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1	/*************************************************************************
2	Copyright (c) 1992-2007 The University of Tennessee. All rights reserved.
3
4	Contributors:
5	* Sergey Bochkanov (ALGLIB project). Translation from FORTRAN to
6	pseudocode.
7
8	See subroutines comments for additional copyrights.
9
10	>>> SOURCE LICENSE >>>
11	This program is free software; you can redistribute it and/or modify
12	it under the terms of the GNU General Public License as published by
13	the Free Software Foundation (www.fsf.org); either version 2 of the
14	License, or (at your option) any later version.
15
16	This program is distributed in the hope that it will be useful,
17	but WITHOUT ANY WARRANTY; without even the implied warranty of
18	MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
19	GNU General Public License for more details.
20
21	A copy of the GNU General Public License is available at
22	http://www.fsf.org/licensing/licenses
23
24	>>> END OF LICENSE >>>
25	*************************************************************************/
26
27	using System;
28
29	namespace alglib
30	{
31	public class ldlt
32	{
33	/*************************************************************************
34	LDLTDecomposition of a symmetric matrix
35
36	The algorithm represents a symmetric matrix (which is not necessarily
37	positive definite) as A=LDL' or A = UDU', where D is a block-diagonal
38	matrix with blocks 1x1 or 2x2, matrix L (matrix U) is a product of lower
39	(upper) triangular matrices with unit diagonal and permutation matrices.
40
41	Input parameters:
42	A - factorized matrix, array with elements [0..N-1, 0..N-1].
43	If IsUpper True, then the upper triangle contains
44	elements of symmetric matrix A, and the lower triangle is
45	not used.
46	The same applies if IsUpper = False.
47	N - size of factorized matrix.
48	IsUpper - parameter which shows a method of matrix definition (lower
49	or upper triangle).
50
51	Output parameters:
52	A - matrices D and U, if IsUpper = True, or L, if IsUpper = False,
53	in compact form, replacing the upper (lower) triangle of
54	matrix A. In that case, the elements under (over) the main
55	diagonal are not used nor modified.
56	Pivots - tables of performed permutations (see below).
57
58	If IsUpper = True, then A = UDU', U = P(n)U(n)...P(k)U(k), where
59	P(k) is the permutation matrix, U(k) - upper triangular matrix with its
60	unit main diagonal and k decreases from n with step s which is equal to
61	1 or 2 (according to the size of the blocks of matrix D).
62
63	( I v 0 ) k-s+1
64	U(k) = ( 0 I 0 ) s
65	( 0 0 I ) n-k-1
66	k-s+1 s n-k-1
67
68	If Pivots[k]>=0, then s=1, P(k) - permutation of rows k and Pivots[k], the
69	vectorv forming matrix U(k) is stored in elements A(0:k-1,k), D(k) replaces
70	A(k,k). If Pivots[k]=Pivots[k-1]<0 then s=2, P(k) - permutation of rows k-1
71	and N+Pivots[k-1], the vector v forming matrix U(k) is stored in elements
72	A(0:k-1,k:k+1), the upper triangle of block D(k) is stored in A(k,k),
73	A(k,k+1) and A(k+1,k+1).
74
75	If IsUpper = False, then A = LDL', L=P(0)L(0)...P(k)L(k), where P(k)
76	is the permutation matrix, L(k) lower triangular matrix with unit main
77	diagonal and k decreases from 1 with step s which is equal to 1 or 2
78	(according to the size of the blocks of matrix D).
79
80	( I 0 0 ) k-1
81	L(k) = ( 0 I 0 ) s
82	( 0 v I ) n-k-s+1
83	k-1 s n-k-s+1
84
85	If Pivots[k]>=0 then s=1, P(k) permutation of rows k and Pivots[k], the
86	vector v forming matrix L(k) is stored in elements A(k+1:n-1,k), D(k)
87	replaces A(k,k). If Pivots[k]=Pivots[k+1]<0 then s=2, P(k) - permutation
88	of rows k+1 and N+Pivots[k+1], the vector v forming matrix L(k) is stored
89	in elements A(k+2:n-1,k:k+1), the lower triangle of block D(k) is stored in
90	A(k,k), A(k+1,k) and A(k+1,k+1).
91
92	-- LAPACK routine (version 3.0) --
93	Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
94	Courant Institute, Argonne National Lab, and Rice University
95	June 30, 1999
96	*************************************************************************/
97	public static void smatrixldlt(ref double[,] a,
98	int n,
99	bool isupper,
100	ref int[] pivots)
101	{
102	int i = 0;
103	int imax = 0;
104	int j = 0;
105	int jmax = 0;
106	int k = 0;
107	int kk = 0;
108	int kp = 0;
109	int kstep = 0;
110	double absakk = 0;
111	double alpha = 0;
112	double colmax = 0;
113	double d11 = 0;
114	double d12 = 0;
115	double d21 = 0;
116	double d22 = 0;
117	double r1 = 0;
118	double rowmax = 0;
119	double t = 0;
120	double wk = 0;
121	double wkm1 = 0;
122	double wkp1 = 0;
123	int ii = 0;
124	int i1 = 0;
125	int i2 = 0;
126	double vv = 0;
127	double[] temp = new double[0];
128	int i_ = 0;
129
130	pivots = new int[n-1+1];
131	temp = new double[n-1+1];
132
133	//
134	// Initialize ALPHA for use in choosing pivot block size.
135	//
136	alpha = (1+Math.Sqrt(17))/8;
137	if( isupper )
138	{
139
140	//
141	// Factorize A as UDU' using the upper triangle of A
142	//
143	//
144	// K is the main loop index, decreasing from N to 1 in steps of
145	// 1 or 2
146	//
147	k = n-1;
148	while( k>=0 )
149	{
150	kstep = 1;
151
152	//
153	// Determine rows and columns to be interchanged and whether
154	// a 1-by-1 or 2-by-2 pivot block will be used
155	//
156	absakk = Math.Abs(a[k,k]);
157
158	//
159	// IMAX is the row-index of the largest off-diagonal element in
160	// column K+1, and COLMAX is its absolute value
161	//
162	if( k>0 )
163	{
164	imax = 1;
165	for(ii=2; ii<=k; ii++)
166	{
167	if( (double)(Math.Abs(a[ii-1,k]))>(double)(Math.Abs(a[imax-1,k])) )
168	{
169	imax = ii;
170	}
171	}
172	colmax = Math.Abs(a[imax-1,k]);
173	}
174	else
175	{
176	colmax = 0;
177	}
178	if( (double)(Math.Max(absakk, colmax))==(double)(0) )
179	{
180
181	//
182	// Column K is zero
183	//
184	kp = k;
185	}
186	else
187	{
188	if( (double)(absakk)>=(double)(alpha*colmax) )
189	{
190
191	//
192	// no interchange, use 1-by-1 pivot block
193	//
194	kp = k;
195	}
196	else
197	{
198
199	//
200	// JMAX is the column-index of the largest off-diagonal
201	// element in row IMAX, and ROWMAX is its absolute value
202	//
203	jmax = imax+1;
204	for(ii=imax+2; ii<=k+1; ii++)
205	{
206	if( (double)(Math.Abs(a[imax-1,ii-1]))>(double)(Math.Abs(a[imax-1,jmax-1])) )
207	{
208	jmax = ii;
209	}
210	}
211	rowmax = Math.Abs(a[imax-1,jmax-1]);
212	if( imax>1 )
213	{
214	jmax = 1;
215	for(ii=2; ii<=imax-1; ii++)
216	{
217	if( (double)(Math.Abs(a[ii-1,imax-1]))>(double)(Math.Abs(a[jmax-1,imax-1])) )
218	{
219	jmax = ii;
220	}
221	}
222	rowmax = Math.Max(rowmax, Math.Abs(a[jmax-1,imax-1]));
223	}
224	vv = colmax/rowmax;
225	if( (double)(absakk)>=(double)(alphacolmaxvv) )
226	{
227
228	//
229	// no interchange, use 1-by-1 pivot block
230	//
231	kp = k;
232	}
233	else
234	{
235	if( (double)(Math.Abs(a[imax-1,imax-1]))>=(double)(alpha*rowmax) )
236	{
237
238	//
239	// interchange rows and columns K and IMAX, use 1-by-1
240	// pivot block
241	//
242	kp = imax-1;
243	}
244	else
245	{
246
247	//
248	// interchange rows and columns K-1 and IMAX, use 2-by-2
249	// pivot block
250	//
251	kp = imax-1;
252	kstep = 2;
253	}
254	}
255	}
256	kk = k+1-kstep;
257	if( kp+1!=kk+1 )
258	{
259
260	//
261	// Interchange rows and columns KK and KP+1 in the leading
262	// submatrix A(0:K,0:K)
263	//
264	for(i_=0; i_<=kp-1;i_++)
265	{
266	temp[i_] = a[i_,kk];
267	}
268	for(i_=0; i_<=kp-1;i_++)
269	{
270	a[i_,kk] = a[i_,kp];
271	}
272	for(i_=0; i_<=kp-1;i_++)
273	{
274	a[i_,kp] = temp[i_];
275	}
276	for(i_=kp+1; i_<=kk-1;i_++)
277	{
278	temp[i_] = a[i_,kk];
279	}
280	for(i_=kp+1; i_<=kk-1;i_++)
281	{
282	a[i_,kk] = a[kp,i_];
283	}
284	for(i_=kp+1; i_<=kk-1;i_++)
285	{
286	a[kp,i_] = temp[i_];
287	}
288	t = a[kk,kk];
289	a[kk,kk] = a[kp,kp];
290	a[kp,kp] = t;
291	if( kstep==2 )
292	{
293	t = a[k-1,k];
294	a[k-1,k] = a[kp,k];
295	a[kp,k] = t;
296	}
297	}
298
299	//
300	// Update the leading submatrix
301	//
302	if( kstep==1 )
303	{
304
305	//
306	// 1-by-1 pivot block D(k): column k now holds
307	//
308	// W(k) = U(k)*D(k)
309	//
310	// where U(k) is the k-th column of U
311	//
312	// Perform a rank-1 update of A(1:k-1,1:k-1) as
313	//
314	// A := A - U(k)D(k)U(k)' = A - W(k)1/D(k)W(k)'
315	//
316	r1 = 1/a[k,k];
317	for(i=0; i<=k-1; i++)
318	{
319	vv = -(r1*a[i,k]);
320	for(i_=i; i_<=k-1;i_++)
321	{
322	a[i,i_] = a[i,i_] + vv*a[i_,k];
323	}
324	}
325
326	//
327	// Store U(K+1) in column K+1
328	//
329	for(i_=0; i_<=k-1;i_++)
330	{
331	a[i_,k] = r1*a[i_,k];
332	}
333	}
334	else
335	{
336
337	//
338	// 2-by-2 pivot block D(k): columns k and k-1 now hold
339	//
340	// ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k)
341	//
342	// where U(k) and U(k-1) are the k-th and (k-1)-th columns
343	// of U
344	//
345	// Perform a rank-2 update of A(1:k-2,1:k-2) as
346	//
347	// A := A - ( U(k-1) U(k) )D(k)( U(k-1) U(k) )'
348	// = A - ( W(k-1) W(k) )inv(D(k))( W(k-1) W(k) )'
349	//
350	if( k>1 )
351	{
352	d12 = a[k-1,k];
353	d22 = a[k-1,k-1]/d12;
354	d11 = a[k,k]/d12;
355	t = 1/(d11*d22-1);
356	d12 = t/d12;
357	for(j=k-2; j>=0; j--)
358	{
359	wkm1 = d12(d11a[j,k-1]-a[j,k]);
360	wk = d12(d22a[j,k]-a[j,k-1]);
361	for(i_=0; i_<=j;i_++)
362	{
363	a[i_,j] = a[i_,j] - wk*a[i_,k];
364	}
365	for(i_=0; i_<=j;i_++)
366	{
367	a[i_,j] = a[i_,j] - wkm1*a[i_,k-1];
368	}
369	a[j,k] = wk;
370	a[j,k-1] = wkm1;
371	}
372	}
373	}
374	}
375
376	//
377	// Store details of the interchanges in IPIV
378	//
379	if( kstep==1 )
380	{
381	pivots[k] = kp;
382	}
383	else
384	{
385	pivots[k] = kp-n;
386	pivots[k-1] = kp-n;
387	}
388
389	//
390	// Decrease K+1 and return to the start of the main loop
391	//
392	k = k-kstep;
393	}
394	}
395	else
396	{
397
398	//
399	// Factorize A as LDL' using the lower triangle of A
400	//
401	// K+1 is the main loop index, increasing from 1 to N in steps of
402	// 1 or 2
403	//
404	k = 0;
405	while( k<=n-1 )
406	{
407	kstep = 1;
408
409	//
410	// Determine rows and columns to be interchanged and whether
411	// a 1-by-1 or 2-by-2 pivot block will be used
412	//
413	absakk = Math.Abs(a[k,k]);
414
415	//
416	// IMAX is the row-index of the largest off-diagonal element in
417	// column K+1, and COLMAX is its absolute value
418	//
419	if( k<n-1 )
420	{
421	imax = k+1+1;
422	for(ii=k+1+2; ii<=n; ii++)
423	{
424	if( (double)(Math.Abs(a[ii-1,k]))>(double)(Math.Abs(a[imax-1,k])) )
425	{
426	imax = ii;
427	}
428	}
429	colmax = Math.Abs(a[imax-1,k]);
430	}
431	else
432	{
433	colmax = 0;
434	}
435	if( (double)(Math.Max(absakk, colmax))==(double)(0) )
436	{
437
438	//
439	// Column K+1 is zero
440	//
441	kp = k;
442	}
443	else
444	{
445	if( (double)(absakk)>=(double)(alpha*colmax) )
446	{
447
448	//
449	// no interchange, use 1-by-1 pivot block
450	//
451	kp = k;
452	}
453	else
454	{
455
456	//
457	// JMAX is the column-index of the largest off-diagonal
458	// element in row IMAX, and ROWMAX is its absolute value
459	//
460	jmax = k+1;
461	for(ii=k+1+1; ii<=imax-1; ii++)
462	{
463	if( (double)(Math.Abs(a[imax-1,ii-1]))>(double)(Math.Abs(a[imax-1,jmax-1])) )
464	{
465	jmax = ii;
466	}
467	}
468	rowmax = Math.Abs(a[imax-1,jmax-1]);
469	if( imax<n )
470	{
471	jmax = imax+1;
472	for(ii=imax+2; ii<=n; ii++)
473	{
474	if( (double)(Math.Abs(a[ii-1,imax-1]))>(double)(Math.Abs(a[jmax-1,imax-1])) )
475	{
476	jmax = ii;
477	}
478	}
479	rowmax = Math.Max(rowmax, Math.Abs(a[jmax-1,imax-1]));
480	}
481	vv = colmax/rowmax;
482	if( (double)(absakk)>=(double)(alphacolmaxvv) )
483	{
484
485	//
486	// no interchange, use 1-by-1 pivot block
487	//
488	kp = k;
489	}
490	else
491	{
492	if( (double)(Math.Abs(a[imax-1,imax-1]))>=(double)(alpha*rowmax) )
493	{
494
495	//
496	// interchange rows and columns K+1 and IMAX, use 1-by-1
497	// pivot block
498	//
499	kp = imax-1;
500	}
501	else
502	{
503
504	//
505	// interchange rows and columns K+1+1 and IMAX, use 2-by-2
506	// pivot block
507	//
508	kp = imax-1;
509	kstep = 2;
510	}
511	}
512	}
513	kk = k+kstep-1;
514	if( kp!=kk )
515	{
516
517	//
518	// Interchange rows and columns KK+1 and KP+1 in the trailing
519	// submatrix A(K+1:n,K+1:n)
520	//
521	if( kp+1<n )
522	{
523	for(i_=kp+1; i_<=n-1;i_++)
524	{
525	temp[i_] = a[i_,kk];
526	}
527	for(i_=kp+1; i_<=n-1;i_++)
528	{
529	a[i_,kk] = a[i_,kp];
530	}
531	for(i_=kp+1; i_<=n-1;i_++)
532	{
533	a[i_,kp] = temp[i_];
534	}
535	}
536	for(i_=kk+1; i_<=kp-1;i_++)
537	{
538	temp[i_] = a[i_,kk];
539	}
540	for(i_=kk+1; i_<=kp-1;i_++)
541	{
542	a[i_,kk] = a[kp,i_];
543	}
544	for(i_=kk+1; i_<=kp-1;i_++)
545	{
546	a[kp,i_] = temp[i_];
547	}
548	t = a[kk,kk];
549	a[kk,kk] = a[kp,kp];
550	a[kp,kp] = t;
551	if( kstep==2 )
552	{
553	t = a[k+1,k];
554	a[k+1,k] = a[kp,k];
555	a[kp,k] = t;
556	}
557	}
558
559	//
560	// Update the trailing submatrix
561	//
562	if( kstep==1 )
563	{
564
565	//
566	// 1-by-1 pivot block D(K+1): column K+1 now holds
567	//
568	// W(K+1) = L(K+1)*D(K+1)
569	//
570	// where L(K+1) is the K+1-th column of L
571	//
572	if( k+1<n )
573	{
574
575	//
576	// Perform a rank-1 update of A(K+1+1:n,K+1+1:n) as
577	//
578	// A := A - L(K+1)D(K+1)L(K+1)' = A - W(K+1)(1/D(K+1))W(K+1)'
579	//
580	d11 = 1/a[k+1-1,k+1-1];
581	for(ii=k+1; ii<=n-1; ii++)
582	{
583	vv = -(d11*a[ii,k]);
584	for(i_=k+1; i_<=ii;i_++)
585	{
586	a[ii,i_] = a[ii,i_] + vv*a[i_,k];
587	}
588	}
589
590	//
591	// Store L(K+1) in column K+1
592	//
593	for(i_=k+1; i_<=n-1;i_++)
594	{
595	a[i_,k] = d11*a[i_,k];
596	}
597	}
598	}
599	else
600	{
601
602	//
603	// 2-by-2 pivot block D(K+1)
604	//
605	if( k<n-2 )
606	{
607
608	//
609	// Perform a rank-2 update of A(K+1+2:n,K+1+2:n) as
610	//
611	// A := A - ( (A(K+1) A(K+1+1))D(K+1)(-1) ) (A(K+1) A(K+1+1))'
612	//
613	// where L(K+1) and L(K+1+1) are the K+1-th and (K+1+1)-th
614	// columns of L
615	//
616	d21 = a[k+1,k];
617	d11 = a[k+1,k+1]/d21;
618	d22 = a[k,k]/d21;
619	t = 1/(d11*d22-1);
620	d21 = t/d21;
621	for(j=k+2; j<=n-1; j++)
622	{
623	wk = d21(d11a[j,k]-a[j,k+1]);
624	wkp1 = d21(d22a[j,k+1]-a[j,k]);
625	for(i_=j; i_<=n-1;i_++)
626	{
627	a[i_,j] = a[i_,j] - wk*a[i_,k];
628	}
629	for(i_=j; i_<=n-1;i_++)
630	{
631	a[i_,j] = a[i_,j] - wkp1*a[i_,k+1];
632	}
633	a[j,k] = wk;
634	a[j,k+1] = wkp1;
635	}
636	}
637	}
638	}
639
640	//
641	// Store details of the interchanges in IPIV
642	//
643	if( kstep==1 )
644	{
645	pivots[k+1-1] = kp+1-1;
646	}
647	else
648	{
649	pivots[k+1-1] = kp+1-1-n;
650	pivots[k+1+1-1] = kp+1-1-n;
651	}
652
653	//
654	// Increase K+1 and return to the start of the main loop
655	//
656	k = k+kstep;
657	}
658	}
659	}
660
661
662	public static void ldltdecomposition(ref double[,] a,
663	int n,
664	bool isupper,
665	ref int[] pivots)
666	{
667	int i = 0;
668	int imax = 0;
669	int j = 0;
670	int jmax = 0;
671	int k = 0;
672	int kk = 0;
673	int kp = 0;
674	int kstep = 0;
675	double absakk = 0;
676	double alpha = 0;
677	double colmax = 0;
678	double d11 = 0;
679	double d12 = 0;
680	double d21 = 0;
681	double d22 = 0;
682	double r1 = 0;
683	double rowmax = 0;
684	double t = 0;
685	double wk = 0;
686	double wkm1 = 0;
687	double wkp1 = 0;
688	int ii = 0;
689	int i1 = 0;
690	int i2 = 0;
691	double vv = 0;
692	double[] temp = new double[0];
693	int i_ = 0;
694
695	pivots = new int[n+1];
696	temp = new double[n+1];
697
698	//
699	// Initialize ALPHA for use in choosing pivot block size.
700	//
701	alpha = (1+Math.Sqrt(17))/8;
702	if( isupper )
703	{
704
705	//
706	// Factorize A as UDU' using the upper triangle of A
707	//
708	//
709	// K is the main loop index, decreasing from N to 1 in steps of
710	// 1 or 2
711	//
712	k = n;
713	while( k>=1 )
714	{
715	kstep = 1;
716
717	//
718	// Determine rows and columns to be interchanged and whether
719	// a 1-by-1 or 2-by-2 pivot block will be used
720	//
721	absakk = Math.Abs(a[k,k]);
722
723	//
724	// IMAX is the row-index of the largest off-diagonal element in
725	// column K, and COLMAX is its absolute value
726	//
727	if( k>1 )
728	{
729	imax = 1;
730	for(ii=2; ii<=k-1; ii++)
731	{
732	if( (double)(Math.Abs(a[ii,k]))>(double)(Math.Abs(a[imax,k])) )
733	{
734	imax = ii;
735	}
736	}
737	colmax = Math.Abs(a[imax,k]);
738	}
739	else
740	{
741	colmax = 0;
742	}
743	if( (double)(Math.Max(absakk, colmax))==(double)(0) )
744	{
745
746	//
747	// Column K is zero
748	//
749	kp = k;
750	}
751	else
752	{
753	if( (double)(absakk)>=(double)(alpha*colmax) )
754	{
755
756	//
757	// no interchange, use 1-by-1 pivot block
758	//
759	kp = k;
760	}
761	else
762	{
763
764	//
765	// JMAX is the column-index of the largest off-diagonal
766	// element in row IMAX, and ROWMAX is its absolute value
767	//
768	jmax = imax+1;
769	for(ii=imax+2; ii<=k; ii++)
770	{
771	if( (double)(Math.Abs(a[imax,ii]))>(double)(Math.Abs(a[imax,jmax])) )
772	{
773	jmax = ii;
774	}
775	}
776	rowmax = Math.Abs(a[imax,jmax]);
777	if( imax>1 )
778	{
779	jmax = 1;
780	for(ii=2; ii<=imax-1; ii++)
781	{
782	if( (double)(Math.Abs(a[ii,imax]))>(double)(Math.Abs(a[jmax,imax])) )
783	{
784	jmax = ii;
785	}
786	}
787	rowmax = Math.Max(rowmax, Math.Abs(a[jmax,imax]));
788	}
789	vv = colmax/rowmax;
790	if( (double)(absakk)>=(double)(alphacolmaxvv) )
791	{
792
793	//
794	// no interchange, use 1-by-1 pivot block
795	//
796	kp = k;
797	}
798	else
799	{
800	if( (double)(Math.Abs(a[imax,imax]))>=(double)(alpha*rowmax) )
801	{
802
803	//
804	// interchange rows and columns K and IMAX, use 1-by-1
805	// pivot block
806	//
807	kp = imax;
808	}
809	else
810	{
811
812	//
813	// interchange rows and columns K-1 and IMAX, use 2-by-2
814	// pivot block
815	//
816	kp = imax;
817	kstep = 2;
818	}
819	}
820	}
821	kk = k-kstep+1;
822	if( kp!=kk )
823	{
824
825	//
826	// Interchange rows and columns KK and KP in the leading
827	// submatrix A(1:k,1:k)
828	//
829	i1 = kp-1;
830	for(i_=1; i_<=i1;i_++)
831	{
832	temp[i_] = a[i_,kk];
833	}
834	for(i_=1; i_<=i1;i_++)
835	{
836	a[i_,kk] = a[i_,kp];
837	}
838	for(i_=1; i_<=i1;i_++)
839	{
840	a[i_,kp] = temp[i_];
841	}
842	i1 = kp+1;
843	i2 = kk-1;
844	for(i_=i1; i_<=i2;i_++)
845	{
846	temp[i_] = a[i_,kk];
847	}
848	for(i_=i1; i_<=i2;i_++)
849	{
850	a[i_,kk] = a[kp,i_];
851	}
852	for(i_=i1; i_<=i2;i_++)
853	{
854	a[kp,i_] = temp[i_];
855	}
856	t = a[kk,kk];
857	a[kk,kk] = a[kp,kp];
858	a[kp,kp] = t;
859	if( kstep==2 )
860	{
861	t = a[k-1,k];
862	a[k-1,k] = a[kp,k];
863	a[kp,k] = t;
864	}
865	}
866
867	//
868	// Update the leading submatrix
869	//
870	if( kstep==1 )
871	{
872
873	//
874	// 1-by-1 pivot block D(k): column k now holds
875	//
876	// W(k) = U(k)*D(k)
877	//
878	// where U(k) is the k-th column of U
879	//
880	// Perform a rank-1 update of A(1:k-1,1:k-1) as
881	//
882	// A := A - U(k)D(k)U(k)' = A - W(k)1/D(k)W(k)'
883	//
884	r1 = 1/a[k,k];
885	for(i=1; i<=k-1; i++)
886	{
887	i2 = k-1;
888	vv = -(r1*a[i,k]);
889	for(i_=i; i_<=i2;i_++)
890	{
891	a[i,i_] = a[i,i_] + vv*a[i_,k];
892	}
893	}
894
895	//
896	// Store U(k) in column k
897	//
898	i2 = k-1;
899	for(i_=1; i_<=i2;i_++)
900	{
901	a[i_,k] = r1*a[i_,k];
902	}
903	}
904	else
905	{
906
907	//
908	// 2-by-2 pivot block D(k): columns k and k-1 now hold
909	//
910	// ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k)
911	//
912	// where U(k) and U(k-1) are the k-th and (k-1)-th columns
913	// of U
914	//
915	// Perform a rank-2 update of A(1:k-2,1:k-2) as
916	//
917	// A := A - ( U(k-1) U(k) )D(k)( U(k-1) U(k) )'
918	// = A - ( W(k-1) W(k) )inv(D(k))( W(k-1) W(k) )'
919	//
920	if( k>2 )
921	{
922	d12 = a[k-1,k];
923	d22 = a[k-1,k-1]/d12;
924	d11 = a[k,k]/d12;
925	t = 1/(d11*d22-1);
926	d12 = t/d12;
927	for(j=k-2; j>=1; j--)
928	{
929	wkm1 = d12(d11a[j,k-1]-a[j,k]);
930	wk = d12(d22a[j,k]-a[j,k-1]);
931	for(i_=1; i_<=j;i_++)
932	{
933	a[i_,j] = a[i_,j] - wk*a[i_,k];
934	}
935	i1 = k-1;
936	for(i_=1; i_<=j;i_++)
937	{
938	a[i_,j] = a[i_,j] - wkm1*a[i_,i1];
939	}
940	a[j,k] = wk;
941	a[j,k-1] = wkm1;
942	}
943	}
944	}
945	}
946
947	//
948	// Store details of the interchanges in IPIV
949	//
950	if( kstep==1 )
951	{
952	pivots[k] = kp;
953	}
954	else
955	{
956	pivots[k] = -kp;
957	pivots[k-1] = -kp;
958	}
959
960	//
961	// Decrease K and return to the start of the main loop
962	//
963	k = k-kstep;
964	}
965	}
966	else
967	{
968
969	//
970	// Factorize A as LDL' using the lower triangle of A
971	//
972	// K is the main loop index, increasing from 1 to N in steps of
973	// 1 or 2
974	//
975	k = 1;
976	while( k<=n )
977	{
978	kstep = 1;
979
980	//
981	// Determine rows and columns to be interchanged and whether
982	// a 1-by-1 or 2-by-2 pivot block will be used
983	//
984	absakk = Math.Abs(a[k,k]);
985
986	//
987	// IMAX is the row-index of the largest off-diagonal element in
988	// column K, and COLMAX is its absolute value
989	//
990	if( k<n )
991	{
992	imax = k+1;
993	for(ii=k+2; ii<=n; ii++)
994	{
995	if( (double)(Math.Abs(a[ii,k]))>(double)(Math.Abs(a[imax,k])) )
996	{
997	imax = ii;
998	}
999	}
1000	colmax = Math.Abs(a[imax,k]);
1001	}
1002	else
1003	{
1004	colmax = 0;
1005	}
1006	if( (double)(Math.Max(absakk, colmax))==(double)(0) )
1007	{
1008
1009	//
1010	// Column K is zero
1011	//
1012	kp = k;
1013	}
1014	else
1015	{
1016	if( (double)(absakk)>=(double)(alpha*colmax) )
1017	{
1018
1019	//
1020	// no interchange, use 1-by-1 pivot block
1021	//
1022	kp = k;
1023	}
1024	else
1025	{
1026
1027	//
1028	// JMAX is the column-index of the largest off-diagonal
1029	// element in row IMAX, and ROWMAX is its absolute value
1030	//
1031	jmax = k;
1032	for(ii=k+1; ii<=imax-1; ii++)
1033	{
1034	if( (double)(Math.Abs(a[imax,ii]))>(double)(Math.Abs(a[imax,jmax])) )
1035	{
1036	jmax = ii;
1037	}
1038	}
1039	rowmax = Math.Abs(a[imax,jmax]);
1040	if( imax<n )
1041	{
1042	jmax = imax+1;
1043	for(ii=imax+2; ii<=n; ii++)
1044	{
1045	if( (double)(Math.Abs(a[ii,imax]))>(double)(Math.Abs(a[jmax,imax])) )
1046	{
1047	jmax = ii;
1048	}
1049	}
1050	rowmax = Math.Max(rowmax, Math.Abs(a[jmax,imax]));
1051	}
1052	vv = colmax/rowmax;
1053	if( (double)(absakk)>=(double)(alphacolmaxvv) )
1054	{
1055
1056	//
1057	// no interchange, use 1-by-1 pivot block
1058	//
1059	kp = k;
1060	}
1061	else
1062	{
1063	if( (double)(Math.Abs(a[imax,imax]))>=(double)(alpha*rowmax) )
1064	{
1065
1066	//
1067	// interchange rows and columns K and IMAX, use 1-by-1
1068	// pivot block
1069	//
1070	kp = imax;
1071	}
1072	else
1073	{
1074
1075	//
1076	// interchange rows and columns K+1 and IMAX, use 2-by-2
1077	// pivot block
1078	//
1079	kp = imax;
1080	kstep = 2;
1081	}
1082	}
1083	}
1084	kk = k+kstep-1;
1085	if( kp!=kk )
1086	{
1087
1088	//
1089	// Interchange rows and columns KK and KP in the trailing
1090	// submatrix A(k:n,k:n)
1091	//
1092	if( kp<n )
1093	{
1094	i1 = kp+1;
1095	for(i_=i1; i_<=n;i_++)
1096	{
1097	temp[i_] = a[i_,kk];
1098	}
1099	for(i_=i1; i_<=n;i_++)
1100	{
1101	a[i_,kk] = a[i_,kp];
1102	}
1103	for(i_=i1; i_<=n;i_++)
1104	{
1105	a[i_,kp] = temp[i_];
1106	}
1107	}
1108	i1 = kk+1;
1109	i2 = kp-1;
1110	for(i_=i1; i_<=i2;i_++)
1111	{
1112	temp[i_] = a[i_,kk];
1113	}
1114	for(i_=i1; i_<=i2;i_++)
1115	{
1116	a[i_,kk] = a[kp,i_];
1117	}
1118	for(i_=i1; i_<=i2;i_++)
1119	{
1120	a[kp,i_] = temp[i_];
1121	}
1122	t = a[kk,kk];
1123	a[kk,kk] = a[kp,kp];
1124	a[kp,kp] = t;
1125	if( kstep==2 )
1126	{
1127	t = a[k+1,k];
1128	a[k+1,k] = a[kp,k];
1129	a[kp,k] = t;
1130	}
1131	}
1132
1133	//
1134	// Update the trailing submatrix
1135	//
1136	if( kstep==1 )
1137	{
1138
1139	//
1140	// 1-by-1 pivot block D(k): column k now holds
1141	//
1142	// W(k) = L(k)*D(k)
1143	//
1144	// where L(k) is the k-th column of L
1145	//
1146	if( k<n )
1147	{
1148
1149	//
1150	// Perform a rank-1 update of A(k+1:n,k+1:n) as
1151	//
1152	// A := A - L(k)D(k)L(k)' = A - W(k)(1/D(k))W(k)'
1153	//
1154	d11 = 1/a[k,k];
1155	for(ii=k+1; ii<=n; ii++)
1156	{
1157	i1 = k+1;
1158	i2 = ii;
1159	vv = -(d11*a[ii,k]);
1160	for(i_=i1; i_<=i2;i_++)
1161	{
1162	a[ii,i_] = a[ii,i_] + vv*a[i_,k];
1163	}
1164	}
1165
1166	//
1167	// Store L(k) in column K
1168	//
1169	i1 = k+1;
1170	for(i_=i1; i_<=n;i_++)
1171	{
1172	a[i_,k] = d11*a[i_,k];
1173	}
1174	}
1175	}
1176	else
1177	{
1178
1179	//
1180	// 2-by-2 pivot block D(k)
1181	//
1182	if( k<n-1 )
1183	{
1184
1185	//
1186	// Perform a rank-2 update of A(k+2:n,k+2:n) as
1187	//
1188	// A := A - ( (A(k) A(k+1))D(k)(-1) ) (A(k) A(k+1))'
1189	//
1190	// where L(k) and L(k+1) are the k-th and (k+1)-th
1191	// columns of L
1192	//
1193	d21 = a[k+1,k];
1194	d11 = a[k+1,k+1]/d21;
1195	d22 = a[k,k]/d21;
1196	t = 1/(d11*d22-1);
1197	d21 = t/d21;
1198	for(j=k+2; j<=n; j++)
1199	{
1200	wk = d21(d11a[j,k]-a[j,k+1]);
1201	wkp1 = d21(d22a[j,k+1]-a[j,k]);
1202	ii = k+1;
1203	for(i_=j; i_<=n;i_++)
1204	{
1205	a[i_,j] = a[i_,j] - wk*a[i_,k];
1206	}
1207	for(i_=j; i_<=n;i_++)
1208	{
1209	a[i_,j] = a[i_,j] - wkp1*a[i_,ii];
1210	}
1211	a[j,k] = wk;
1212	a[j,k+1] = wkp1;
1213	}
1214	}
1215	}
1216	}
1217
1218	//
1219	// Store details of the interchanges in IPIV
1220	//
1221	if( kstep==1 )
1222	{
1223	pivots[k] = kp;
1224	}
1225	else
1226	{
1227	pivots[k] = -kp;
1228	pivots[k+1] = -kp;
1229	}
1230
1231	//
1232	// Increase K and return to the start of the main loop
1233	//
1234	k = k+kstep;
1235	}
1236	}
1237	}
1238	}
1239	}

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