LAPACK 3.12.0
LAPACK: Linear Algebra PACKage
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◆ dsytrf_rk()

subroutine dsytrf_rk ( character  uplo,
integer  n,
double precision, dimension( lda, * )  a,
integer  lda,
double precision, dimension( * )  e,
integer, dimension( * )  ipiv,
double precision, dimension( * )  work,
integer  lwork,
integer  info 
)

DSYTRF_RK computes the factorization of a real symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS3 blocked algorithm).

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

Purpose:
 DSYTRF_RK computes the factorization of a real symmetric matrix A
 using the bounded Bunch-Kaufman (rook) diagonal pivoting method:

    A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T),

 where U (or L) is unit upper (or lower) triangular matrix,
 U**T (or L**T) is the transpose of U (or L), P is a permutation
 matrix, P**T is the transpose of P, and D is symmetric and block
 diagonal with 1-by-1 and 2-by-2 diagonal blocks.

 This is the blocked version of the algorithm, calling Level 3 BLAS.
 For more information see Further Details section.
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 DOUBLE PRECISION 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, contains:
            a) ONLY diagonal elements of the symmetric block diagonal
               matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
               (superdiagonal (or subdiagonal) elements of D
                are stored on exit in array E), and
            b) If UPLO = 'U': factor U in the superdiagonal part of A.
               If UPLO = 'L': factor L in the subdiagonal part of A.
[in]LDA
          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).
[out]E
          E is DOUBLE PRECISION array, dimension (N)
          On exit, contains the superdiagonal (or subdiagonal)
          elements of the symmetric block diagonal matrix D
          with 1-by-1 or 2-by-2 diagonal blocks, where
          If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
          If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0.

          NOTE: For 1-by-1 diagonal block D(k), where
          1 <= k <= N, the element E(k) is set to 0 in both
          UPLO = 'U' or UPLO = 'L' cases.
[out]IPIV
          IPIV is INTEGER array, dimension (N)
          IPIV describes the permutation matrix P in the factorization
          of matrix A as follows. The absolute value of IPIV(k)
          represents the index of row and column that were
          interchanged with the k-th row and column. The value of UPLO
          describes the order in which the interchanges were applied.
          Also, the sign of IPIV represents the block structure of
          the symmetric block diagonal matrix D with 1-by-1 or 2-by-2
          diagonal blocks which correspond to 1 or 2 interchanges
          at each factorization step. For more info see Further
          Details section.

          If UPLO = 'U',
          ( in factorization order, k decreases from N to 1 ):
            a) A single positive entry IPIV(k) > 0 means:
               D(k,k) is a 1-by-1 diagonal block.
               If IPIV(k) != k, rows and columns k and IPIV(k) were
               interchanged in the matrix A(1:N,1:N);
               If IPIV(k) = k, no interchange occurred.

            b) A pair of consecutive negative entries
               IPIV(k) < 0 and IPIV(k-1) < 0 means:
               D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
               (NOTE: negative entries in IPIV appear ONLY in pairs).
               1) If -IPIV(k) != k, rows and columns
                  k and -IPIV(k) were interchanged
                  in the matrix A(1:N,1:N).
                  If -IPIV(k) = k, no interchange occurred.
               2) If -IPIV(k-1) != k-1, rows and columns
                  k-1 and -IPIV(k-1) were interchanged
                  in the matrix A(1:N,1:N).
                  If -IPIV(k-1) = k-1, no interchange occurred.

            c) In both cases a) and b), always ABS( IPIV(k) ) <= k.

            d) NOTE: Any entry IPIV(k) is always NONZERO on output.

          If UPLO = 'L',
          ( in factorization order, k increases from 1 to N ):
            a) A single positive entry IPIV(k) > 0 means:
               D(k,k) is a 1-by-1 diagonal block.
               If IPIV(k) != k, rows and columns k and IPIV(k) were
               interchanged in the matrix A(1:N,1:N).
               If IPIV(k) = k, no interchange occurred.

            b) A pair of consecutive negative entries
               IPIV(k) < 0 and IPIV(k+1) < 0 means:
               D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
               (NOTE: negative entries in IPIV appear ONLY in pairs).
               1) If -IPIV(k) != k, rows and columns
                  k and -IPIV(k) were interchanged
                  in the matrix A(1:N,1:N).
                  If -IPIV(k) = k, no interchange occurred.
               2) If -IPIV(k+1) != k+1, rows and columns
                  k-1 and -IPIV(k-1) were interchanged
                  in the matrix A(1:N,1:N).
                  If -IPIV(k+1) = k+1, no interchange occurred.

            c) In both cases a) and b), always ABS( IPIV(k) ) >= k.

            d) NOTE: Any entry IPIV(k) is always NONZERO on output.
[out]WORK
          WORK is DOUBLE PRECISION array, dimension ( MAX(1,LWORK) ).
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
[in]LWORK
          LWORK is INTEGER
          The length of WORK.  LWORK >=1.  For best performance
          LWORK >= N*NB, where NB is the block size returned
          by ILAENV.

          If LWORK = -1, then a workspace query is assumed;
          the routine only calculates the optimal size of the WORK
          array, returns this value as the first entry of the WORK
          array, and no error message related to LWORK is issued
          by XERBLA.
[out]INFO
          INFO is INTEGER
          = 0: successful exit

          < 0: If INFO = -k, the k-th argument had an illegal value

          > 0: If INFO = k, the matrix A is singular, because:
                 If UPLO = 'U': column k in the upper
                 triangular part of A contains all zeros.
                 If UPLO = 'L': column k in the lower
                 triangular part of A contains all zeros.

               Therefore D(k,k) is exactly zero, and superdiagonal
               elements of column k of U (or subdiagonal elements of
               column k of L ) are all zeros. 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.

               NOTE: INFO only stores the first occurrence of
               a singularity, any subsequent occurrence of singularity
               is not stored in INFO even though the factorization
               always completes.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
 TODO: put correct description
Contributors:
  December 2016,  Igor Kozachenko,
                  Computer Science Division,
                  University of California, Berkeley

  September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
                  School of Mathematics,
                  University of Manchester

Definition at line 257 of file dsytrf_rk.f.

259*
260* -- LAPACK computational routine --
261* -- LAPACK is a software package provided by Univ. of Tennessee, --
262* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
263*
264* .. Scalar Arguments ..
265 CHARACTER UPLO
266 INTEGER INFO, LDA, LWORK, N
267* ..
268* .. Array Arguments ..
269 INTEGER IPIV( * )
270 DOUBLE PRECISION A( LDA, * ), E( * ), WORK( * )
271* ..
272*
273* =====================================================================
274*
275* .. Local Scalars ..
276 LOGICAL LQUERY, UPPER
277 INTEGER I, IINFO, IP, IWS, K, KB, LDWORK, LWKOPT,
278 $ NB, NBMIN
279* ..
280* .. External Functions ..
281 LOGICAL LSAME
282 INTEGER ILAENV
283 EXTERNAL lsame, ilaenv
284* ..
285* .. External Subroutines ..
286 EXTERNAL dlasyf_rk, dsytf2_rk, dswap, xerbla
287* ..
288* .. Intrinsic Functions ..
289 INTRINSIC abs, max
290* ..
291* .. Executable Statements ..
292*
293* Test the input parameters.
294*
295 info = 0
296 upper = lsame( uplo, 'U' )
297 lquery = ( lwork.EQ.-1 )
298 IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
299 info = -1
300 ELSE IF( n.LT.0 ) THEN
301 info = -2
302 ELSE IF( lda.LT.max( 1, n ) ) THEN
303 info = -4
304 ELSE IF( lwork.LT.1 .AND. .NOT.lquery ) THEN
305 info = -8
306 END IF
307*
308 IF( info.EQ.0 ) THEN
309*
310* Determine the block size
311*
312 nb = ilaenv( 1, 'DSYTRF_RK', uplo, n, -1, -1, -1 )
313 lwkopt = max( 1, n*nb )
314 work( 1 ) = lwkopt
315 END IF
316*
317 IF( info.NE.0 ) THEN
318 CALL xerbla( 'DSYTRF_RK', -info )
319 RETURN
320 ELSE IF( lquery ) THEN
321 RETURN
322 END IF
323*
324 nbmin = 2
325 ldwork = n
326 IF( nb.GT.1 .AND. nb.LT.n ) THEN
327 iws = ldwork*nb
328 IF( lwork.LT.iws ) THEN
329 nb = max( lwork / ldwork, 1 )
330 nbmin = max( 2, ilaenv( 2, 'DSYTRF_RK',
331 $ uplo, n, -1, -1, -1 ) )
332 END IF
333 ELSE
334 iws = 1
335 END IF
336 IF( nb.LT.nbmin )
337 $ nb = n
338*
339 IF( upper ) THEN
340*
341* Factorize A as U*D*U**T using the upper triangle of A
342*
343* K is the main loop index, decreasing from N to 1 in steps of
344* KB, where KB is the number of columns factorized by DLASYF_RK;
345* KB is either NB or NB-1, or K for the last block
346*
347 k = n
348 10 CONTINUE
349*
350* If K < 1, exit from loop
351*
352 IF( k.LT.1 )
353 $ GO TO 15
354*
355 IF( k.GT.nb ) THEN
356*
357* Factorize columns k-kb+1:k of A and use blocked code to
358* update columns 1:k-kb
359*
360 CALL dlasyf_rk( uplo, k, nb, kb, a, lda, e,
361 $ ipiv, work, ldwork, iinfo )
362 ELSE
363*
364* Use unblocked code to factorize columns 1:k of A
365*
366 CALL dsytf2_rk( uplo, k, a, lda, e, ipiv, iinfo )
367 kb = k
368 END IF
369*
370* Set INFO on the first occurrence of a zero pivot
371*
372 IF( info.EQ.0 .AND. iinfo.GT.0 )
373 $ info = iinfo
374*
375* No need to adjust IPIV
376*
377*
378* Apply permutations to the leading panel 1:k-1
379*
380* Read IPIV from the last block factored, i.e.
381* indices k-kb+1:k and apply row permutations to the
382* last k+1 colunms k+1:N after that block
383* (We can do the simple loop over IPIV with decrement -1,
384* since the ABS value of IPIV( I ) represents the row index
385* of the interchange with row i in both 1x1 and 2x2 pivot cases)
386*
387 IF( k.LT.n ) THEN
388 DO i = k, ( k - kb + 1 ), -1
389 ip = abs( ipiv( i ) )
390 IF( ip.NE.i ) THEN
391 CALL dswap( n-k, a( i, k+1 ), lda,
392 $ a( ip, k+1 ), lda )
393 END IF
394 END DO
395 END IF
396*
397* Decrease K and return to the start of the main loop
398*
399 k = k - kb
400 GO TO 10
401*
402* This label is the exit from main loop over K decreasing
403* from N to 1 in steps of KB
404*
405 15 CONTINUE
406*
407 ELSE
408*
409* Factorize A as L*D*L**T using the lower triangle of A
410*
411* K is the main loop index, increasing from 1 to N in steps of
412* KB, where KB is the number of columns factorized by DLASYF_RK;
413* KB is either NB or NB-1, or N-K+1 for the last block
414*
415 k = 1
416 20 CONTINUE
417*
418* If K > N, exit from loop
419*
420 IF( k.GT.n )
421 $ GO TO 35
422*
423 IF( k.LE.n-nb ) THEN
424*
425* Factorize columns k:k+kb-1 of A and use blocked code to
426* update columns k+kb:n
427*
428 CALL dlasyf_rk( uplo, n-k+1, nb, kb, a( k, k ), lda, e( k ),
429 $ ipiv( k ), work, ldwork, iinfo )
430
431
432 ELSE
433*
434* Use unblocked code to factorize columns k:n of A
435*
436 CALL dsytf2_rk( uplo, n-k+1, a( k, k ), lda, e( k ),
437 $ ipiv( k ), iinfo )
438 kb = n - k + 1
439*
440 END IF
441*
442* Set INFO on the first occurrence of a zero pivot
443*
444 IF( info.EQ.0 .AND. iinfo.GT.0 )
445 $ info = iinfo + k - 1
446*
447* Adjust IPIV
448*
449 DO i = k, k + kb - 1
450 IF( ipiv( i ).GT.0 ) THEN
451 ipiv( i ) = ipiv( i ) + k - 1
452 ELSE
453 ipiv( i ) = ipiv( i ) - k + 1
454 END IF
455 END DO
456*
457* Apply permutations to the leading panel 1:k-1
458*
459* Read IPIV from the last block factored, i.e.
460* indices k:k+kb-1 and apply row permutations to the
461* first k-1 colunms 1:k-1 before that block
462* (We can do the simple loop over IPIV with increment 1,
463* since the ABS value of IPIV( I ) represents the row index
464* of the interchange with row i in both 1x1 and 2x2 pivot cases)
465*
466 IF( k.GT.1 ) THEN
467 DO i = k, ( k + kb - 1 ), 1
468 ip = abs( ipiv( i ) )
469 IF( ip.NE.i ) THEN
470 CALL dswap( k-1, a( i, 1 ), lda,
471 $ a( ip, 1 ), lda )
472 END IF
473 END DO
474 END IF
475*
476* Increase K and return to the start of the main loop
477*
478 k = k + kb
479 GO TO 20
480*
481* This label is the exit from main loop over K increasing
482* from 1 to N in steps of KB
483*
484 35 CONTINUE
485*
486* End Lower
487*
488 END IF
489*
490 work( 1 ) = lwkopt
491 RETURN
492*
493* End of DSYTRF_RK
494*
subroutine xerbla(srname, info)
Definition cblat2.f:3285
subroutine dsytf2_rk(uplo, n, a, lda, e, ipiv, info)
DSYTF2_RK computes the factorization of a real symmetric indefinite matrix using the bounded Bunch-Ka...
Definition dsytf2_rk.f:241
integer function ilaenv(ispec, name, opts, n1, n2, n3, n4)
ILAENV
Definition ilaenv.f:162
subroutine dlasyf_rk(uplo, n, nb, kb, a, lda, e, ipiv, w, ldw, info)
DLASYF_RK computes a partial factorization of a real symmetric indefinite matrix using bounded Bunch-...
Definition dlasyf_rk.f:262
logical function lsame(ca, cb)
LSAME
Definition lsame.f:48
subroutine dswap(n, dx, incx, dy, incy)
DSWAP
Definition dswap.f:82
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