LAPACK  3.10.0
LAPACK: Linear Algebra PACKage

◆ zsytf2()

subroutine zsytf2 ( character  UPLO,
integer  N,
complex*16, dimension( lda, * )  A,
integer  LDA,
integer, dimension( * )  IPIV,
integer  INFO 
)

ZSYTF2 computes the factorization of a real symmetric indefinite matrix, using the diagonal pivoting method (unblocked algorithm).

Download ZSYTF2 + dependencies [TGZ] [ZIP] [TXT]

Purpose:
 ZSYTF2 computes the factorization of a complex symmetric matrix A
 using the Bunch-Kaufman diagonal pivoting method:

    A = U*D*U**T  or  A = L*D*L**T

 where U (or L) is a product of permutation and unit upper (lower)
 triangular matrices, U**T is the transpose of U, and D is symmetric 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
          symmetric 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 symmetric 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.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
  If UPLO = 'U', then A = U*D*U**T, 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**T, 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.209 and l.377
         IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
    by
         IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. DISNAN(ABSAKK) ) THEN

  1-96 - Based on modifications by J. Lewis, Boeing Computer Services
         Company

Definition at line 190 of file zsytf2.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  COMPLEX*16 CONE
213  parameter( cone = ( 1.0d+0, 0.0d+0 ) )
214 * ..
215 * .. Local Scalars ..
216  LOGICAL UPPER
217  INTEGER I, IMAX, J, JMAX, K, KK, KP, KSTEP
218  DOUBLE PRECISION ABSAKK, ALPHA, COLMAX, ROWMAX
219  COMPLEX*16 D11, D12, D21, D22, R1, T, WK, WKM1, WKP1, Z
220 * ..
221 * .. External Functions ..
222  LOGICAL DISNAN, LSAME
223  INTEGER IZAMAX
224  EXTERNAL disnan, lsame, izamax
225 * ..
226 * .. External Subroutines ..
227  EXTERNAL xerbla, zscal, zswap, zsyr
228 * ..
229 * .. Intrinsic Functions ..
230  INTRINSIC abs, dble, dimag, max, sqrt
231 * ..
232 * .. Statement Functions ..
233  DOUBLE PRECISION CABS1
234 * ..
235 * .. Statement Function definitions ..
236  cabs1( z ) = abs( dble( z ) ) + abs( dimag( z ) )
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( 'ZSYTF2', -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**T 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 70
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 = cabs1( 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  ELSE
301  IF( absakk.GE.alpha*colmax ) THEN
302 *
303 * no interchange, use 1-by-1 pivot block
304 *
305  kp = k
306  ELSE
307 *
308 * JMAX is the column-index of the largest off-diagonal
309 * element in row IMAX, and ROWMAX is its absolute value
310 *
311  jmax = imax + izamax( k-imax, a( imax, imax+1 ), lda )
312  rowmax = cabs1( a( imax, jmax ) )
313  IF( imax.GT.1 ) THEN
314  jmax = izamax( imax-1, a( 1, imax ), 1 )
315  rowmax = max( rowmax, cabs1( a( jmax, imax ) ) )
316  END IF
317 *
318  IF( absakk.GE.alpha*colmax*( colmax / rowmax ) ) THEN
319 *
320 * no interchange, use 1-by-1 pivot block
321 *
322  kp = k
323  ELSE IF( cabs1( a( imax, imax ) ).GE.alpha*rowmax ) THEN
324 *
325 * interchange rows and columns K and IMAX, use 1-by-1
326 * pivot block
327 *
328  kp = imax
329  ELSE
330 *
331 * interchange rows and columns K-1 and IMAX, use 2-by-2
332 * pivot block
333 *
334  kp = imax
335  kstep = 2
336  END IF
337  END IF
338 *
339  kk = k - kstep + 1
340  IF( kp.NE.kk ) THEN
341 *
342 * Interchange rows and columns KK and KP in the leading
343 * submatrix A(1:k,1:k)
344 *
345  CALL zswap( kp-1, a( 1, kk ), 1, a( 1, kp ), 1 )
346  CALL zswap( kk-kp-1, a( kp+1, kk ), 1, a( kp, kp+1 ),
347  $ lda )
348  t = a( kk, kk )
349  a( kk, kk ) = a( kp, kp )
350  a( kp, kp ) = t
351  IF( kstep.EQ.2 ) THEN
352  t = a( k-1, k )
353  a( k-1, k ) = a( kp, k )
354  a( kp, k ) = t
355  END IF
356  END IF
357 *
358 * Update the leading submatrix
359 *
360  IF( kstep.EQ.1 ) THEN
361 *
362 * 1-by-1 pivot block D(k): column k now holds
363 *
364 * W(k) = U(k)*D(k)
365 *
366 * where U(k) is the k-th column of U
367 *
368 * Perform a rank-1 update of A(1:k-1,1:k-1) as
369 *
370 * A := A - U(k)*D(k)*U(k)**T = A - W(k)*1/D(k)*W(k)**T
371 *
372  r1 = cone / a( k, k )
373  CALL zsyr( uplo, k-1, -r1, a( 1, k ), 1, a, lda )
374 *
375 * Store U(k) in column k
376 *
377  CALL zscal( k-1, r1, a( 1, k ), 1 )
378  ELSE
379 *
380 * 2-by-2 pivot block D(k): columns k and k-1 now hold
381 *
382 * ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k)
383 *
384 * where U(k) and U(k-1) are the k-th and (k-1)-th columns
385 * of U
386 *
387 * Perform a rank-2 update of A(1:k-2,1:k-2) as
388 *
389 * A := A - ( U(k-1) U(k) )*D(k)*( U(k-1) U(k) )**T
390 * = A - ( W(k-1) W(k) )*inv(D(k))*( W(k-1) W(k) )**T
391 *
392  IF( k.GT.2 ) THEN
393 *
394  d12 = a( k-1, k )
395  d22 = a( k-1, k-1 ) / d12
396  d11 = a( k, k ) / d12
397  t = cone / ( d11*d22-cone )
398  d12 = t / d12
399 *
400  DO 30 j = k - 2, 1, -1
401  wkm1 = d12*( d11*a( j, k-1 )-a( j, k ) )
402  wk = d12*( d22*a( j, k )-a( j, k-1 ) )
403  DO 20 i = j, 1, -1
404  a( i, j ) = a( i, j ) - a( i, k )*wk -
405  $ a( i, k-1 )*wkm1
406  20 CONTINUE
407  a( j, k ) = wk
408  a( j, k-1 ) = wkm1
409  30 CONTINUE
410 *
411  END IF
412 *
413  END IF
414  END IF
415 *
416 * Store details of the interchanges in IPIV
417 *
418  IF( kstep.EQ.1 ) THEN
419  ipiv( k ) = kp
420  ELSE
421  ipiv( k ) = -kp
422  ipiv( k-1 ) = -kp
423  END IF
424 *
425 * Decrease K and return to the start of the main loop
426 *
427  k = k - kstep
428  GO TO 10
429 *
430  ELSE
431 *
432 * Factorize A as L*D*L**T using the lower triangle of A
433 *
434 * K is the main loop index, increasing from 1 to N in steps of
435 * 1 or 2
436 *
437  k = 1
438  40 CONTINUE
439 *
440 * If K > N, exit from loop
441 *
442  IF( k.GT.n )
443  $ GO TO 70
444  kstep = 1
445 *
446 * Determine rows and columns to be interchanged and whether
447 * a 1-by-1 or 2-by-2 pivot block will be used
448 *
449  absakk = cabs1( a( k, k ) )
450 *
451 * IMAX is the row-index of the largest off-diagonal element in
452 * column K, and COLMAX is its absolute value.
453 * Determine both COLMAX and IMAX.
454 *
455  IF( k.LT.n ) THEN
456  imax = k + izamax( n-k, a( k+1, k ), 1 )
457  colmax = cabs1( a( imax, k ) )
458  ELSE
459  colmax = zero
460  END IF
461 *
462  IF( max( absakk, colmax ).EQ.zero .OR. disnan(absakk) ) THEN
463 *
464 * Column K is zero or underflow, or contains a NaN:
465 * set INFO and continue
466 *
467  IF( info.EQ.0 )
468  $ info = k
469  kp = k
470  ELSE
471  IF( absakk.GE.alpha*colmax ) THEN
472 *
473 * no interchange, use 1-by-1 pivot block
474 *
475  kp = k
476  ELSE
477 *
478 * JMAX is the column-index of the largest off-diagonal
479 * element in row IMAX, and ROWMAX is its absolute value
480 *
481  jmax = k - 1 + izamax( imax-k, a( imax, k ), lda )
482  rowmax = cabs1( a( imax, jmax ) )
483  IF( imax.LT.n ) THEN
484  jmax = imax + izamax( n-imax, a( imax+1, imax ), 1 )
485  rowmax = max( rowmax, cabs1( a( jmax, imax ) ) )
486  END IF
487 *
488  IF( absakk.GE.alpha*colmax*( colmax / rowmax ) ) THEN
489 *
490 * no interchange, use 1-by-1 pivot block
491 *
492  kp = k
493  ELSE IF( cabs1( a( imax, imax ) ).GE.alpha*rowmax ) THEN
494 *
495 * interchange rows and columns K and IMAX, use 1-by-1
496 * pivot block
497 *
498  kp = imax
499  ELSE
500 *
501 * interchange rows and columns K+1 and IMAX, use 2-by-2
502 * pivot block
503 *
504  kp = imax
505  kstep = 2
506  END IF
507  END IF
508 *
509  kk = k + kstep - 1
510  IF( kp.NE.kk ) THEN
511 *
512 * Interchange rows and columns KK and KP in the trailing
513 * submatrix A(k:n,k:n)
514 *
515  IF( kp.LT.n )
516  $ CALL zswap( n-kp, a( kp+1, kk ), 1, a( kp+1, kp ), 1 )
517  CALL zswap( kp-kk-1, a( kk+1, kk ), 1, a( kp, kk+1 ),
518  $ lda )
519  t = a( kk, kk )
520  a( kk, kk ) = a( kp, kp )
521  a( kp, kp ) = t
522  IF( kstep.EQ.2 ) THEN
523  t = a( k+1, k )
524  a( k+1, k ) = a( kp, k )
525  a( kp, k ) = t
526  END IF
527  END IF
528 *
529 * Update the trailing submatrix
530 *
531  IF( kstep.EQ.1 ) THEN
532 *
533 * 1-by-1 pivot block D(k): column k now holds
534 *
535 * W(k) = L(k)*D(k)
536 *
537 * where L(k) is the k-th column of L
538 *
539  IF( k.LT.n ) THEN
540 *
541 * Perform a rank-1 update of A(k+1:n,k+1:n) as
542 *
543 * A := A - L(k)*D(k)*L(k)**T = A - W(k)*(1/D(k))*W(k)**T
544 *
545  r1 = cone / a( k, k )
546  CALL zsyr( uplo, n-k, -r1, a( k+1, k ), 1,
547  $ a( k+1, k+1 ), lda )
548 *
549 * Store L(k) in column K
550 *
551  CALL zscal( n-k, r1, a( k+1, k ), 1 )
552  END IF
553  ELSE
554 *
555 * 2-by-2 pivot block D(k)
556 *
557  IF( k.LT.n-1 ) THEN
558 *
559 * Perform a rank-2 update of A(k+2:n,k+2:n) as
560 *
561 * A := A - ( L(k) L(k+1) )*D(k)*( L(k) L(k+1) )**T
562 * = A - ( W(k) W(k+1) )*inv(D(k))*( W(k) W(k+1) )**T
563 *
564 * where L(k) and L(k+1) are the k-th and (k+1)-th
565 * columns of L
566 *
567  d21 = a( k+1, k )
568  d11 = a( k+1, k+1 ) / d21
569  d22 = a( k, k ) / d21
570  t = cone / ( d11*d22-cone )
571  d21 = t / d21
572 *
573  DO 60 j = k + 2, n
574  wk = d21*( d11*a( j, k )-a( j, k+1 ) )
575  wkp1 = d21*( d22*a( j, k+1 )-a( j, k ) )
576  DO 50 i = j, n
577  a( i, j ) = a( i, j ) - a( i, k )*wk -
578  $ a( i, k+1 )*wkp1
579  50 CONTINUE
580  a( j, k ) = wk
581  a( j, k+1 ) = wkp1
582  60 CONTINUE
583  END IF
584  END IF
585  END IF
586 *
587 * Store details of the interchanges in IPIV
588 *
589  IF( kstep.EQ.1 ) THEN
590  ipiv( k ) = kp
591  ELSE
592  ipiv( k ) = -kp
593  ipiv( k+1 ) = -kp
594  END IF
595 *
596 * Increase K and return to the start of the main loop
597 *
598  k = k + kstep
599  GO TO 40
600 *
601  END IF
602 *
603  70 CONTINUE
604  RETURN
605 *
606 * End of ZSYTF2
607 *
logical function disnan(DIN)
DISNAN tests input for NaN.
Definition: disnan.f:59
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 zscal(N, ZA, ZX, INCX)
ZSCAL
Definition: zscal.f:78
subroutine zsyr(UPLO, N, ALPHA, X, INCX, A, LDA)
ZSYR performs the symmetric rank-1 update of a complex symmetric matrix.
Definition: zsyr.f:135
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