LAPACK 3.11.0
LAPACK: Linear Algebra PACKage
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csptrf.f
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1*> \brief \b CSPTRF
2*
3* =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6* http://www.netlib.org/lapack/explore-html/
7*
8*> \htmlonly
9*> Download CSPTRF + dependencies
10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/csptrf.f">
11*> [TGZ]</a>
12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/csptrf.f">
13*> [ZIP]</a>
14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/csptrf.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18* Definition:
19* ===========
20*
21* SUBROUTINE CSPTRF( UPLO, N, AP, IPIV, INFO )
22*
23* .. Scalar Arguments ..
24* CHARACTER UPLO
25* INTEGER INFO, N
26* ..
27* .. Array Arguments ..
28* INTEGER IPIV( * )
29* COMPLEX AP( * )
30* ..
31*
32*
33*> \par Purpose:
34* =============
35*>
36*> \verbatim
37*>
38*> CSPTRF computes the factorization of a complex symmetric matrix A
39*> stored in packed format using the Bunch-Kaufman diagonal pivoting
40*> method:
41*>
42*> A = U*D*U**T or A = L*D*L**T
43*>
44*> where U (or L) is a product of permutation and unit upper (lower)
45*> triangular matrices, and D is symmetric and block diagonal with
46*> 1-by-1 and 2-by-2 diagonal blocks.
47*> \endverbatim
48*
49* Arguments:
50* ==========
51*
52*> \param[in] UPLO
53*> \verbatim
54*> UPLO is CHARACTER*1
55*> = 'U': Upper triangle of A is stored;
56*> = 'L': Lower triangle of A is stored.
57*> \endverbatim
58*>
59*> \param[in] N
60*> \verbatim
61*> N is INTEGER
62*> The order of the matrix A. N >= 0.
63*> \endverbatim
64*>
65*> \param[in,out] AP
66*> \verbatim
67*> AP is COMPLEX array, dimension (N*(N+1)/2)
68*> On entry, the upper or lower triangle of the symmetric matrix
69*> A, packed columnwise in a linear array. The j-th column of A
70*> is stored in the array AP as follows:
71*> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
72*> if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
73*>
74*> On exit, the block diagonal matrix D and the multipliers used
75*> to obtain the factor U or L, stored as a packed triangular
76*> matrix overwriting A (see below for further details).
77*> \endverbatim
78*>
79*> \param[out] IPIV
80*> \verbatim
81*> IPIV is INTEGER array, dimension (N)
82*> Details of the interchanges and the block structure of D.
83*> If IPIV(k) > 0, then rows and columns k and IPIV(k) were
84*> interchanged and D(k,k) is a 1-by-1 diagonal block.
85*> If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
86*> columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
87*> is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) =
88*> IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
89*> interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
90*> \endverbatim
91*>
92*> \param[out] INFO
93*> \verbatim
94*> INFO is INTEGER
95*> = 0: successful exit
96*> < 0: if INFO = -i, the i-th argument had an illegal value
97*> > 0: if INFO = i, D(i,i) is exactly zero. The factorization
98*> has been completed, but the block diagonal matrix D is
99*> exactly singular, and division by zero will occur if it
100*> is used to solve a system of equations.
101*> \endverbatim
102*
103* Authors:
104* ========
105*
106*> \author Univ. of Tennessee
107*> \author Univ. of California Berkeley
108*> \author Univ. of Colorado Denver
109*> \author NAG Ltd.
110*
111*> \ingroup complexOTHERcomputational
112*
113*> \par Further Details:
114* =====================
115*>
116*> \verbatim
117*>
118*> 5-96 - Based on modifications by J. Lewis, Boeing Computer Services
119*> Company
120*>
121*> If UPLO = 'U', then A = U*D*U**T, where
122*> U = P(n)*U(n)* ... *P(k)U(k)* ...,
123*> i.e., U is a product of terms P(k)*U(k), where k decreases from n to
124*> 1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
125*> and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
126*> defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
127*> that if the diagonal block D(k) is of order s (s = 1 or 2), then
128*>
129*> ( I v 0 ) k-s
130*> U(k) = ( 0 I 0 ) s
131*> ( 0 0 I ) n-k
132*> k-s s n-k
133*>
134*> If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
135*> If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
136*> and A(k,k), and v overwrites A(1:k-2,k-1:k).
137*>
138*> If UPLO = 'L', then A = L*D*L**T, where
139*> L = P(1)*L(1)* ... *P(k)*L(k)* ...,
140*> i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
141*> n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
142*> and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
143*> defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
144*> that if the diagonal block D(k) is of order s (s = 1 or 2), then
145*>
146*> ( I 0 0 ) k-1
147*> L(k) = ( 0 I 0 ) s
148*> ( 0 v I ) n-k-s+1
149*> k-1 s n-k-s+1
150*>
151*> If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
152*> If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
153*> and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
154*> \endverbatim
155*>
156* =====================================================================
157 SUBROUTINE csptrf( UPLO, N, AP, IPIV, INFO )
158*
159* -- LAPACK computational routine --
160* -- LAPACK is a software package provided by Univ. of Tennessee, --
161* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
162*
163* .. Scalar Arguments ..
164 CHARACTER UPLO
165 INTEGER INFO, N
166* ..
167* .. Array Arguments ..
168 INTEGER IPIV( * )
169 COMPLEX AP( * )
170* ..
171*
172* =====================================================================
173*
174* .. Parameters ..
175 REAL ZERO, ONE
176 parameter( zero = 0.0e+0, one = 1.0e+0 )
177 REAL EIGHT, SEVTEN
178 parameter( eight = 8.0e+0, sevten = 17.0e+0 )
179 COMPLEX CONE
180 parameter( cone = ( 1.0e+0, 0.0e+0 ) )
181* ..
182* .. Local Scalars ..
183 LOGICAL UPPER
184 INTEGER I, IMAX, J, JMAX, K, KC, KK, KNC, KP, KPC,
185 $ KSTEP, KX, NPP
186 REAL ABSAKK, ALPHA, COLMAX, ROWMAX
187 COMPLEX D11, D12, D21, D22, R1, T, WK, WKM1, WKP1, ZDUM
188* ..
189* .. External Functions ..
190 LOGICAL LSAME
191 INTEGER ICAMAX
192 EXTERNAL lsame, icamax
193* ..
194* .. External Subroutines ..
195 EXTERNAL cscal, cspr, cswap, xerbla
196* ..
197* .. Intrinsic Functions ..
198 INTRINSIC abs, aimag, max, real, sqrt
199* ..
200* .. Statement Functions ..
201 REAL CABS1
202* ..
203* .. Statement Function definitions ..
204 cabs1( zdum ) = abs( real( zdum ) ) + abs( aimag( zdum ) )
205* ..
206* .. Executable Statements ..
207*
208* Test the input parameters.
209*
210 info = 0
211 upper = lsame( uplo, 'U' )
212 IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
213 info = -1
214 ELSE IF( n.LT.0 ) THEN
215 info = -2
216 END IF
217 IF( info.NE.0 ) THEN
218 CALL xerbla( 'CSPTRF', -info )
219 RETURN
220 END IF
221*
222* Initialize ALPHA for use in choosing pivot block size.
223*
224 alpha = ( one+sqrt( sevten ) ) / eight
225*
226 IF( upper ) THEN
227*
228* Factorize A as U*D*U**T using the upper triangle of A
229*
230* K is the main loop index, decreasing from N to 1 in steps of
231* 1 or 2
232*
233 k = n
234 kc = ( n-1 )*n / 2 + 1
235 10 CONTINUE
236 knc = kc
237*
238* If K < 1, exit from loop
239*
240 IF( k.LT.1 )
241 $ GO TO 110
242 kstep = 1
243*
244* Determine rows and columns to be interchanged and whether
245* a 1-by-1 or 2-by-2 pivot block will be used
246*
247 absakk = cabs1( ap( kc+k-1 ) )
248*
249* IMAX is the row-index of the largest off-diagonal element in
250* column K, and COLMAX is its absolute value
251*
252 IF( k.GT.1 ) THEN
253 imax = icamax( k-1, ap( kc ), 1 )
254 colmax = cabs1( ap( kc+imax-1 ) )
255 ELSE
256 colmax = zero
257 END IF
258*
259 IF( max( absakk, colmax ).EQ.zero ) THEN
260*
261* Column K is zero: set INFO and continue
262*
263 IF( info.EQ.0 )
264 $ info = k
265 kp = k
266 ELSE
267 IF( absakk.GE.alpha*colmax ) THEN
268*
269* no interchange, use 1-by-1 pivot block
270*
271 kp = k
272 ELSE
273*
274 rowmax = zero
275 jmax = imax
276 kx = imax*( imax+1 ) / 2 + imax
277 DO 20 j = imax + 1, k
278 IF( cabs1( ap( kx ) ).GT.rowmax ) THEN
279 rowmax = cabs1( ap( kx ) )
280 jmax = j
281 END IF
282 kx = kx + j
283 20 CONTINUE
284 kpc = ( imax-1 )*imax / 2 + 1
285 IF( imax.GT.1 ) THEN
286 jmax = icamax( imax-1, ap( kpc ), 1 )
287 rowmax = max( rowmax, cabs1( ap( kpc+jmax-1 ) ) )
288 END IF
289*
290 IF( absakk.GE.alpha*colmax*( colmax / rowmax ) ) THEN
291*
292* no interchange, use 1-by-1 pivot block
293*
294 kp = k
295 ELSE IF( cabs1( ap( kpc+imax-1 ) ).GE.alpha*rowmax ) THEN
296*
297* interchange rows and columns K and IMAX, use 1-by-1
298* pivot block
299*
300 kp = imax
301 ELSE
302*
303* interchange rows and columns K-1 and IMAX, use 2-by-2
304* pivot block
305*
306 kp = imax
307 kstep = 2
308 END IF
309 END IF
310*
311 kk = k - kstep + 1
312 IF( kstep.EQ.2 )
313 $ knc = knc - k + 1
314 IF( kp.NE.kk ) THEN
315*
316* Interchange rows and columns KK and KP in the leading
317* submatrix A(1:k,1:k)
318*
319 CALL cswap( kp-1, ap( knc ), 1, ap( kpc ), 1 )
320 kx = kpc + kp - 1
321 DO 30 j = kp + 1, kk - 1
322 kx = kx + j - 1
323 t = ap( knc+j-1 )
324 ap( knc+j-1 ) = ap( kx )
325 ap( kx ) = t
326 30 CONTINUE
327 t = ap( knc+kk-1 )
328 ap( knc+kk-1 ) = ap( kpc+kp-1 )
329 ap( kpc+kp-1 ) = t
330 IF( kstep.EQ.2 ) THEN
331 t = ap( kc+k-2 )
332 ap( kc+k-2 ) = ap( kc+kp-1 )
333 ap( kc+kp-1 ) = t
334 END IF
335 END IF
336*
337* Update the leading submatrix
338*
339 IF( kstep.EQ.1 ) THEN
340*
341* 1-by-1 pivot block D(k): column k now holds
342*
343* W(k) = U(k)*D(k)
344*
345* where U(k) is the k-th column of U
346*
347* Perform a rank-1 update of A(1:k-1,1:k-1) as
348*
349* A := A - U(k)*D(k)*U(k)**T = A - W(k)*1/D(k)*W(k)**T
350*
351 r1 = cone / ap( kc+k-1 )
352 CALL cspr( uplo, k-1, -r1, ap( kc ), 1, ap )
353*
354* Store U(k) in column k
355*
356 CALL cscal( k-1, r1, ap( kc ), 1 )
357 ELSE
358*
359* 2-by-2 pivot block D(k): columns k and k-1 now hold
360*
361* ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k)
362*
363* where U(k) and U(k-1) are the k-th and (k-1)-th columns
364* of U
365*
366* Perform a rank-2 update of A(1:k-2,1:k-2) as
367*
368* A := A - ( U(k-1) U(k) )*D(k)*( U(k-1) U(k) )**T
369* = A - ( W(k-1) W(k) )*inv(D(k))*( W(k-1) W(k) )**T
370*
371 IF( k.GT.2 ) THEN
372*
373 d12 = ap( k-1+( k-1 )*k / 2 )
374 d22 = ap( k-1+( k-2 )*( k-1 ) / 2 ) / d12
375 d11 = ap( k+( k-1 )*k / 2 ) / d12
376 t = cone / ( d11*d22-cone )
377 d12 = t / d12
378*
379 DO 50 j = k - 2, 1, -1
380 wkm1 = d12*( d11*ap( j+( k-2 )*( k-1 ) / 2 )-
381 $ ap( j+( k-1 )*k / 2 ) )
382 wk = d12*( d22*ap( j+( k-1 )*k / 2 )-
383 $ ap( j+( k-2 )*( k-1 ) / 2 ) )
384 DO 40 i = j, 1, -1
385 ap( i+( j-1 )*j / 2 ) = ap( i+( j-1 )*j / 2 ) -
386 $ ap( i+( k-1 )*k / 2 )*wk -
387 $ ap( i+( k-2 )*( k-1 ) / 2 )*wkm1
388 40 CONTINUE
389 ap( j+( k-1 )*k / 2 ) = wk
390 ap( j+( k-2 )*( k-1 ) / 2 ) = wkm1
391 50 CONTINUE
392*
393 END IF
394 END IF
395 END IF
396*
397* Store details of the interchanges in IPIV
398*
399 IF( kstep.EQ.1 ) THEN
400 ipiv( k ) = kp
401 ELSE
402 ipiv( k ) = -kp
403 ipiv( k-1 ) = -kp
404 END IF
405*
406* Decrease K and return to the start of the main loop
407*
408 k = k - kstep
409 kc = knc - k
410 GO TO 10
411*
412 ELSE
413*
414* Factorize A as L*D*L**T using the lower triangle of A
415*
416* K is the main loop index, increasing from 1 to N in steps of
417* 1 or 2
418*
419 k = 1
420 kc = 1
421 npp = n*( n+1 ) / 2
422 60 CONTINUE
423 knc = kc
424*
425* If K > N, exit from loop
426*
427 IF( k.GT.n )
428 $ GO TO 110
429 kstep = 1
430*
431* Determine rows and columns to be interchanged and whether
432* a 1-by-1 or 2-by-2 pivot block will be used
433*
434 absakk = cabs1( ap( kc ) )
435*
436* IMAX is the row-index of the largest off-diagonal element in
437* column K, and COLMAX is its absolute value
438*
439 IF( k.LT.n ) THEN
440 imax = k + icamax( n-k, ap( kc+1 ), 1 )
441 colmax = cabs1( ap( kc+imax-k ) )
442 ELSE
443 colmax = zero
444 END IF
445*
446 IF( max( absakk, colmax ).EQ.zero ) THEN
447*
448* Column K is zero: set INFO and continue
449*
450 IF( info.EQ.0 )
451 $ info = k
452 kp = k
453 ELSE
454 IF( absakk.GE.alpha*colmax ) THEN
455*
456* no interchange, use 1-by-1 pivot block
457*
458 kp = k
459 ELSE
460*
461* JMAX is the column-index of the largest off-diagonal
462* element in row IMAX, and ROWMAX is its absolute value
463*
464 rowmax = zero
465 kx = kc + imax - k
466 DO 70 j = k, imax - 1
467 IF( cabs1( ap( kx ) ).GT.rowmax ) THEN
468 rowmax = cabs1( ap( kx ) )
469 jmax = j
470 END IF
471 kx = kx + n - j
472 70 CONTINUE
473 kpc = npp - ( n-imax+1 )*( n-imax+2 ) / 2 + 1
474 IF( imax.LT.n ) THEN
475 jmax = imax + icamax( n-imax, ap( kpc+1 ), 1 )
476 rowmax = max( rowmax, cabs1( ap( kpc+jmax-imax ) ) )
477 END IF
478*
479 IF( absakk.GE.alpha*colmax*( colmax / rowmax ) ) THEN
480*
481* no interchange, use 1-by-1 pivot block
482*
483 kp = k
484 ELSE IF( cabs1( ap( kpc ) ).GE.alpha*rowmax ) THEN
485*
486* interchange rows and columns K and IMAX, use 1-by-1
487* pivot block
488*
489 kp = imax
490 ELSE
491*
492* interchange rows and columns K+1 and IMAX, use 2-by-2
493* pivot block
494*
495 kp = imax
496 kstep = 2
497 END IF
498 END IF
499*
500 kk = k + kstep - 1
501 IF( kstep.EQ.2 )
502 $ knc = knc + n - k + 1
503 IF( kp.NE.kk ) THEN
504*
505* Interchange rows and columns KK and KP in the trailing
506* submatrix A(k:n,k:n)
507*
508 IF( kp.LT.n )
509 $ CALL cswap( n-kp, ap( knc+kp-kk+1 ), 1, ap( kpc+1 ),
510 $ 1 )
511 kx = knc + kp - kk
512 DO 80 j = kk + 1, kp - 1
513 kx = kx + n - j + 1
514 t = ap( knc+j-kk )
515 ap( knc+j-kk ) = ap( kx )
516 ap( kx ) = t
517 80 CONTINUE
518 t = ap( knc )
519 ap( knc ) = ap( kpc )
520 ap( kpc ) = t
521 IF( kstep.EQ.2 ) THEN
522 t = ap( kc+1 )
523 ap( kc+1 ) = ap( kc+kp-k )
524 ap( kc+kp-k ) = t
525 END IF
526 END IF
527*
528* Update the trailing submatrix
529*
530 IF( kstep.EQ.1 ) THEN
531*
532* 1-by-1 pivot block D(k): column k now holds
533*
534* W(k) = L(k)*D(k)
535*
536* where L(k) is the k-th column of L
537*
538 IF( k.LT.n ) THEN
539*
540* Perform a rank-1 update of A(k+1:n,k+1:n) as
541*
542* A := A - L(k)*D(k)*L(k)**T = A - W(k)*(1/D(k))*W(k)**T
543*
544 r1 = cone / ap( kc )
545 CALL cspr( uplo, n-k, -r1, ap( kc+1 ), 1,
546 $ ap( kc+n-k+1 ) )
547*
548* Store L(k) in column K
549*
550 CALL cscal( n-k, r1, ap( kc+1 ), 1 )
551 END IF
552 ELSE
553*
554* 2-by-2 pivot block D(k): columns K and K+1 now hold
555*
556* ( W(k) W(k+1) ) = ( L(k) L(k+1) )*D(k)
557*
558* where L(k) and L(k+1) are the k-th and (k+1)-th columns
559* of L
560*
561 IF( k.LT.n-1 ) THEN
562*
563* Perform a rank-2 update of A(k+2:n,k+2:n) as
564*
565* A := A - ( L(k) L(k+1) )*D(k)*( L(k) L(k+1) )**T
566* = A - ( W(k) W(k+1) )*inv(D(k))*( W(k) W(k+1) )**T
567*
568* where L(k) and L(k+1) are the k-th and (k+1)-th
569* columns of L
570*
571 d21 = ap( k+1+( k-1 )*( 2*n-k ) / 2 )
572 d11 = ap( k+1+k*( 2*n-k-1 ) / 2 ) / d21
573 d22 = ap( k+( k-1 )*( 2*n-k ) / 2 ) / d21
574 t = cone / ( d11*d22-cone )
575 d21 = t / d21
576*
577 DO 100 j = k + 2, n
578 wk = d21*( d11*ap( j+( k-1 )*( 2*n-k ) / 2 )-
579 $ ap( j+k*( 2*n-k-1 ) / 2 ) )
580 wkp1 = d21*( d22*ap( j+k*( 2*n-k-1 ) / 2 )-
581 $ ap( j+( k-1 )*( 2*n-k ) / 2 ) )
582 DO 90 i = j, n
583 ap( i+( j-1 )*( 2*n-j ) / 2 ) = ap( i+( j-1 )*
584 $ ( 2*n-j ) / 2 ) - ap( i+( k-1 )*( 2*n-k ) /
585 $ 2 )*wk - ap( i+k*( 2*n-k-1 ) / 2 )*wkp1
586 90 CONTINUE
587 ap( j+( k-1 )*( 2*n-k ) / 2 ) = wk
588 ap( j+k*( 2*n-k-1 ) / 2 ) = wkp1
589 100 CONTINUE
590 END IF
591 END IF
592 END IF
593*
594* Store details of the interchanges in IPIV
595*
596 IF( kstep.EQ.1 ) THEN
597 ipiv( k ) = kp
598 ELSE
599 ipiv( k ) = -kp
600 ipiv( k+1 ) = -kp
601 END IF
602*
603* Increase K and return to the start of the main loop
604*
605 k = k + kstep
606 kc = knc + n - k + 2
607 GO TO 60
608*
609 END IF
610*
611 110 CONTINUE
612 RETURN
613*
614* End of CSPTRF
615*
616 END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine cswap(N, CX, INCX, CY, INCY)
CSWAP
Definition: cswap.f:81
subroutine cscal(N, CA, CX, INCX)
CSCAL
Definition: cscal.f:78
subroutine cspr(UPLO, N, ALPHA, X, INCX, AP)
CSPR performs the symmetrical rank-1 update of a complex symmetric packed matrix.
Definition: cspr.f:132
subroutine csptrf(UPLO, N, AP, IPIV, INFO)
CSPTRF
Definition: csptrf.f:158