LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ zhetf2()

 subroutine zhetf2 ( character UPLO, integer N, complex*16, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, integer INFO )

ZHETF2 computes the factorization of a complex Hermitian matrix, using the diagonal pivoting method (unblocked algorithm, calling Level 2 BLAS).

Purpose:
``` ZHETF2 computes the factorization of a complex Hermitian matrix A
using the Bunch-Kaufman diagonal pivoting method:

A = U*D*U**H  or  A = L*D*L**H

where U (or L) is a product of permutation and unit upper (lower)
triangular matrices, U**H is the conjugate transpose of U, and D is
Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.

This is the unblocked version of the algorithm, calling Level 2 BLAS.```
Parameters
 [in] UPLO ``` UPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the Hermitian matrix A is stored: = 'U': Upper triangular = 'L': Lower triangular``` [in] N ``` N is INTEGER The order of the matrix A. N >= 0.``` [in,out] A ``` A is COMPLEX*16 array, dimension (LDA,N) On entry, the Hermitian matrix A. If UPLO = 'U', the leading n-by-n upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading n-by-n lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced. On exit, the block diagonal matrix D and the multipliers used to obtain the factor U or L (see below for further details).``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).``` [out] IPIV ``` IPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D. If UPLO = 'U': If IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal block. If IPIV(k) = IPIV(k-1) < 0, then rows and columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k) is a 2-by-2 diagonal block. If UPLO = 'L': If IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal block. If IPIV(k) = IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.``` [out] INFO ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -k, the k-th argument had an illegal value > 0: if INFO = k, D(k,k) is exactly zero. The factorization has been completed, but the block diagonal matrix D is exactly singular, and division by zero will occur if it is used to solve a system of equations.```
Further Details:
```  If UPLO = 'U', then A = U*D*U**H, where
U = P(n)*U(n)* ... *P(k)U(k)* ...,
i.e., U is a product of terms P(k)*U(k), where k decreases from n to
1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
that if the diagonal block D(k) is of order s (s = 1 or 2), then

(   I    v    0   )   k-s
U(k) =  (   0    I    0   )   s
(   0    0    I   )   n-k
k-s   s   n-k

If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
and A(k,k), and v overwrites A(1:k-2,k-1:k).

If UPLO = 'L', then A = L*D*L**H, where
L = P(1)*L(1)* ... *P(k)*L(k)* ...,
i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
that if the diagonal block D(k) is of order s (s = 1 or 2), then

(   I    0     0   )  k-1
L(k) =  (   0    I     0   )  s
(   0    v     I   )  n-k-s+1
k-1   s  n-k-s+1

If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).```
Contributors:
```  09-29-06 - patch from
Bobby Cheng, MathWorks

Replace l.210 and l.393
IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
by
IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. DISNAN(ABSAKK) ) THEN

01-01-96 - Based on modifications by
J. Lewis, Boeing Computer Services Company
A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA```

Definition at line 190 of file zhetf2.f.

191 *
192 * -- LAPACK computational routine --
193 * -- LAPACK is a software package provided by Univ. of Tennessee, --
194 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
195 *
196 * .. Scalar Arguments ..
197  CHARACTER UPLO
198  INTEGER INFO, LDA, N
199 * ..
200 * .. Array Arguments ..
201  INTEGER IPIV( * )
202  COMPLEX*16 A( LDA, * )
203 * ..
204 *
205 * =====================================================================
206 *
207 * .. Parameters ..
208  DOUBLE PRECISION ZERO, ONE
209  parameter( zero = 0.0d+0, one = 1.0d+0 )
210  DOUBLE PRECISION EIGHT, SEVTEN
211  parameter( eight = 8.0d+0, sevten = 17.0d+0 )
212 * ..
213 * .. Local Scalars ..
214  LOGICAL UPPER
215  INTEGER I, IMAX, J, JMAX, K, KK, KP, KSTEP
216  DOUBLE PRECISION ABSAKK, ALPHA, COLMAX, D, D11, D22, R1, ROWMAX,
217  \$ TT
218  COMPLEX*16 D12, D21, T, WK, WKM1, WKP1, ZDUM
219 * ..
220 * .. External Functions ..
221  LOGICAL LSAME, DISNAN
222  INTEGER IZAMAX
223  DOUBLE PRECISION DLAPY2
224  EXTERNAL lsame, izamax, dlapy2, disnan
225 * ..
226 * .. External Subroutines ..
227  EXTERNAL xerbla, zdscal, zher, zswap
228 * ..
229 * .. Intrinsic Functions ..
230  INTRINSIC abs, dble, dcmplx, dconjg, dimag, max, sqrt
231 * ..
232 * .. Statement Functions ..
233  DOUBLE PRECISION CABS1
234 * ..
235 * .. Statement Function definitions ..
236  cabs1( zdum ) = abs( dble( zdum ) ) + abs( dimag( zdum ) )
237 * ..
238 * .. Executable Statements ..
239 *
240 * Test the input parameters.
241 *
242  info = 0
243  upper = lsame( uplo, 'U' )
244  IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
245  info = -1
246  ELSE IF( n.LT.0 ) THEN
247  info = -2
248  ELSE IF( lda.LT.max( 1, n ) ) THEN
249  info = -4
250  END IF
251  IF( info.NE.0 ) THEN
252  CALL xerbla( 'ZHETF2', -info )
253  RETURN
254  END IF
255 *
256 * Initialize ALPHA for use in choosing pivot block size.
257 *
258  alpha = ( one+sqrt( sevten ) ) / eight
259 *
260  IF( upper ) THEN
261 *
262 * Factorize A as U*D*U**H using the upper triangle of A
263 *
264 * K is the main loop index, decreasing from N to 1 in steps of
265 * 1 or 2
266 *
267  k = n
268  10 CONTINUE
269 *
270 * If K < 1, exit from loop
271 *
272  IF( k.LT.1 )
273  \$ GO TO 90
274  kstep = 1
275 *
276 * Determine rows and columns to be interchanged and whether
277 * a 1-by-1 or 2-by-2 pivot block will be used
278 *
279  absakk = abs( dble( a( k, k ) ) )
280 *
281 * IMAX is the row-index of the largest off-diagonal element in
282 * column K, and COLMAX is its absolute value.
283 * Determine both COLMAX and IMAX.
284 *
285  IF( k.GT.1 ) THEN
286  imax = izamax( k-1, a( 1, k ), 1 )
287  colmax = cabs1( a( imax, k ) )
288  ELSE
289  colmax = zero
290  END IF
291 *
292  IF( (max( absakk, colmax ).EQ.zero) .OR. disnan(absakk) ) THEN
293 *
294 * Column K is zero or underflow, or contains a NaN:
295 * set INFO and continue
296 *
297  IF( info.EQ.0 )
298  \$ info = k
299  kp = k
300  a( k, k ) = dble( a( k, k ) )
301  ELSE
302 *
303 * ============================================================
304 *
305 * Test for interchange
306 *
307  IF( absakk.GE.alpha*colmax ) THEN
308 *
309 * no interchange, use 1-by-1 pivot block
310 *
311  kp = k
312  ELSE
313 *
314 * JMAX is the column-index of the largest off-diagonal
315 * element in row IMAX, and ROWMAX is its absolute value.
316 * Determine only ROWMAX.
317 *
318  jmax = imax + izamax( k-imax, a( imax, imax+1 ), lda )
319  rowmax = cabs1( a( imax, jmax ) )
320  IF( imax.GT.1 ) THEN
321  jmax = izamax( imax-1, a( 1, imax ), 1 )
322  rowmax = max( rowmax, cabs1( a( jmax, imax ) ) )
323  END IF
324 *
325  IF( absakk.GE.alpha*colmax*( colmax / rowmax ) ) THEN
326 *
327 * no interchange, use 1-by-1 pivot block
328 *
329  kp = k
330 *
331  ELSE IF( abs( dble( a( imax, imax ) ) ).GE.alpha*rowmax )
332  \$ THEN
333 *
334 * interchange rows and columns K and IMAX, use 1-by-1
335 * pivot block
336 *
337  kp = imax
338  ELSE
339 *
340 * interchange rows and columns K-1 and IMAX, use 2-by-2
341 * pivot block
342 *
343  kp = imax
344  kstep = 2
345  END IF
346 *
347  END IF
348 *
349 * ============================================================
350 *
351  kk = k - kstep + 1
352  IF( kp.NE.kk ) THEN
353 *
354 * Interchange rows and columns KK and KP in the leading
355 * submatrix A(1:k,1:k)
356 *
357  CALL zswap( kp-1, a( 1, kk ), 1, a( 1, kp ), 1 )
358  DO 20 j = kp + 1, kk - 1
359  t = dconjg( a( j, kk ) )
360  a( j, kk ) = dconjg( a( kp, j ) )
361  a( kp, j ) = t
362  20 CONTINUE
363  a( kp, kk ) = dconjg( a( kp, kk ) )
364  r1 = dble( a( kk, kk ) )
365  a( kk, kk ) = dble( a( kp, kp ) )
366  a( kp, kp ) = r1
367  IF( kstep.EQ.2 ) THEN
368  a( k, k ) = dble( a( k, k ) )
369  t = a( k-1, k )
370  a( k-1, k ) = a( kp, k )
371  a( kp, k ) = t
372  END IF
373  ELSE
374  a( k, k ) = dble( a( k, k ) )
375  IF( kstep.EQ.2 )
376  \$ a( k-1, k-1 ) = dble( a( k-1, k-1 ) )
377  END IF
378 *
379 * Update the leading submatrix
380 *
381  IF( kstep.EQ.1 ) THEN
382 *
383 * 1-by-1 pivot block D(k): column k now holds
384 *
385 * W(k) = U(k)*D(k)
386 *
387 * where U(k) is the k-th column of U
388 *
389 * Perform a rank-1 update of A(1:k-1,1:k-1) as
390 *
391 * A := A - U(k)*D(k)*U(k)**H = A - W(k)*1/D(k)*W(k)**H
392 *
393  r1 = one / dble( a( k, k ) )
394  CALL zher( uplo, k-1, -r1, a( 1, k ), 1, a, lda )
395 *
396 * Store U(k) in column k
397 *
398  CALL zdscal( k-1, r1, a( 1, k ), 1 )
399  ELSE
400 *
401 * 2-by-2 pivot block D(k): columns k and k-1 now hold
402 *
403 * ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k)
404 *
405 * where U(k) and U(k-1) are the k-th and (k-1)-th columns
406 * of U
407 *
408 * Perform a rank-2 update of A(1:k-2,1:k-2) as
409 *
410 * A := A - ( U(k-1) U(k) )*D(k)*( U(k-1) U(k) )**H
411 * = A - ( W(k-1) W(k) )*inv(D(k))*( W(k-1) W(k) )**H
412 *
413  IF( k.GT.2 ) THEN
414 *
415  d = dlapy2( dble( a( k-1, k ) ),
416  \$ dimag( a( k-1, k ) ) )
417  d22 = dble( a( k-1, k-1 ) ) / d
418  d11 = dble( a( k, k ) ) / d
419  tt = one / ( d11*d22-one )
420  d12 = a( k-1, k ) / d
421  d = tt / d
422 *
423  DO 40 j = k - 2, 1, -1
424  wkm1 = d*( d11*a( j, k-1 )-dconjg( d12 )*
425  \$ a( j, k ) )
426  wk = d*( d22*a( j, k )-d12*a( j, k-1 ) )
427  DO 30 i = j, 1, -1
428  a( i, j ) = a( i, j ) - a( i, k )*dconjg( wk ) -
429  \$ a( i, k-1 )*dconjg( wkm1 )
430  30 CONTINUE
431  a( j, k ) = wk
432  a( j, k-1 ) = wkm1
433  a( j, j ) = dcmplx( dble( a( j, j ) ), 0.0d+0 )
434  40 CONTINUE
435 *
436  END IF
437 *
438  END IF
439  END IF
440 *
441 * Store details of the interchanges in IPIV
442 *
443  IF( kstep.EQ.1 ) THEN
444  ipiv( k ) = kp
445  ELSE
446  ipiv( k ) = -kp
447  ipiv( k-1 ) = -kp
448  END IF
449 *
450 * Decrease K and return to the start of the main loop
451 *
452  k = k - kstep
453  GO TO 10
454 *
455  ELSE
456 *
457 * Factorize A as L*D*L**H using the lower triangle of A
458 *
459 * K is the main loop index, increasing from 1 to N in steps of
460 * 1 or 2
461 *
462  k = 1
463  50 CONTINUE
464 *
465 * If K > N, exit from loop
466 *
467  IF( k.GT.n )
468  \$ GO TO 90
469  kstep = 1
470 *
471 * Determine rows and columns to be interchanged and whether
472 * a 1-by-1 or 2-by-2 pivot block will be used
473 *
474  absakk = abs( dble( a( k, k ) ) )
475 *
476 * IMAX is the row-index of the largest off-diagonal element in
477 * column K, and COLMAX is its absolute value.
478 * Determine both COLMAX and IMAX.
479 *
480  IF( k.LT.n ) THEN
481  imax = k + izamax( n-k, a( k+1, k ), 1 )
482  colmax = cabs1( a( imax, k ) )
483  ELSE
484  colmax = zero
485  END IF
486 *
487  IF( (max( absakk, colmax ).EQ.zero) .OR. disnan(absakk) ) THEN
488 *
489 * Column K is zero or underflow, or contains a NaN:
490 * set INFO and continue
491 *
492  IF( info.EQ.0 )
493  \$ info = k
494  kp = k
495  a( k, k ) = dble( a( k, k ) )
496  ELSE
497 *
498 * ============================================================
499 *
500 * Test for interchange
501 *
502  IF( absakk.GE.alpha*colmax ) THEN
503 *
504 * no interchange, use 1-by-1 pivot block
505 *
506  kp = k
507  ELSE
508 *
509 * JMAX is the column-index of the largest off-diagonal
510 * element in row IMAX, and ROWMAX is its absolute value.
511 * Determine only ROWMAX.
512 *
513  jmax = k - 1 + izamax( imax-k, a( imax, k ), lda )
514  rowmax = cabs1( a( imax, jmax ) )
515  IF( imax.LT.n ) THEN
516  jmax = imax + izamax( n-imax, a( imax+1, imax ), 1 )
517  rowmax = max( rowmax, cabs1( a( jmax, imax ) ) )
518  END IF
519 *
520  IF( absakk.GE.alpha*colmax*( colmax / rowmax ) ) THEN
521 *
522 * no interchange, use 1-by-1 pivot block
523 *
524  kp = k
525 *
526  ELSE IF( abs( dble( a( imax, imax ) ) ).GE.alpha*rowmax )
527  \$ THEN
528 *
529 * interchange rows and columns K and IMAX, use 1-by-1
530 * pivot block
531 *
532  kp = imax
533  ELSE
534 *
535 * interchange rows and columns K+1 and IMAX, use 2-by-2
536 * pivot block
537 *
538  kp = imax
539  kstep = 2
540  END IF
541 *
542  END IF
543 *
544 * ============================================================
545 *
546  kk = k + kstep - 1
547  IF( kp.NE.kk ) THEN
548 *
549 * Interchange rows and columns KK and KP in the trailing
550 * submatrix A(k:n,k:n)
551 *
552  IF( kp.LT.n )
553  \$ CALL zswap( n-kp, a( kp+1, kk ), 1, a( kp+1, kp ), 1 )
554  DO 60 j = kk + 1, kp - 1
555  t = dconjg( a( j, kk ) )
556  a( j, kk ) = dconjg( a( kp, j ) )
557  a( kp, j ) = t
558  60 CONTINUE
559  a( kp, kk ) = dconjg( a( kp, kk ) )
560  r1 = dble( a( kk, kk ) )
561  a( kk, kk ) = dble( a( kp, kp ) )
562  a( kp, kp ) = r1
563  IF( kstep.EQ.2 ) THEN
564  a( k, k ) = dble( a( k, k ) )
565  t = a( k+1, k )
566  a( k+1, k ) = a( kp, k )
567  a( kp, k ) = t
568  END IF
569  ELSE
570  a( k, k ) = dble( a( k, k ) )
571  IF( kstep.EQ.2 )
572  \$ a( k+1, k+1 ) = dble( a( k+1, k+1 ) )
573  END IF
574 *
575 * Update the trailing submatrix
576 *
577  IF( kstep.EQ.1 ) THEN
578 *
579 * 1-by-1 pivot block D(k): column k now holds
580 *
581 * W(k) = L(k)*D(k)
582 *
583 * where L(k) is the k-th column of L
584 *
585  IF( k.LT.n ) THEN
586 *
587 * Perform a rank-1 update of A(k+1:n,k+1:n) as
588 *
589 * A := A - L(k)*D(k)*L(k)**H = A - W(k)*(1/D(k))*W(k)**H
590 *
591  r1 = one / dble( a( k, k ) )
592  CALL zher( uplo, n-k, -r1, a( k+1, k ), 1,
593  \$ a( k+1, k+1 ), lda )
594 *
595 * Store L(k) in column K
596 *
597  CALL zdscal( n-k, r1, a( k+1, k ), 1 )
598  END IF
599  ELSE
600 *
601 * 2-by-2 pivot block D(k)
602 *
603  IF( k.LT.n-1 ) THEN
604 *
605 * Perform a rank-2 update of A(k+2:n,k+2:n) as
606 *
607 * A := A - ( L(k) L(k+1) )*D(k)*( L(k) L(k+1) )**H
608 * = A - ( W(k) W(k+1) )*inv(D(k))*( W(k) W(k+1) )**H
609 *
610 * where L(k) and L(k+1) are the k-th and (k+1)-th
611 * columns of L
612 *
613  d = dlapy2( dble( a( k+1, k ) ),
614  \$ dimag( a( k+1, k ) ) )
615  d11 = dble( a( k+1, k+1 ) ) / d
616  d22 = dble( a( k, k ) ) / d
617  tt = one / ( d11*d22-one )
618  d21 = a( k+1, k ) / d
619  d = tt / d
620 *
621  DO 80 j = k + 2, n
622  wk = d*( d11*a( j, k )-d21*a( j, k+1 ) )
623  wkp1 = d*( d22*a( j, k+1 )-dconjg( d21 )*
624  \$ a( j, k ) )
625  DO 70 i = j, n
626  a( i, j ) = a( i, j ) - a( i, k )*dconjg( wk ) -
627  \$ a( i, k+1 )*dconjg( wkp1 )
628  70 CONTINUE
629  a( j, k ) = wk
630  a( j, k+1 ) = wkp1
631  a( j, j ) = dcmplx( dble( a( j, j ) ), 0.0d+0 )
632  80 CONTINUE
633  END IF
634  END IF
635  END IF
636 *
637 * Store details of the interchanges in IPIV
638 *
639  IF( kstep.EQ.1 ) THEN
640  ipiv( k ) = kp
641  ELSE
642  ipiv( k ) = -kp
643  ipiv( k+1 ) = -kp
644  END IF
645 *
646 * Increase K and return to the start of the main loop
647 *
648  k = k + kstep
649  GO TO 50
650 *
651  END IF
652 *
653  90 CONTINUE
654  RETURN
655 *
656 * End of ZHETF2
657 *
logical function disnan(DIN)
DISNAN tests input for NaN.
Definition: disnan.f:59
double precision function dlapy2(X, Y)
DLAPY2 returns sqrt(x2+y2).
Definition: dlapy2.f:63
integer function izamax(N, ZX, INCX)
IZAMAX
Definition: izamax.f:71
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine zswap(N, ZX, INCX, ZY, INCY)
ZSWAP
Definition: zswap.f:81
subroutine zdscal(N, DA, ZX, INCX)
ZDSCAL
Definition: zdscal.f:78
subroutine zher(UPLO, N, ALPHA, X, INCX, A, LDA)
ZHER
Definition: zher.f:135
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