1426	/// Greater than comparison for complex numbers
1427	/// </summary>
1428	/// <param name="A">Left side</param>
1429	/// <param name="B">Right side</param>
1430	/// <returns>true if real part of A is greater than real part of B, false otherwise</returns>
1431	/// <remarks>Only the real parts are compared!</remarks>
1432	public static bool operator > ( Int32 A, complex B) {
1433	return (A > B.real);
1434	}
1435	/// <summary>
1436	/// Lower than comparison for complex numbers
1437	/// </summary>
1438	/// <param name="A">Left side</param>
1439	/// <param name="B">Right side</param>
1440	/// <returns>true if real part of A is lower then real part of B, false otherwise</returns>
1441	/// <remarks>Only the real parts are compared!</remarks>
1442	public static bool operator < ( Int32 A, complex B) {
1443	return (A < B.real);
1444	}
1445	/// <summary>
1446	/// Greater than or equal to comparison for complex numbers
1447	/// </summary>
1448	/// <param name="A">Left side</param>
1449	/// <param name="B">Right side</param>
1450	/// <returns>true if real part of A is greater than real part of B, false otherwise</returns>
1451	/// <remarks>Only the real parts are compared!</remarks>
1452	public static bool operator >=( Int32 A, complex B) {
1453	return (A >= B.real);
1454	}
1455	/// <summary>
1456	/// Lower than or equal to comparison for complex numbers
1457	/// </summary>
1458	/// <param name="A">Left side</param>
1459	/// <param name="B">Right side</param>
1460	/// <returns>true if real part of A is lower then real part of B, false otherwise</returns>
1461	/// <remarks>Only the real parts are compared!</remarks>
1462	public static bool operator <=( Int32 A, complex B) {
1463	return (A <= B.real);
1464	}
1465
1466	/// <summary>
1467	/// Add two complex values
1468	/// </summary>
1469	/// <param name="A">First summand</param>
1470	/// <param name="B">Second summand</param>
1471	/// <returns>Result</returns>
1472	public static complex operator +( float A, complex B) {
1473	complex ret;
1474	ret.real = (double) (A + B.real);
1475	ret.imag = (double) B.imag;
1476	return ret;
1477	}
1478	/// <summary>
1479	/// Subtract two values
1480	/// </summary>
1481	/// <param name="A">Minuend</param>
1482	/// <param name="B">Subtrahend</param>
1483	/// <returns>Result</returns>
1484	public static complex operator -( float A, complex B) {
1485	complex ret;
1486	ret.real = (double) (A - B.real);
1487	ret.imag = - (double) B.imag;
1488	return ret;
1489	}
1490	/// <summary>
1491	/// Multiply two values
1492	/// </summary>
1493	/// <param name="A">First factor</param>
1494	/// <param name="B">Second factor</param>
1495	/// <returns>Result</returns>
1496	public static complex operator *( float A, complex B) {
1497	complex ret;
1498	ret.real = (double) (A * B.real);
1499	ret.imag = (double) (A * B.imag);
1500	return ret;
1501	}
1502	/// <summary>
1503	/// Divide two values
1504	/// </summary>
1505	/// <param name="A">Divident</param>
1506	/// <param name="B">Divisor</param>
1507	/// <returns>Result</returns>
1508	public static complex operator /( float A, complex B) {
1509	complex ret;
1510	if (A == 0) {
1511	if (IsInfinity(B)) return NaN;
1512	} else {
1513	if (IsInfinity(B)) return ( complex )0;
1514	}
1515	if (B.real == 0 && B.imag == 0) {
1516	return INF;
1517	}
1518	// this algorithm is taken from [1]. The one described in [2] was not taken. Tests
1519	// did not show any advantage when using double precision floating point arithmetic.
1520	double tmp;
1521	if (Math.Abs(B.real) >= Math.Abs(B.imag)) {
1522	tmp = (double) (B.imag * (1/B.real));
1523	ret.imag = (double) (B.real + B.imag*tmp);
1524	ret.real = (double) A/ret.imag;
1525	ret.imag = - (double) (A*tmp)/ret.imag;
1526	} else {
1527	tmp = (double) (B.real * (1/B.imag));
1528	ret.imag = (double) (B.imag + B.real*tmp);
1529	ret.real = (double) (A*tmp)/ret.imag;
1530	ret.imag = - (double) A/ret.imag;
1531	}
1532	return ret;
1533	}
1534	/// <summary>
1535	/// Equality comparison for complex numbers
1536	/// </summary>
1537	/// <param name="A">Left side</param>
1538	/// <param name="B">Right side</param>
1539	/// <returns>Result</returns>
1540	/// <remarks>Real inputs are converted to a complex number and the result is compared to the complex input.</remarks>
1541	public static bool operator ==( float A, complex B) {
1542	return (B.real == A && B.imag == 0.0);
1543	}
1544	/// <summary>
1545	/// Unequality comparison for complex numbers
1546	/// </summary>
1547	/// <param name="A">Left side</param>
1548	/// <param name="B">Right side</param>
1549	/// <returns>true if real and imaginary parts of A and B are not equal, false otherwise</returns>
1550	/// <remarks>Real inputs are converted to a complex number and the result is compared to the complex input.</remarks>
1551	public static bool operator !=( float A, complex B) {
1552	return (B.imag != 0.0) \|\| (B.real != A);
1553	}
1554	/// <summary>
1555	/// Greater than comparison for complex numbers
1556	/// </summary>
1557	/// <param name="A">Left side</param>
1558	/// <param name="B">Right side</param>
1559	/// <returns>true if real part of A is greater than real part of B, false otherwise</returns>
1560	/// <remarks>Only the real parts are compared!</remarks>
1561	public static bool operator > ( float A, complex B) {
1562	return (A > B.real);
1563	}
1564	/// <summary>
1565	/// Lower than comparison for complex numbers
1566	/// </summary>
1567	/// <param name="A">Left side</param>
1568	/// <param name="B">Right side</param>
1569	/// <returns>true if real part of A is lower then real part of B, false otherwise</returns>
1570	/// <remarks>Only the real parts are compared!</remarks>
1571	public static bool operator < ( float A, complex B) {
1572	return (A < B.real);
1573	}
1574	/// <summary>
1575	/// Greater than or equal to comparison for complex numbers
1576	/// </summary>
1577	/// <param name="A">Left side</param>
1578	/// <param name="B">Right side</param>
1579	/// <returns>true if real part of A is greater than real part of B, false otherwise</returns>
1580	/// <remarks>Only the real parts are compared!</remarks>
1581	public static bool operator >=( float A, complex B) {
1582	return (A >= B.real);
1583	}
1584	/// <summary>
1585	/// Lower than or equal to comparison for complex numbers
1586	/// </summary>
1587	/// <param name="A">Left side</param>
1588	/// <param name="B">Right side</param>
1589	/// <returns>true if real part of A is lower then real part of B, false otherwise</returns>
1590	/// <remarks>Only the real parts are compared!</remarks>
1591	public static bool operator <=( float A, complex B) {
1592	return (A <= B.real);
1593	}
1594
1595	/// <summary>
1596	/// Add two complex values
1597	/// </summary>
1598	/// <param name="A">First summand</param>
1599	/// <param name="B">Second summand</param>
1600	/// <returns>Result</returns>
1601	public static complex operator +( byte A, complex B) {
1602	complex ret;
1603	ret.real = (double) (A + B.real);
1604	ret.imag = (double) B.imag;
1605	return ret;
1606	}
1607	/// <summary>
1608	/// Subtract two values
1609	/// </summary>
1610	/// <param name="A">Minuend</param>
1611	/// <param name="B">Subtrahend</param>
1612	/// <returns>Result</returns>
1613	public static complex operator -( byte A, complex B) {
1614	complex ret;
1615	ret.real = (double) (A - B.real);
1616	ret.imag = - (double) B.imag;
1617	return ret;
1618	}
1619	/// <summary>
1620	/// Multiply two values
1621	/// </summary>
1622	/// <param name="A">First factor</param>
1623	/// <param name="B">Second factor</param>
1624	/// <returns>Result</returns>
1625	public static complex operator *( byte A, complex B) {
1626	complex ret;
1627	ret.real = (double) (A * B.real);
1628	ret.imag = (double) (A * B.imag);
1629	return ret;
1630	}
1631	/// <summary>
1632	/// Divide two values
1633	/// </summary>
1634	/// <param name="A">Divident</param>
1635	/// <param name="B">Divisor</param>
1636	/// <returns>Result</returns>
1637	public static complex operator /( byte A, complex B) {
1638	complex ret;
1639	if (A == 0) {
1640	if (IsInfinity(B)) return NaN;
1641	} else {
1642	if (IsInfinity(B)) return ( complex )0;
1643	}
1644	if (B.real == 0 && B.imag == 0) {
1645	return INF;
1646	}
1647	// this algorithm is taken from [1]. The one described in [2] was not taken. Tests
1648	// did not show any advantage when using double precision floating point arithmetic.
1649	double tmp;
1650	if (Math.Abs(B.real) >= Math.Abs(B.imag)) {
1651	tmp = (double) (B.imag * (1/B.real));
1652	ret.imag = (double) (B.real + B.imag*tmp);
1653	ret.real = (double) A/ret.imag;
1654	ret.imag = - (double) (A*tmp)/ret.imag;
1655	} else {
1656	tmp = (double) (B.real * (1/B.imag));
1657	ret.imag = (double) (B.imag + B.real*tmp);
1658	ret.real = (double) (A*tmp)/ret.imag;
1659	ret.imag = - (double) A/ret.imag;
1660	}
1661	return ret;
1662	}
1663	/// <summary>
1664	/// Equality comparison for complex numbers
1665	/// </summary>
1666	/// <param name="A">Left side</param>
1667	/// <param name="B">Right side</param>
1668	/// <returns>Result</returns>
1669	/// <remarks>Real inputs are converted to a complex number and the result is compared to the complex input.</remarks>
1670	public static bool operator ==( byte A, complex B) {
1671	return (B.real == A && B.imag == 0.0);
1672	}
1673	/// <summary>
1674	/// Unequality comparison for complex numbers
1675	/// </summary>
1676	/// <param name="A">Left side</param>
1677	/// <param name="B">Right side</param>
1678	/// <returns>true if real and imaginary parts of A and B are not equal, false otherwise</returns>
1679	/// <remarks>Real inputs are converted to a complex number and the result is compared to the complex input.</remarks>
1680	public static bool operator !=( byte A, complex B) {
1681	return (B.imag != 0.0) \|\| (B.real != A);
1682	}
1683	/// <summary>
1684	/// Greater than comparison for complex numbers
1685	/// </summary>
1686	/// <param name="A">Left side</param>
1687	/// <param name="B">Right side</param>
1688	/// <returns>true if real part of A is greater than real part of B, false otherwise</returns>
1689	/// <remarks>Only the real parts are compared!</remarks>
1690	public static bool operator > ( byte A, complex B) {
1691	return (A > B.real);
1692	}
1693	/// <summary>
1694	/// Lower than comparison for complex numbers
1695	/// </summary>
1696	/// <param name="A">Left side</param>
1697	/// <param name="B">Right side</param>
1698	/// <returns>true if real part of A is lower then real part of B, false otherwise</returns>
1699	/// <remarks>Only the real parts are compared!</remarks>
1700	public static bool operator < ( byte A, complex B) {
1701	return (A < B.real);
1702	}
1703	/// <summary>
1704	/// Greater than or equal to comparison for complex numbers
1705	/// </summary>
1706	/// <param name="A">Left side</param>
1707	/// <param name="B">Right side</param>
1708	/// <returns>true if real part of A is greater than real part of B, false otherwise</returns>
1709	/// <remarks>Only the real parts are compared!</remarks>
1710	public static bool operator >=( byte A, complex B) {
1711	return (A >= B.real);
1712	}
1713	/// <summary>
1714	/// Lower than or equal to comparison for complex numbers
1715	/// </summary>
1716	/// <param name="A">Left side</param>
1717	/// <param name="B">Right side</param>
1718	/// <returns>true if real part of A is lower then real part of B, false otherwise</returns>
1719	/// <remarks>Only the real parts are compared!</remarks>
1720	public static bool operator <=( byte A, complex B) {
1721	return (A <= B.real);
1722	}
1723
1724	#endregion HYCALPER AUTO GENERATED CODE
1725
1726	#region unary minus
1727	/// <summary>
1728	/// Unary minus operator
1729	/// </summary>
1730	/// <param name="A">Complex input</param>
1731	/// <returns>Complex number similar to A, having real and imag part negated</returns>
1732	public static complex operator -( complex A) {
1733	complex ret = new complex();
1734	ret.imag = -A.imag;
1735	ret.real = -A.real;
1736	return ret;
1737	}
1738	#endregion
1739
1740	#region CAST_OPERATORS
1741	/// <summary>
1742	/// Cast value to complex number
1743	/// </summary>
1744	/// <param name="a">Value to cast</param>
1745	/// <returns>Complex number with the real part having the same value as a and the imaginary part is 0.</returns>
1746	public static implicit operator complex(double a) {
1747	return new complex(a, 0.0);
1748	}
1749	/// <summary>
1750	/// Cast value to complex number
1751	/// </summary>
1752	/// <param name="a">Value to cast</param>
1753	/// <returns>Complex number with the real part having the same value as the a and the imaginary part is 0.</returns>
1754	public static implicit operator complex(float a) {
1755	return new complex(a, 0.0);
1756	}
1757	/// <summary>
1758	/// Cast value to complex number
1759	/// </summary>
1760	/// <param name="a">Value to cast</param>
1761	/// <returns>Complex number being a copy of the real and imaginary parts of a.</returns>
1762	public static implicit operator complex(fcomplex a) {
1763	return new complex(a.real, a.imag);
1764	}
1765	/// <summary>
1766	/// Cast value to complex number
1767	/// </summary>
1768	/// <param name="a">Value to cast</param>
1769	/// <returns>Complex number with the real part having the same value as a and the imaginary part is 0.</returns>
1770	public static implicit operator complex(byte a) {
1771	return new complex(a, 0.0);
1772	}
1773	/// <summary>
1774	/// Cast value to complex number
1775	/// </summary>
1776	/// <param name="a">Value to cast</param>
1777	/// <returns>Complex number with the real part having the same value as a and the imaginary part is 0.</returns>
1778	public static implicit operator complex(Int32 a) {
1779	return new complex(a, 0.0);
1780	}
1781	/// <summary>
1782	/// Cast value to complex number
1783	/// </summary>
1784	/// <param name="a">Value to cast</param>
1785	/// <returns>Complex number with the real part having the same value as a and the imaginary part is 0.</returns>
1786	public static implicit operator complex(Int64 a) {
1787	return new complex(a, 0.0);
1788	}
1789
1790	/// <summary>
1791	/// Cast value from complex number
1792	/// </summary>
1793	/// <param name="a">Complex value to cast</param>
1794	/// <returns>Double number with the real part of a </returns>
1795	public static explicit operator double(complex a) {
1796	return a.real;
1797	}
1798	/// <summary>
1799	/// Cast value from complex number
1800	/// </summary>
1801	/// <param name="a">Complex value to cast</param>
1802	/// <returns>number with the real part of a </returns>
1803	/// <remarks>the return value is the result of a cast from double to float.</remarks>
1804	public static explicit operator float(complex a) {
1805	return (float)a.real;
1806	}
1807	/// <summary>
1808	/// Cast value from complex number
1809	/// </summary>
1810	/// <param name="a">Complex value to cast</param>
1811	/// <returns>float complex number with the real and imaginary parts being a copy of a </returns>
1812	/// <remarks>The real and imaginary parts are the result of a cast to float.</remarks>
1813	public static explicit operator fcomplex(complex a) {
1814	return new fcomplex((float)a.real, (float)a.imag);
1815	}
1816	/// <summary>
1817	/// Cast value from complex number
1818	/// </summary>
1819	/// <param name="a">Complex value to cast</param>
1820	/// <returns>Number with the real part of a </returns>
1821	/// <remarks>The return value is the result of a cast to byte.</remarks>
1822	public static explicit operator byte(complex a) {
1823	return (byte) a.real;
1824	}
1825	/// <summary>
1826	/// Cast value from complex number
1827	/// </summary>
1828	/// <param name="a">Complex value to cast</param>
1829	/// <returns>Number with the real part of a </returns>
1830	/// <remarks>The return value is the result of a cast to Int32.</remarks>
1831	public static explicit operator Int32(complex a) {
1832	return (Int32) a.real;
1833	}
1834	/// <summary>
1835	/// Cast value from complex number
1836	/// </summary>
1837	/// <param name="a">Complex value to cast</param>
1838	/// <returns>number with the real part of a </returns>
1839	/// <remarks>the return value is the result of a cast to Int64.</remarks>
1840	public static explicit operator Int64(complex a) {
1841	return (Int64) a.real;
1842	}
1843
1844	public static implicit operator complex(System.Numerics.Complex a) {
1845	return new complex(a.Real, a.Imaginary);
1846	}
1847	#endregion CAST_OPERATORS
1848
1849	#region Functions Basic Math
1850	/// <summary>
1851	/// Absolute value of input
1852	/// </summary>
1853	/// <param name="input">Input value</param>
1854	/// <returns>The absolute value of the input</returns>
1855	public static double Abs(complex input) {
1856	return Math.Sqrt(input.real * input.real + input.imag * input.imag);
1857	}
1858	/// <summary>
1859	/// Phase angle of complex number
1860	/// </summary>
1861	/// <param name="input">Input value</param>
1862	/// <returns>The phase angle of the input</returns>
1863	/// <remarks>For the result the Atan2 function of the <see cref="Math"/> class is used.</remarks>
1864	public static double Angle(complex input) {
1865	return Math.Atan2(input.imag, input.real);
1866	}
1867	/// <summary>
1868	/// Arcus tangens of complex input
1869	/// </summary>
1870	/// <param name="input">Complex input</param>
1871	/// <returns>Arcus tangens of complex input</returns>
1872	/// <remarks></remarks>
1873	public static complex Atan(complex input) {
1874	complex ret = new complex(0, (float)0.5);
1875	return (ret * Log((complex.i + input) / (complex.i - input)));
1876	}
1877	/// <summary>
1878	/// Arcus cosinus of complex input
1879	/// </summary>
1880	/// <param name="input">Complex input</param>
1881	/// <returns>Arcus cosinus of input</returns>
1882	public static complex Acos(complex input) {
1883	complex ni = complex.i * -1.0;
1884	return complex.Log(complex.Sqrt(input * input - 1)
1885	+ input) * ni;
1886	}
1887	/// <summary>
1888	/// Arcus cosinus of input
1889	/// </summary>
1890	/// <param name="input">Input value</param>
1891	/// <returns>Arcus cosinus of input</returns>
1892	public static complex Acos(double input) {
1893	if (Math.Abs(input) <= 1.0)
1894	return new complex(Math.Acos(input), 0.0);
1895	else {
1896	return Acos((complex)input);
1897	}
1898	}
1899	/// <summary>
1900	/// Arcus sinus of complex input
1901	/// </summary>
1902	/// <param name="input">Input value</param>
1903	/// <returns>Arcus sinus of input</returns>
1904	public static complex Asin(double input) {
1905	if (Math.Abs(input) <= 1.0)
1906	return new complex(Math.Asin(input), 0.0);
1907	else {
1908	return Asin((complex)input);
1909	}
1910	}
1911	/// <summary>
1912	/// Arcus sinus of input
1913	/// </summary>
1914	/// <param name="input">Input value</param>
1915	/// <returns>Arcus sinus of input</returns>
1916	public static complex Asin(complex input) {
1917	complex ret = Acos(input);
1918	ret.real = Math.PI / 2 - ret.real;
1919	return ret;
1920	}
1921	/// <summary>
1922	/// Round towards positive infinity
1923	/// </summary>
1924	/// <param name="input">Input value</param>
1925	/// <returns>Result is the next integer value greater then input</returns>
1926	/// <remarks>ILMath.Ceiling operates in both: real and imaginary parts seperately</remarks>
1927	public static complex Ceiling (complex input){
1928	return new complex(
1929	Math.Ceiling(input.real),
1930	Math.Ceiling(input.imag)
1931	);
1932	}
1933	/// <summary>
1934	/// Round towards negative infinity
1935	/// </summary>
1936	/// <param name="input">Input value</param>
1937	/// <returns>Result is the next integer value lower then input</returns>
1938	/// <remarks>ILMath.Floor operates in both: real and imaginary parts seperately</remarks>
1939	public static complex Floor (complex input){
1940	return new complex(
1941	Math.Floor(input.real),
1942	Math.Floor(input.imag)
1943	);
1944	}
1945	/// <summary>
1946	/// Rounds towards nearest integer
1947	/// </summary>
1948	/// <param name="input">Input value</param>
1949	/// <returns>Result is the nearest integer value for input</returns>
1950	/// <remarks>ILMath.Round operates in both: real and imaginary parts deperately</remarks>
1951	public static complex Round (complex input){
1952	return new complex(
1953	Math.Round(input.real),
1954	Math.Round(input.imag)
1955	);
1956	}
1957	/// <summary>
1958	/// Signum function
1959	/// </summary>
1960	/// <param name="input">Complex input </param>
1961	/// <returns>Sesult as input / Abs(input)</returns>
1962	/// <remarks>Sign(input) with input being complex returns the projection onto
1963	/// the unit circle. If input is 0+0i the result will be 0+0i.</remarks>
1964	public static complex Sign (complex input){
1965	if (input.real == 0.0 && input.imag == 0.0)
1966	return new complex();
1967	else {
1968	double mag = Math.Sqrt(input.real * input.real + input.imag * input.imag);
1969	return new complex(
1970	input.real / mag,
1971	input.imag / mag);
1972	}
1973	}
1974	/// <summary>
1975	/// Truncate a floating point complex value
1976	/// </summary>
1977	/// <param name="input">Input value</param>
1978	/// <returns>Integer part of input</returns>
1979	/// <remarks>Operates on real and imaginary parts seperately.</remarks>
1980	public static complex Truncate (complex input){
1981	return new complex(
1982	Math.Truncate(input.real),
1983	Math.Truncate(input.imag)
1984	);
1985	}
1986	/// <summary>
1987	/// Cosinus
1988	/// </summary>
1989	/// <param name="input">Input value</param>
1990	/// <returns>Cosine of input</returns>
1991	/// <remarks><para>The cosine is computed by the trigonometric euler equation: </para>
1992	/// <para>0.5 * [exp(i input) + exp(-i input)]</para></remarks>
1993	public static complex Cos(complex input) {
1994	complex i = new complex(0, 1.0);
1995	complex mi = new complex(0, -1.0);
1996	return (Exp(i * input) + Exp(mi * input)) / 2.0;
1997	}
1998	/// <summary>
1999	/// Cosinus hyperbolicus
2000	/// </summary>
2001	/// <param name="input">Input</param>
2002	/// <returns>Cosine hyperbolicus of input</returns>
2003	/// <remarks><para>The cosine is computed by the trigonometric euler equation: </para>
2004	/// <para>(Exp(input) + Exp(-1.0 * input)) / 2.0</para></remarks>
2005	public static complex Cosh(complex input) {
2006	return (Exp(input) + Exp(-1.0 * input)) / 2.0;
2007	}
2008	/// <summary>
2009	/// Sinus
2010	/// </summary>
2011	/// <param name="input">Input value</param>
2012	/// <returns>Sinus of input</returns>
2013	/// <remarks><para>The sinus is computed by the trigonometric euler equation: </para>
2014	/// <para>(Exp(i * input) - Exp(-1.0 * i * input)) / (2.0 * i)</para></remarks>
2015	public static complex Sin(complex input) {
2016	complex i = new complex(0, 1.0);
2017	complex mi = new complex(0, -1.0);
2018	return (Exp(i * input) - Exp(mi * input)) / (2.0 * i);
2019	}
2020	/// <summary>
2021	/// Sinus hyperbolicus
2022	/// </summary>
2023	/// <param name="input">Input</param>
2024	/// <returns>Sinus hyperbolicus of input</returns>
2025	/// <remarks><para>The sinus hyperbolicus is computed by the trigonometric euler equation: </para>
2026	/// <para>(Exp(input) - Exp(-1.0 * input)) / 2.0</para></remarks>
2027	public static complex Sinh(complex input) {
2028	return (Exp(input) - Exp(-1.0 * input)) / 2.0;
2029	}
2030	/// <summary>
2031	/// Complex exponent
2032	/// </summary>
2033	/// <param name="exponent">Exponent</param>
2034	/// <returns>Result of exp(exponent)</returns>
2035	/// <remarks>For complex exponents, exp(exponent) is computed by
2036	/// <para>complex.FromPol(Math.Exp(exponent.real), exponent.imag)</para></remarks>
2037	public static complex Exp(complex exponent) {
2038	return complex.FromPol(Math.Exp(exponent.real), exponent.imag);
2039	}
2040	/// <summary>
2041	/// Complex power for real exponent
2042	/// </summary>
2043	/// <param name="input">Basis</param>
2044	/// <param name="exponent">Exponent</param>
2045	/// <returns>Result of input power exponent</returns>
2046	/// <remarks>The computation will be carried out by
2047	/// <para>exp(log(input) * exponent)</para></remarks>
2048	public static complex Pow(complex input, double exponent) {
2049	complex ret = input.Log();
2050	ret.imag *= exponent;
2051	ret.real *= exponent;
2052	return ret.Exp();
2053	}
2054	/// <summary>
2055	/// Complex power - real basis, real exponent
2056	/// </summary>
2057	/// <param name="basis">Basis</param>
2058	/// <param name="exponent">Exponent</param>
2059	/// <returns>Complex number.</returns>
2060	/// <remarks>The result will be a complex number. For negative basis
2061	/// the basis will be converted to a complex number and the power
2062	/// will be computed in the complex plane.</remarks>
2063	public static complex Pow(double basis, double exponent) {
2064	if (basis >= 0.0)
2065	return Math.Pow(basis, exponent);
2066	else
2067	return Pow((complex)basis, exponent);
2068	}
2069	/// <summary>
2070	/// Complex power - complex exponent
2071	/// </summary>
2072	/// <param name="basis">Basis</param>
2073	/// <param name="exponent">Exponent</param>
2074	/// <returns>Complex number exp(log(basis) * exponent).</returns>
2075	/// <remarks>The result will be the complex number exp(log(basis) * exponent). </remarks>
2076	public static complex Pow(complex basis, complex exponent) {
2077	complex ret = (basis.Log() * exponent);
2078	return ret.Exp();
2079	}
2080	/// <summary>
2081	/// Square root
2082	/// </summary>
2083	/// <param name="input">Input value</param>
2084	/// <returns>The square root of input</returns>
2085	/// <remarks>If input is smaller than 0.0, the computation will be done in the complex plane. </remarks>
2086	public static complex Sqrt(double input) {
2087	if (input > 0.0)
2088	return new complex(Math.Sqrt(input), 0.0);
2089	else
2090	return Sqrt(input);
2091	}
2092	/// <summary>
2093	/// Square root
2094	/// </summary>
2095	/// <param name="input">Input value</param>
2096	/// <returns>The square root of input</returns>
2097	/// <remarks>Numerical recipes in C: Appendix C </remarks>
2098	public static complex Sqrt(complex input) {
2099	// Reference : numerical recipes in C: Appendix C
2100	complex ret = new complex();
2101	double x, y, w, r;
2102	if (input.real == 0.0 && input.imag == 0.0)
2103	return ret;
2104	else {
2105	x = (double)Math.Abs(input.real);
2106	y = (double)Math.Abs(input.imag);
2107	if (x >= y) {
2108	r = y / x;
2109	w = Math.Sqrt(x) * Math.Sqrt(0.5 * (1.0 + Math.Sqrt(1.0 + r * r)));
2110	} else {
2111	r = x / y;
2112	w = Math.Sqrt(y) * Math.Sqrt(0.5 * (r + Math.Sqrt(1.0 + r * r)));
2113	}
2114	if (input.real >= 0.0) {
2115	ret.real = w;
2116	ret.imag = input.imag / (2.0 * w);
2117	} else {
2118	ret.imag = (input.imag >= 0) ? w : -w;
2119	ret.real = input.imag / (2.0 * ret.imag);
2120	}
2121	return ret;
2122	}
2123	}
2124	/// <summary>
2125	/// Tangens
2126	/// </summary>
2127	/// <param name="input">Input value</param>
2128	/// <returns>Tangens of input</returns>
2129	/// <remarks>The tangens is
2130	/// <para>sin(input) / cos(input)</para>
2131	/// if cos(input) == 0.0+0.0i, INF will be returned.</remarks>
2132	public static complex Tan(complex input) {
2133	complex ci = Cos(input);
2134	if (ci.real == 0.0 && ci.imag == 0.0)
2135	return INF;
2136	return (Sin(input) / ci);
2137	}
2138	/// <summary>
2139	/// Tangens hyperbolicus
2140	/// </summary>
2141	/// <param name="input">Input value</param>
2142	/// <returns>Tangens hyperbolicus</returns>
2143	/// <remarks>The tangens hyperbolicus is
2144	/// <para>sinh(input) / cosh(input)</para>
2145	/// if cosh(input) == 0.0+0.0i, INF will be returned.</remarks>
2146	public static complex Tanh(complex input) {
2147	complex si = Cosh(input);
2148	if (si.real == 0.0 && si.imag == 0.0)
2149	return INF;
2150	return (Sinh(input) / si);
2151	}
2152	/// <summary>
2153	/// Complex logarithm
2154	/// </summary>
2155	/// <param name="input">Input value</param>
2156	/// <returns>Complex logarithm of input</returns>
2157	/// <remarks>The real part of the logarithm is computed by
2158	/// <para>log (abs (input))</para>
2159	/// <para>The imaginary part holds the phase of input.</para>
2160	/// </remarks>
2161	public static complex Log(complex input) {
2162	complex ret = new complex();
2163	ret.real = Math.Log(Math.Sqrt(input.real * input.real + input.imag * input.imag));
2164	ret.imag = Math.Atan2(input.imag, input.real);
2165	return ret;
2166	}
2167	/// <summary>
2168	/// Logarithm of real input
2169	/// </summary>
2170	/// <param name="input">Input value - may be negative</param>
2171	/// <returns>Complex logarithm</returns>
2172	public static complex Log(double input) {
2173	return Log (new complex(input,0.0));
2174	}
2175	/// <summary>
2176	/// Logarithm of base 10 of real input
2177	/// </summary>
2178	/// <param name="input">Input value - may be negative</param>
2179	/// <returns>Complex logarithm of base 10</returns>
2180	public static complex Log10(double input) {
2181	return Log(new complex(input,0.0)) * 0.43429448190325176;
2182	}
2183	/// <summary>
2184	/// Logarithm of base 2 of real input
2185	/// </summary>
2186	/// <param name="input">Input value - may be negative</param>
2187	/// <returns>Complex logarithm of base 2</returns>
2188	public static complex Log2(double input) {
2189	return Log(new complex(input,0.0)) * 1.4426950408889641;
2190	}
2191	/// <summary>
2192	/// Logarithm of base 10
2193	/// </summary>
2194	/// <param name="input">Input value</param>
2195	/// <returns>Logarithm of base 10</returns>
2196	/// <seealso cref="ILNumerics.complex.Log(complex)"/>
2197	public static complex Log10(complex input) {
2198	return Log(input) * 0.43429448190325176;
2199	}
2200	/// <summary>
2201	/// Logarithm of base 2
2202	/// </summary>
2203	/// <param name="input">Input value</param>
2204	/// <returns>Logarithm of base 2.</returns>
2205	/// <seealso cref="ILNumerics.complex.Log(complex)"/>
2206	public static complex Log2(complex input) {
2207	return Log(input) * 1.4426950408889634;
2208	}
2209	/// <summary>
2210	/// Convert polar notation into cartesian notation
2211	/// </summary>
2212	/// <param name="magnitude">Magnitude</param>
2213	/// <param name="angle">Phase</param>
2214	/// <returns>Complex value having magnitude and phase</returns>
2215	public static complex FromPol(double magnitude, double angle) {
2216	return new complex(
2217	magnitude * Math.Cos(angle),
2218	magnitude * Math.Sin(angle)
2219	);
2220	}
2221	/// <summary>
2222	/// Convert to string
2223	/// </summary>
2224	/// <returns>String displaying the comlex number (full precision)</returns>
2225	public override String ToString() {
2226	if (imag>=0)
2227	return String.Format("{0} + {1}i",real,imag);
2228	else
2229	return String.Format("{0} - {1}i",real,-imag);
2230	}
2231	private static string m_precSpecI = "";
2232	private static string m_precSpecR = "";
2233	private static int m_lastDigits = 0;
2234	/// <summary>
2235	/// Print formated output of this number, determine number of digits
2236	/// </summary>
2237	/// <param name="digits">Number of digits</param>
2238	/// <returns>Formated output</returns>
2239	public string ToString(int digits) {
2240	if (digits < 1) return "";
2241	if (digits != m_lastDigits) {
2242	m_lastDigits = digits;
2243	m_precSpecR = String.Format("{{0:f{0}}}",digits);
2244	m_precSpecI = String.Format("{{1:f{0}}}i",digits);
2245	}
2246	if (imag >= 0) {
2247	return String.Format(m_precSpecR+"+"+m_precSpecI,real,imag);
2248	} else {
2249	return String.Format(m_precSpecR+"-"+m_precSpecI,real,-imag);
2250	}
2251	}
2252	/// <summary>
2253	/// Magnitude of this complex instance
2254	/// </summary>
2255	/// <returns>Magnitude</returns>
2256	public double Abs() {
2257	return Math.Sqrt(real * real + imag * imag);
2258	}
2259	/// <summary>
2260	/// Phase of this complex instance
2261	/// </summary>
2262	/// <returns>Phase</returns>
2263	public double Angle() {
2264	return Math.Atan2(imag, real);
2265	}
2266	/// <summary>
2267	/// Arcus cosinus of this complex instance
2268	/// </summary>
2269	/// <returns>Arcus cosinus</returns>
2270	public complex Acos() {
2271	complex ret = new complex(0, -1);
2272	return complex.Log(complex.Sqrt(this * this - 1)
2273	+ this) * ret;
2274	}
2275	/// <summary>
2276	/// Arcus sinus of this complex instance
2277	/// </summary>
2278	/// <returns>arcus sinus</returns>
2279	public complex Asin() {
2280	complex ret = Acos(this);
2281	ret.real = Math.PI / 2 - ret.real;
2282	return ret;
2283	}
2284	/// <summary>
2285	/// Exponential / power of base e
2286	/// </summary>
2287	/// <returns>Power of base e</returns>
2288	public complex Exp() {
2289	return complex.FromPol(Math.Exp(real), imag);
2290	}
2291	/// <summary>
2292	/// Complex power real exponent
2293	/// </summary>
2294	/// <param name="exponent">Exponent</param>
2295	/// <returns>New complex number with result</returns>
2296	/// <remarks>If this instance is a and the exponent is e than
2297	/// the result will be the complex number exp(log(a) * e). </remarks>
2298	public complex Pow(double exponent) {
2299	complex ret = Log();
2300	ret.imag *= exponent;
2301	ret.real *= exponent;
2302	return ret.Exp();
2303	}
2304	/// <summary>
2305	/// Complex power - complex exponent
2306	/// </summary>
2307	/// <param name="exponent">Exponent</param>
2308	/// <returns>Complex number exp(log(this) * exponent).</returns>
2309	/// <remarks>If this instance is a than
2310	/// the result will be the complex number exp(log(a) * exponent). </remarks>
2311	public complex Pow(complex exponent) {
2312	complex ret = (Log() * exponent);
2313	return ret.Exp();
2314	}
2315	/// <summary>
2316	/// Square root of this complex value
2317	/// </summary>
2318	/// <returns>Square root of this complex value</returns>
2319	public complex Sqrt() {
2320	// Reference : numerical recipes in C: Appendix C
2321	complex ret = new complex();
2322	double x, y, w, r;
2323	if (real == 0.0 && imag == 0.0)
2324	return ret;
2325	else {
2326	x = (double)Math.Abs(real);
2327	y = (double)Math.Abs(imag);
2328	if (x >= y) {
2329	r = y / x;
2330	w = Math.Sqrt(x) * Math.Sqrt(0.5 * (1.0 + Math.Sqrt(1.0 + r * r)));
2331	} else {
2332	r = x / y;
2333	w = Math.Sqrt(y) * Math.Sqrt(0.5 * (r + Math.Sqrt(1.0 + r * r)));
2334	}
2335	if (real >= 0.0) {
2336	ret.real = w;
2337	ret.imag = imag / (2.0 * w);
2338	} else {
2339	ret.imag = (imag >= 0) ? w : -w;
2340	ret.real = imag / ( 2.0 * ret.imag );
2341	}
2342	return ret;
2343	}
2344	}
2345	/// <summary>
2346	/// Logarithm of base e
2347	/// </summary>
2348	/// <returns>Logarithm of base e</returns>
2349	/// <remarks>The logarithm of a complex number A is defined as follows: <br />
2350	/// <list type="none"><item>real part: log(abs(A))</item>
2351	/// <item>imag part: Atan2(imag(A),real(A))</item></list>
2352	/// </remarks>
2353	public complex Log() {
2354	complex ret = new complex();
2355	ret.real = Math.Log(Math.Sqrt(real * real + imag * imag));
2356	ret.imag = Math.Atan2(imag, real);
2357	return ret;
2358	}
2359	/// <summary>
2360	/// Test if any of real or imaginary parts are NAN's
2361	/// </summary>
2362	/// <param name="input">Complex number to test</param>
2363	/// <returns>true if any of real or imag part is not a number</returns>
2364	public static bool IsNaN(complex input) {
2365	if (double.IsNaN(input.real) \|\| double.IsNaN(input.imag))
2366	return true;
2367	else
2368	return false;
2369	}
2370	/// <summary>
2371	/// Test if any of real or imaginary parts are infinite
2372	/// </summary>
2373	/// <param name="input">Complex number to test</param>
2374	/// <returns>true if any of real or imag part is infinite</returns>
2375	public static bool IsInfinity(complex input) {
2376	if (double.IsInfinity(input.real) \|\| double.IsInfinity(input.imag))
2377	return true;
2378	else
2379	return false;
2380	}
2381	/// <summary>
2382	/// Test if any of real or imaginary parts are pos.nfinite
2383	/// </summary>
2384	/// <param name="input">Complex number to test</param>
2385	/// <returns>true if any of real or imag part is positive infinite</returns>
2386	public static bool IsPositiveInfinity(complex input) {
2387	if (double.IsPositiveInfinity(input.real) \|\| double.IsPositiveInfinity(input.imag))
2388	return true;
2389	else
2390	return false;
2391	}
2392	/// <summary>
2393	/// Test if any of real or imaginary parts are neg. infinite
2394	/// </summary>
2395	/// <param name="input">Complex number to test</param>
2396	/// <returns>true if any of real or imag part is negative infinite</returns>
2397	public static bool IsNegativeInfinity(complex input) {
2398	if (double.IsNegativeInfinity(input.real) \|\| double.IsNegativeInfinity(input.imag))
2399	return true;
2400	else
2401	return false;
2402	}
2403	/// <summary>
2404	/// Test if any of real or imaginary parts are finite
2405	/// </summary>
2406	/// <param name="input">Complex number to test</param>
2407	/// <returns>true if any of real and imag part is finite</returns>
2408	public static bool IsFinite (complex input) {
2409	if (ILMath.isfinite(input.real) && ILMath.isfinite(input.imag))
2410	return true;
2411	else
2412	return false;
2413	}
2414	/// <summary>
2415	/// Test if both of real or imaginary parts are 0
2416	/// </summary>
2417	/// <returns>true if real and imag part is 0</returns>
2418	public bool iszero() {
2419	if (real == 0.0 && imag == 0.0)
2420	return true;
2421	else
2422	return false;
2423	}
2424	#endregion Functions Basic Math
2425	}
2426
2427	}

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