LAPACK  3.10.0
LAPACK: Linear Algebra PACKage

◆ zhetrf()

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

ZHETRF

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

Purpose:
 ZHETRF computes the factorization of a complex Hermitian matrix A
 using the Bunch-Kaufman diagonal pivoting method.  The form of the
 factorization is

    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, and D is Hermitian 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.
Parameters
[in]UPLO
          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.
[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 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 UPLO = 'U' and 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' and 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]WORK
          WORK is COMPLEX*16 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.
[out]INFO
          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, D(i,i) 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**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).

Definition at line 176 of file zhetrf.f.

177 *
178 * -- LAPACK computational routine --
179 * -- LAPACK is a software package provided by Univ. of Tennessee, --
180 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
181 *
182 * .. Scalar Arguments ..
183  CHARACTER UPLO
184  INTEGER INFO, LDA, LWORK, N
185 * ..
186 * .. Array Arguments ..
187  INTEGER IPIV( * )
188  COMPLEX*16 A( LDA, * ), WORK( * )
189 * ..
190 *
191 * =====================================================================
192 *
193 * .. Local Scalars ..
194  LOGICAL LQUERY, UPPER
195  INTEGER IINFO, IWS, J, K, KB, LDWORK, LWKOPT, NB, NBMIN
196 * ..
197 * .. External Functions ..
198  LOGICAL LSAME
199  INTEGER ILAENV
200  EXTERNAL lsame, ilaenv
201 * ..
202 * .. External Subroutines ..
203  EXTERNAL xerbla, zhetf2, zlahef
204 * ..
205 * .. Intrinsic Functions ..
206  INTRINSIC max
207 * ..
208 * .. Executable Statements ..
209 *
210 * Test the input parameters.
211 *
212  info = 0
213  upper = lsame( uplo, 'U' )
214  lquery = ( lwork.EQ.-1 )
215  IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
216  info = -1
217  ELSE IF( n.LT.0 ) THEN
218  info = -2
219  ELSE IF( lda.LT.max( 1, n ) ) THEN
220  info = -4
221  ELSE IF( lwork.LT.1 .AND. .NOT.lquery ) THEN
222  info = -7
223  END IF
224 *
225  IF( info.EQ.0 ) THEN
226 *
227 * Determine the block size
228 *
229  nb = ilaenv( 1, 'ZHETRF', uplo, n, -1, -1, -1 )
230  lwkopt = n*nb
231  work( 1 ) = lwkopt
232  END IF
233 *
234  IF( info.NE.0 ) THEN
235  CALL xerbla( 'ZHETRF', -info )
236  RETURN
237  ELSE IF( lquery ) THEN
238  RETURN
239  END IF
240 *
241  nbmin = 2
242  ldwork = n
243  IF( nb.GT.1 .AND. nb.LT.n ) THEN
244  iws = ldwork*nb
245  IF( lwork.LT.iws ) THEN
246  nb = max( lwork / ldwork, 1 )
247  nbmin = max( 2, ilaenv( 2, 'ZHETRF', uplo, n, -1, -1, -1 ) )
248  END IF
249  ELSE
250  iws = 1
251  END IF
252  IF( nb.LT.nbmin )
253  $ nb = n
254 *
255  IF( upper ) THEN
256 *
257 * Factorize A as U*D*U**H using the upper triangle of A
258 *
259 * K is the main loop index, decreasing from N to 1 in steps of
260 * KB, where KB is the number of columns factorized by ZLAHEF;
261 * KB is either NB or NB-1, or K for the last block
262 *
263  k = n
264  10 CONTINUE
265 *
266 * If K < 1, exit from loop
267 *
268  IF( k.LT.1 )
269  $ GO TO 40
270 *
271  IF( k.GT.nb ) THEN
272 *
273 * Factorize columns k-kb+1:k of A and use blocked code to
274 * update columns 1:k-kb
275 *
276  CALL zlahef( uplo, k, nb, kb, a, lda, ipiv, work, n, iinfo )
277  ELSE
278 *
279 * Use unblocked code to factorize columns 1:k of A
280 *
281  CALL zhetf2( uplo, k, a, lda, ipiv, iinfo )
282  kb = k
283  END IF
284 *
285 * Set INFO on the first occurrence of a zero pivot
286 *
287  IF( info.EQ.0 .AND. iinfo.GT.0 )
288  $ info = iinfo
289 *
290 * Decrease K and return to the start of the main loop
291 *
292  k = k - kb
293  GO TO 10
294 *
295  ELSE
296 *
297 * Factorize A as L*D*L**H using the lower triangle of A
298 *
299 * K is the main loop index, increasing from 1 to N in steps of
300 * KB, where KB is the number of columns factorized by ZLAHEF;
301 * KB is either NB or NB-1, or N-K+1 for the last block
302 *
303  k = 1
304  20 CONTINUE
305 *
306 * If K > N, exit from loop
307 *
308  IF( k.GT.n )
309  $ GO TO 40
310 *
311  IF( k.LE.n-nb ) THEN
312 *
313 * Factorize columns k:k+kb-1 of A and use blocked code to
314 * update columns k+kb:n
315 *
316  CALL zlahef( uplo, n-k+1, nb, kb, a( k, k ), lda, ipiv( k ),
317  $ work, n, iinfo )
318  ELSE
319 *
320 * Use unblocked code to factorize columns k:n of A
321 *
322  CALL zhetf2( uplo, n-k+1, a( k, k ), lda, ipiv( k ), iinfo )
323  kb = n - k + 1
324  END IF
325 *
326 * Set INFO on the first occurrence of a zero pivot
327 *
328  IF( info.EQ.0 .AND. iinfo.GT.0 )
329  $ info = iinfo + k - 1
330 *
331 * Adjust IPIV
332 *
333  DO 30 j = k, k + kb - 1
334  IF( ipiv( j ).GT.0 ) THEN
335  ipiv( j ) = ipiv( j ) + k - 1
336  ELSE
337  ipiv( j ) = ipiv( j ) - k + 1
338  END IF
339  30 CONTINUE
340 *
341 * Increase K and return to the start of the main loop
342 *
343  k = k + kb
344  GO TO 20
345 *
346  END IF
347 *
348  40 CONTINUE
349  work( 1 ) = lwkopt
350  RETURN
351 *
352 * End of ZHETRF
353 *
integer function ilaenv(ISPEC, NAME, OPTS, N1, N2, N3, N4)
ILAENV
Definition: ilaenv.f:162
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine zhetf2(UPLO, N, A, LDA, IPIV, INFO)
ZHETF2 computes the factorization of a complex Hermitian matrix, using the diagonal pivoting method (...
Definition: zhetf2.f:191
subroutine zlahef(UPLO, N, NB, KB, A, LDA, IPIV, W, LDW, INFO)
ZLAHEF computes a partial factorization of a complex Hermitian indefinite matrix using the Bunch-Kauf...
Definition: zlahef.f:177
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