LAPACK  3.6.0 LAPACK: Linear Algebra PACKage
complex16
Collaboration diagram for complex16:


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## Functions

subroutine zcposv (UPLO, N, NRHS, A, LDA, B, LDB, X, LDX, WORK, SWORK, RWORK, ITER, INFO)
ZCPOSV computes the solution to system of linear equations A * X = B for PO matrices More...

subroutine zposv (UPLO, N, NRHS, A, LDA, B, LDB, INFO)
ZPOSV computes the solution to system of linear equations A * X = B for PO matrices More...

subroutine zposvx (FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, RWORK, INFO)
ZPOSVX computes the solution to system of linear equations A * X = B for PO matrices More...

subroutine zposvxx (FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED, S, B, LDB, X, LDX, RCOND, RPVGRW, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, RWORK, INFO)
ZPOSVXX computes the solution to system of linear equations A * X = B for PO matrices More...

## Detailed Description

This is the group of complex16 solve driver functions for PO matrices

## Function Documentation

 subroutine zcposv ( character UPLO, integer N, integer NRHS, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension( ldx, * ) X, integer LDX, complex*16, dimension( n, * ) WORK, complex, dimension( * ) SWORK, double precision, dimension( * ) RWORK, integer ITER, integer INFO )

ZCPOSV computes the solution to system of linear equations A * X = B for PO matrices

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

Purpose:
``` ZCPOSV computes the solution to a complex system of linear equations
A * X = B,
where A is an N-by-N Hermitian positive definite matrix and X and B
are N-by-NRHS matrices.

ZCPOSV first attempts to factorize the matrix in COMPLEX and use this
factorization within an iterative refinement procedure to produce a
solution with COMPLEX*16 normwise backward error quality (see below).
If the approach fails the method switches to a COMPLEX*16
factorization and solve.

The iterative refinement is not going to be a winning strategy if
the ratio COMPLEX performance over COMPLEX*16 performance is too
small. A reasonable strategy should take the number of right-hand
sides and the size of the matrix into account. This might be done
with a call to ILAENV in the future. Up to now, we always try
iterative refinement.

The iterative refinement process is stopped if
ITER > ITERMAX
or for all the RHS we have:
RNRM < SQRT(N)*XNRM*ANRM*EPS*BWDMAX
where
o ITER is the number of the current iteration in the iterative
refinement process
o RNRM is the infinity-norm of the residual
o XNRM is the infinity-norm of the solution
o ANRM is the infinity-operator-norm of the matrix A
o EPS is the machine epsilon returned by DLAMCH('Epsilon')
The value ITERMAX and BWDMAX are fixed to 30 and 1.0D+00
respectively.```
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 number of linear equations, i.e., the order of the matrix A. N >= 0.``` [in] NRHS ``` NRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 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. Note that the imaginary parts of the diagonal elements need not be set and are assumed to be zero. On exit, if iterative refinement has been successfully used (INFO.EQ.0 and ITER.GE.0, see description below), then A is unchanged, if double precision factorization has been used (INFO.EQ.0 and ITER.LT.0, see description below), then the array A contains the factor U or L from the Cholesky factorization A = U**H*U or A = L*L**H.``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).``` [in] B ``` B is COMPLEX*16 array, dimension (LDB,NRHS) The N-by-NRHS right hand side matrix B.``` [in] LDB ``` LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).``` [out] X ``` X is COMPLEX*16 array, dimension (LDX,NRHS) If INFO = 0, the N-by-NRHS solution matrix X.``` [in] LDX ``` LDX is INTEGER The leading dimension of the array X. LDX >= max(1,N).``` [out] WORK ``` WORK is COMPLEX*16 array, dimension (N*NRHS) This array is used to hold the residual vectors.``` [out] SWORK ``` SWORK is COMPLEX array, dimension (N*(N+NRHS)) This array is used to use the single precision matrix and the right-hand sides or solutions in single precision.``` [out] RWORK ` RWORK is DOUBLE PRECISION array, dimension (N)` [out] ITER ``` ITER is INTEGER < 0: iterative refinement has failed, COMPLEX*16 factorization has been performed -1 : the routine fell back to full precision for implementation- or machine-specific reasons -2 : narrowing the precision induced an overflow, the routine fell back to full precision -3 : failure of CPOTRF -31: stop the iterative refinement after the 30th iterations > 0: iterative refinement has been sucessfully used. Returns the number of iterations``` [out] INFO ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, the leading minor of order i of (COMPLEX*16) A is not positive definite, so the factorization could not be completed, and the solution has not been computed.```
Date
November 2011

Definition at line 211 of file zcposv.f.

211 *
212 * -- LAPACK driver routine (version 3.4.0) --
213 * -- LAPACK is a software package provided by Univ. of Tennessee, --
214 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
215 * November 2011
216 *
217 * .. Scalar Arguments ..
218  CHARACTER uplo
219  INTEGER info, iter, lda, ldb, ldx, n, nrhs
220 * ..
221 * .. Array Arguments ..
222  DOUBLE PRECISION rwork( * )
223  COMPLEX swork( * )
224  COMPLEX*16 a( lda, * ), b( ldb, * ), work( n, * ),
225  \$ x( ldx, * )
226 * ..
227 *
228 * =====================================================================
229 *
230 * .. Parameters ..
231  LOGICAL doitref
232  parameter( doitref = .true. )
233 *
234  INTEGER itermax
235  parameter( itermax = 30 )
236 *
237  DOUBLE PRECISION bwdmax
238  parameter( bwdmax = 1.0e+00 )
239 *
240  COMPLEX*16 negone, one
241  parameter( negone = ( -1.0d+00, 0.0d+00 ),
242  \$ one = ( 1.0d+00, 0.0d+00 ) )
243 *
244 * .. Local Scalars ..
245  INTEGER i, iiter, ptsa, ptsx
246  DOUBLE PRECISION anrm, cte, eps, rnrm, xnrm
247  COMPLEX*16 zdum
248 *
249 * .. External Subroutines ..
250  EXTERNAL zaxpy, zhemm, zlacpy, zlat2c, zlag2c, clag2z,
251  \$ cpotrf, cpotrs, xerbla
252 * ..
253 * .. External Functions ..
254  INTEGER izamax
255  DOUBLE PRECISION dlamch, zlanhe
256  LOGICAL lsame
257  EXTERNAL izamax, dlamch, zlanhe, lsame
258 * ..
259 * .. Intrinsic Functions ..
260  INTRINSIC abs, dble, max, sqrt
261 * .. Statement Functions ..
262  DOUBLE PRECISION cabs1
263 * ..
264 * .. Statement Function definitions ..
265  cabs1( zdum ) = abs( dble( zdum ) ) + abs( dimag( zdum ) )
266 * ..
267 * .. Executable Statements ..
268 *
269  info = 0
270  iter = 0
271 *
272 * Test the input parameters.
273 *
274  IF( .NOT.lsame( uplo, 'U' ) .AND. .NOT.lsame( uplo, 'L' ) ) THEN
275  info = -1
276  ELSE IF( n.LT.0 ) THEN
277  info = -2
278  ELSE IF( nrhs.LT.0 ) THEN
279  info = -3
280  ELSE IF( lda.LT.max( 1, n ) ) THEN
281  info = -5
282  ELSE IF( ldb.LT.max( 1, n ) ) THEN
283  info = -7
284  ELSE IF( ldx.LT.max( 1, n ) ) THEN
285  info = -9
286  END IF
287  IF( info.NE.0 ) THEN
288  CALL xerbla( 'ZCPOSV', -info )
289  RETURN
290  END IF
291 *
292 * Quick return if (N.EQ.0).
293 *
294  IF( n.EQ.0 )
295  \$ RETURN
296 *
297 * Skip single precision iterative refinement if a priori slower
298 * than double precision factorization.
299 *
300  IF( .NOT.doitref ) THEN
301  iter = -1
302  GO TO 40
303  END IF
304 *
305 * Compute some constants.
306 *
307  anrm = zlanhe( 'I', uplo, n, a, lda, rwork )
308  eps = dlamch( 'Epsilon' )
309  cte = anrm*eps*sqrt( dble( n ) )*bwdmax
310 *
311 * Set the indices PTSA, PTSX for referencing SA and SX in SWORK.
312 *
313  ptsa = 1
314  ptsx = ptsa + n*n
315 *
316 * Convert B from double precision to single precision and store the
317 * result in SX.
318 *
319  CALL zlag2c( n, nrhs, b, ldb, swork( ptsx ), n, info )
320 *
321  IF( info.NE.0 ) THEN
322  iter = -2
323  GO TO 40
324  END IF
325 *
326 * Convert A from double precision to single precision and store the
327 * result in SA.
328 *
329  CALL zlat2c( uplo, n, a, lda, swork( ptsa ), n, info )
330 *
331  IF( info.NE.0 ) THEN
332  iter = -2
333  GO TO 40
334  END IF
335 *
336 * Compute the Cholesky factorization of SA.
337 *
338  CALL cpotrf( uplo, n, swork( ptsa ), n, info )
339 *
340  IF( info.NE.0 ) THEN
341  iter = -3
342  GO TO 40
343  END IF
344 *
345 * Solve the system SA*SX = SB.
346 *
347  CALL cpotrs( uplo, n, nrhs, swork( ptsa ), n, swork( ptsx ), n,
348  \$ info )
349 *
350 * Convert SX back to COMPLEX*16
351 *
352  CALL clag2z( n, nrhs, swork( ptsx ), n, x, ldx, info )
353 *
354 * Compute R = B - AX (R is WORK).
355 *
356  CALL zlacpy( 'All', n, nrhs, b, ldb, work, n )
357 *
358  CALL zhemm( 'Left', uplo, n, nrhs, negone, a, lda, x, ldx, one,
359  \$ work, n )
360 *
361 * Check whether the NRHS normwise backward errors satisfy the
362 * stopping criterion. If yes, set ITER=0 and return.
363 *
364  DO i = 1, nrhs
365  xnrm = cabs1( x( izamax( n, x( 1, i ), 1 ), i ) )
366  rnrm = cabs1( work( izamax( n, work( 1, i ), 1 ), i ) )
367  IF( rnrm.GT.xnrm*cte )
368  \$ GO TO 10
369  END DO
370 *
371 * If we are here, the NRHS normwise backward errors satisfy the
372 * stopping criterion. We are good to exit.
373 *
374  iter = 0
375  RETURN
376 *
377  10 CONTINUE
378 *
379  DO 30 iiter = 1, itermax
380 *
381 * Convert R (in WORK) from double precision to single precision
382 * and store the result in SX.
383 *
384  CALL zlag2c( n, nrhs, work, n, swork( ptsx ), n, info )
385 *
386  IF( info.NE.0 ) THEN
387  iter = -2
388  GO TO 40
389  END IF
390 *
391 * Solve the system SA*SX = SR.
392 *
393  CALL cpotrs( uplo, n, nrhs, swork( ptsa ), n, swork( ptsx ), n,
394  \$ info )
395 *
396 * Convert SX back to double precision and update the current
397 * iterate.
398 *
399  CALL clag2z( n, nrhs, swork( ptsx ), n, work, n, info )
400 *
401  DO i = 1, nrhs
402  CALL zaxpy( n, one, work( 1, i ), 1, x( 1, i ), 1 )
403  END DO
404 *
405 * Compute R = B - AX (R is WORK).
406 *
407  CALL zlacpy( 'All', n, nrhs, b, ldb, work, n )
408 *
409  CALL zhemm( 'L', uplo, n, nrhs, negone, a, lda, x, ldx, one,
410  \$ work, n )
411 *
412 * Check whether the NRHS normwise backward errors satisfy the
413 * stopping criterion. If yes, set ITER=IITER>0 and return.
414 *
415  DO i = 1, nrhs
416  xnrm = cabs1( x( izamax( n, x( 1, i ), 1 ), i ) )
417  rnrm = cabs1( work( izamax( n, work( 1, i ), 1 ), i ) )
418  IF( rnrm.GT.xnrm*cte )
419  \$ GO TO 20
420  END DO
421 *
422 * If we are here, the NRHS normwise backward errors satisfy the
423 * stopping criterion, we are good to exit.
424 *
425  iter = iiter
426 *
427  RETURN
428 *
429  20 CONTINUE
430 *
431  30 CONTINUE
432 *
433 * If we are at this place of the code, this is because we have
434 * performed ITER=ITERMAX iterations and never satisified the
435 * stopping criterion, set up the ITER flag accordingly and follow
436 * up on double precision routine.
437 *
438  iter = -itermax - 1
439 *
440  40 CONTINUE
441 *
442 * Single-precision iterative refinement failed to converge to a
443 * satisfactory solution, so we resort to double precision.
444 *
445  CALL zpotrf( uplo, n, a, lda, info )
446 *
447  IF( info.NE.0 )
448  \$ RETURN
449 *
450  CALL zlacpy( 'All', n, nrhs, b, ldb, x, ldx )
451  CALL zpotrs( uplo, n, nrhs, a, lda, x, ldx, info )
452 *
453  RETURN
454 *
455 * End of ZCPOSV.
456 *
subroutine zpotrf(UPLO, N, A, LDA, INFO)
ZPOTRF VARIANT: right looking block version of the algorithm, calling Level 3 BLAS.
Definition: zpotrf.f:102
integer function izamax(N, ZX, INCX)
IZAMAX
Definition: izamax.f:53
subroutine zlacpy(UPLO, M, N, A, LDA, B, LDB)
ZLACPY copies all or part of one two-dimensional array to another.
Definition: zlacpy.f:105
subroutine zpotrs(UPLO, N, NRHS, A, LDA, B, LDB, INFO)
ZPOTRS
Definition: zpotrs.f:112
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
double precision function zlanhe(NORM, UPLO, N, A, LDA, WORK)
ZLANHE returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a complex Hermitian matrix.
Definition: zlanhe.f:126
subroutine zhemm(SIDE, UPLO, M, N, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
ZHEMM
Definition: zhemm.f:193
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:65
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
subroutine zaxpy(N, ZA, ZX, INCX, ZY, INCY)
ZAXPY
Definition: zaxpy.f:53
subroutine cpotrs(UPLO, N, NRHS, A, LDA, B, LDB, INFO)
CPOTRS
Definition: cpotrs.f:112
subroutine cpotrf(UPLO, N, A, LDA, INFO)
CPOTRF
Definition: cpotrf.f:109
subroutine zlat2c(UPLO, N, A, LDA, SA, LDSA, INFO)
ZLAT2C converts a double complex triangular matrix to a complex triangular matrix.
Definition: zlat2c.f:113
subroutine clag2z(M, N, SA, LDSA, A, LDA, INFO)
CLAG2Z converts a complex single precision matrix to a complex double precision matrix.
Definition: clag2z.f:105
subroutine zlag2c(M, N, A, LDA, SA, LDSA, INFO)
ZLAG2C converts a complex double precision matrix to a complex single precision matrix.
Definition: zlag2c.f:109

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 subroutine zposv ( character UPLO, integer N, integer NRHS, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldb, * ) B, integer LDB, integer INFO )

ZPOSV computes the solution to system of linear equations A * X = B for PO matrices

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

Purpose:
``` ZPOSV computes the solution to a complex system of linear equations
A * X = B,
where A is an N-by-N Hermitian positive definite matrix and X and B
are N-by-NRHS matrices.

The Cholesky decomposition is used to factor A as
A = U**H* U,  if UPLO = 'U', or
A = L * L**H,  if UPLO = 'L',
where U is an upper triangular matrix and  L is a lower triangular
matrix.  The factored form of A is then used to solve the system of
equations A * X = B.```
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 number of linear equations, i.e., the order of the matrix A. N >= 0.``` [in] NRHS ``` NRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 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, if INFO = 0, the factor U or L from the Cholesky factorization A = U**H *U or A = L*L**H.``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).``` [in,out] B ``` B is COMPLEX*16 array, dimension (LDB,NRHS) On entry, the N-by-NRHS right hand side matrix B. On exit, if INFO = 0, the N-by-NRHS solution matrix X.``` [in] LDB ``` LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).``` [out] INFO ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, the leading minor of order i of A is not positive definite, so the factorization could not be completed, and the solution has not been computed.```
Date
November 2011

Definition at line 132 of file zposv.f.

132 *
133 * -- LAPACK driver routine (version 3.4.0) --
134 * -- LAPACK is a software package provided by Univ. of Tennessee, --
135 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
136 * November 2011
137 *
138 * .. Scalar Arguments ..
139  CHARACTER uplo
140  INTEGER info, lda, ldb, n, nrhs
141 * ..
142 * .. Array Arguments ..
143  COMPLEX*16 a( lda, * ), b( ldb, * )
144 * ..
145 *
146 * =====================================================================
147 *
148 * .. External Functions ..
149  LOGICAL lsame
150  EXTERNAL lsame
151 * ..
152 * .. External Subroutines ..
153  EXTERNAL xerbla, zpotrf, zpotrs
154 * ..
155 * .. Intrinsic Functions ..
156  INTRINSIC max
157 * ..
158 * .. Executable Statements ..
159 *
160 * Test the input parameters.
161 *
162  info = 0
163  IF( .NOT.lsame( uplo, 'U' ) .AND. .NOT.lsame( uplo, 'L' ) ) THEN
164  info = -1
165  ELSE IF( n.LT.0 ) THEN
166  info = -2
167  ELSE IF( nrhs.LT.0 ) THEN
168  info = -3
169  ELSE IF( lda.LT.max( 1, n ) ) THEN
170  info = -5
171  ELSE IF( ldb.LT.max( 1, n ) ) THEN
172  info = -7
173  END IF
174  IF( info.NE.0 ) THEN
175  CALL xerbla( 'ZPOSV ', -info )
176  RETURN
177  END IF
178 *
179 * Compute the Cholesky factorization A = U**H *U or A = L*L**H.
180 *
181  CALL zpotrf( uplo, n, a, lda, info )
182  IF( info.EQ.0 ) THEN
183 *
184 * Solve the system A*X = B, overwriting B with X.
185 *
186  CALL zpotrs( uplo, n, nrhs, a, lda, b, ldb, info )
187 *
188  END IF
189  RETURN
190 *
191 * End of ZPOSV
192 *
subroutine zpotrf(UPLO, N, A, LDA, INFO)
ZPOTRF VARIANT: right looking block version of the algorithm, calling Level 3 BLAS.
Definition: zpotrf.f:102
subroutine zpotrs(UPLO, N, NRHS, A, LDA, B, LDB, INFO)
ZPOTRS
Definition: zpotrs.f:112
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55

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 subroutine zposvx ( character FACT, character UPLO, integer N, integer NRHS, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldaf, * ) AF, integer LDAF, character EQUED, double precision, dimension( * ) S, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension( ldx, * ) X, integer LDX, double precision RCOND, double precision, dimension( * ) FERR, double precision, dimension( * ) BERR, complex*16, dimension( * ) WORK, double precision, dimension( * ) RWORK, integer INFO )

ZPOSVX computes the solution to system of linear equations A * X = B for PO matrices

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

Purpose:
``` ZPOSVX uses the Cholesky factorization A = U**H*U or A = L*L**H to
compute the solution to a complex system of linear equations
A * X = B,
where A is an N-by-N Hermitian positive definite matrix and X and B
are N-by-NRHS matrices.

Error bounds on the solution and a condition estimate are also
provided.```
Description:
``` The following steps are performed:

1. If FACT = 'E', real scaling factors are computed to equilibrate
the system:
diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B
Whether or not the system will be equilibrated depends on the
scaling of the matrix A, but if equilibration is used, A is
overwritten by diag(S)*A*diag(S) and B by diag(S)*B.

2. If FACT = 'N' or 'E', the Cholesky decomposition is used to
factor the matrix A (after equilibration if FACT = 'E') as
A = U**H* U,  if UPLO = 'U', or
A = L * L**H,  if UPLO = 'L',
where U is an upper triangular matrix and L is a lower triangular
matrix.

3. If the leading i-by-i principal minor is not positive definite,
then the routine returns with INFO = i. Otherwise, the factored
form of A is used to estimate the condition number of the matrix
A.  If the reciprocal of the condition number is less than machine
precision, INFO = N+1 is returned as a warning, but the routine
still goes on to solve for X and compute error bounds as
described below.

4. The system of equations is solved for X using the factored form
of A.

5. Iterative refinement is applied to improve the computed solution
matrix and calculate error bounds and backward error estimates
for it.

6. If equilibration was used, the matrix X is premultiplied by
diag(S) so that it solves the original system before
equilibration.```
Parameters
 [in] FACT ``` FACT is CHARACTER*1 Specifies whether or not the factored form of the matrix A is supplied on entry, and if not, whether the matrix A should be equilibrated before it is factored. = 'F': On entry, AF contains the factored form of A. If EQUED = 'Y', the matrix A has been equilibrated with scaling factors given by S. A and AF will not be modified. = 'N': The matrix A will be copied to AF and factored. = 'E': The matrix A will be equilibrated if necessary, then copied to AF and factored.``` [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 number of linear equations, i.e., the order of the matrix A. N >= 0.``` [in] NRHS ``` NRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0.``` [in,out] A ``` A is COMPLEX*16 array, dimension (LDA,N) On entry, the Hermitian matrix A, except if FACT = 'F' and EQUED = 'Y', then A must contain the equilibrated matrix diag(S)*A*diag(S). 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. A is not modified if FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit. On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by diag(S)*A*diag(S).``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).``` [in,out] AF ``` AF is COMPLEX*16 array, dimension (LDAF,N) If FACT = 'F', then AF is an input argument and on entry contains the triangular factor U or L from the Cholesky factorization A = U**H *U or A = L*L**H, in the same storage format as A. If EQUED .ne. 'N', then AF is the factored form of the equilibrated matrix diag(S)*A*diag(S). If FACT = 'N', then AF is an output argument and on exit returns the triangular factor U or L from the Cholesky factorization A = U**H *U or A = L*L**H of the original matrix A. If FACT = 'E', then AF is an output argument and on exit returns the triangular factor U or L from the Cholesky factorization A = U**H *U or A = L*L**H of the equilibrated matrix A (see the description of A for the form of the equilibrated matrix).``` [in] LDAF ``` LDAF is INTEGER The leading dimension of the array AF. LDAF >= max(1,N).``` [in,out] EQUED ``` EQUED is CHARACTER*1 Specifies the form of equilibration that was done. = 'N': No equilibration (always true if FACT = 'N'). = 'Y': Equilibration was done, i.e., A has been replaced by diag(S) * A * diag(S). EQUED is an input argument if FACT = 'F'; otherwise, it is an output argument.``` [in,out] S ``` S is DOUBLE PRECISION array, dimension (N) The scale factors for A; not accessed if EQUED = 'N'. S is an input argument if FACT = 'F'; otherwise, S is an output argument. If FACT = 'F' and EQUED = 'Y', each element of S must be positive.``` [in,out] B ``` B is COMPLEX*16 array, dimension (LDB,NRHS) On entry, the N-by-NRHS righthand side matrix B. On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y', B is overwritten by diag(S) * B.``` [in] LDB ``` LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).``` [out] X ``` X is COMPLEX*16 array, dimension (LDX,NRHS) If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to the original system of equations. Note that if EQUED = 'Y', A and B are modified on exit, and the solution to the equilibrated system is inv(diag(S))*X.``` [in] LDX ``` LDX is INTEGER The leading dimension of the array X. LDX >= max(1,N).``` [out] RCOND ``` RCOND is DOUBLE PRECISION The estimate of the reciprocal condition number of the matrix A after equilibration (if done). If RCOND is less than the machine precision (in particular, if RCOND = 0), the matrix is singular to working precision. This condition is indicated by a return code of INFO > 0.``` [out] FERR ``` FERR is DOUBLE PRECISION array, dimension (NRHS) The estimated forward error bound for each solution vector X(j) (the j-th column of the solution matrix X). If XTRUE is the true solution corresponding to X(j), FERR(j) is an estimated upper bound for the magnitude of the largest element in (X(j) - XTRUE) divided by the magnitude of the largest element in X(j). The estimate is as reliable as the estimate for RCOND, and is almost always a slight overestimate of the true error.``` [out] BERR ``` BERR is DOUBLE PRECISION array, dimension (NRHS) The componentwise relative backward error of each solution vector X(j) (i.e., the smallest relative change in any element of A or B that makes X(j) an exact solution).``` [out] WORK ` WORK is COMPLEX*16 array, dimension (2*N)` [out] RWORK ` RWORK is DOUBLE PRECISION array, dimension (N)` [out] INFO ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, and i is <= N: the leading minor of order i of A is not positive definite, so the factorization could not be completed, and the solution has not been computed. RCOND = 0 is returned. = N+1: U is nonsingular, but RCOND is less than machine precision, meaning that the matrix is singular to working precision. Nevertheless, the solution and error bounds are computed because there are a number of situations where the computed solution can be more accurate than the value of RCOND would suggest.```
Date
April 2012

Definition at line 308 of file zposvx.f.

308 *
309 * -- LAPACK driver routine (version 3.4.1) --
310 * -- LAPACK is a software package provided by Univ. of Tennessee, --
311 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
312 * April 2012
313 *
314 * .. Scalar Arguments ..
315  CHARACTER equed, fact, uplo
316  INTEGER info, lda, ldaf, ldb, ldx, n, nrhs
317  DOUBLE PRECISION rcond
318 * ..
319 * .. Array Arguments ..
320  DOUBLE PRECISION berr( * ), ferr( * ), rwork( * ), s( * )
321  COMPLEX*16 a( lda, * ), af( ldaf, * ), b( ldb, * ),
322  \$ work( * ), x( ldx, * )
323 * ..
324 *
325 * =====================================================================
326 *
327 * .. Parameters ..
328  DOUBLE PRECISION zero, one
329  parameter( zero = 0.0d+0, one = 1.0d+0 )
330 * ..
331 * .. Local Scalars ..
332  LOGICAL equil, nofact, rcequ
333  INTEGER i, infequ, j
334  DOUBLE PRECISION amax, anorm, bignum, scond, smax, smin, smlnum
335 * ..
336 * .. External Functions ..
337  LOGICAL lsame
338  DOUBLE PRECISION dlamch, zlanhe
339  EXTERNAL lsame, dlamch, zlanhe
340 * ..
341 * .. External Subroutines ..
342  EXTERNAL xerbla, zlacpy, zlaqhe, zpocon, zpoequ, zporfs,
343  \$ zpotrf, zpotrs
344 * ..
345 * .. Intrinsic Functions ..
346  INTRINSIC max, min
347 * ..
348 * .. Executable Statements ..
349 *
350  info = 0
351  nofact = lsame( fact, 'N' )
352  equil = lsame( fact, 'E' )
353  IF( nofact .OR. equil ) THEN
354  equed = 'N'
355  rcequ = .false.
356  ELSE
357  rcequ = lsame( equed, 'Y' )
358  smlnum = dlamch( 'Safe minimum' )
359  bignum = one / smlnum
360  END IF
361 *
362 * Test the input parameters.
363 *
364  IF( .NOT.nofact .AND. .NOT.equil .AND. .NOT.lsame( fact, 'F' ) )
365  \$ THEN
366  info = -1
367  ELSE IF( .NOT.lsame( uplo, 'U' ) .AND. .NOT.lsame( uplo, 'L' ) )
368  \$ THEN
369  info = -2
370  ELSE IF( n.LT.0 ) THEN
371  info = -3
372  ELSE IF( nrhs.LT.0 ) THEN
373  info = -4
374  ELSE IF( lda.LT.max( 1, n ) ) THEN
375  info = -6
376  ELSE IF( ldaf.LT.max( 1, n ) ) THEN
377  info = -8
378  ELSE IF( lsame( fact, 'F' ) .AND. .NOT.
379  \$ ( rcequ .OR. lsame( equed, 'N' ) ) ) THEN
380  info = -9
381  ELSE
382  IF( rcequ ) THEN
383  smin = bignum
384  smax = zero
385  DO 10 j = 1, n
386  smin = min( smin, s( j ) )
387  smax = max( smax, s( j ) )
388  10 CONTINUE
389  IF( smin.LE.zero ) THEN
390  info = -10
391  ELSE IF( n.GT.0 ) THEN
392  scond = max( smin, smlnum ) / min( smax, bignum )
393  ELSE
394  scond = one
395  END IF
396  END IF
397  IF( info.EQ.0 ) THEN
398  IF( ldb.LT.max( 1, n ) ) THEN
399  info = -12
400  ELSE IF( ldx.LT.max( 1, n ) ) THEN
401  info = -14
402  END IF
403  END IF
404  END IF
405 *
406  IF( info.NE.0 ) THEN
407  CALL xerbla( 'ZPOSVX', -info )
408  RETURN
409  END IF
410 *
411  IF( equil ) THEN
412 *
413 * Compute row and column scalings to equilibrate the matrix A.
414 *
415  CALL zpoequ( n, a, lda, s, scond, amax, infequ )
416  IF( infequ.EQ.0 ) THEN
417 *
418 * Equilibrate the matrix.
419 *
420  CALL zlaqhe( uplo, n, a, lda, s, scond, amax, equed )
421  rcequ = lsame( equed, 'Y' )
422  END IF
423  END IF
424 *
425 * Scale the right hand side.
426 *
427  IF( rcequ ) THEN
428  DO 30 j = 1, nrhs
429  DO 20 i = 1, n
430  b( i, j ) = s( i )*b( i, j )
431  20 CONTINUE
432  30 CONTINUE
433  END IF
434 *
435  IF( nofact .OR. equil ) THEN
436 *
437 * Compute the Cholesky factorization A = U**H *U or A = L*L**H.
438 *
439  CALL zlacpy( uplo, n, n, a, lda, af, ldaf )
440  CALL zpotrf( uplo, n, af, ldaf, info )
441 *
442 * Return if INFO is non-zero.
443 *
444  IF( info.GT.0 )THEN
445  rcond = zero
446  RETURN
447  END IF
448  END IF
449 *
450 * Compute the norm of the matrix A.
451 *
452  anorm = zlanhe( '1', uplo, n, a, lda, rwork )
453 *
454 * Compute the reciprocal of the condition number of A.
455 *
456  CALL zpocon( uplo, n, af, ldaf, anorm, rcond, work, rwork, info )
457 *
458 * Compute the solution matrix X.
459 *
460  CALL zlacpy( 'Full', n, nrhs, b, ldb, x, ldx )
461  CALL zpotrs( uplo, n, nrhs, af, ldaf, x, ldx, info )
462 *
463 * Use iterative refinement to improve the computed solution and
464 * compute error bounds and backward error estimates for it.
465 *
466  CALL zporfs( uplo, n, nrhs, a, lda, af, ldaf, b, ldb, x, ldx,
467  \$ ferr, berr, work, rwork, info )
468 *
469 * Transform the solution matrix X to a solution of the original
470 * system.
471 *
472  IF( rcequ ) THEN
473  DO 50 j = 1, nrhs
474  DO 40 i = 1, n
475  x( i, j ) = s( i )*x( i, j )
476  40 CONTINUE
477  50 CONTINUE
478  DO 60 j = 1, nrhs
479  ferr( j ) = ferr( j ) / scond
480  60 CONTINUE
481  END IF
482 *
483 * Set INFO = N+1 if the matrix is singular to working precision.
484 *
485  IF( rcond.LT.dlamch( 'Epsilon' ) )
486  \$ info = n + 1
487 *
488  RETURN
489 *
490 * End of ZPOSVX
491 *
subroutine zpotrf(UPLO, N, A, LDA, INFO)
ZPOTRF VARIANT: right looking block version of the algorithm, calling Level 3 BLAS.
Definition: zpotrf.f:102
subroutine zlacpy(UPLO, M, N, A, LDA, B, LDB)
ZLACPY copies all or part of one two-dimensional array to another.
Definition: zlacpy.f:105
subroutine zpotrs(UPLO, N, NRHS, A, LDA, B, LDB, INFO)
ZPOTRS
Definition: zpotrs.f:112
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
double precision function zlanhe(NORM, UPLO, N, A, LDA, WORK)
ZLANHE returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a complex Hermitian matrix.
Definition: zlanhe.f:126
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:65
subroutine zporfs(UPLO, N, NRHS, A, LDA, AF, LDAF, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO)
ZPORFS
Definition: zporfs.f:185
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
subroutine zlaqhe(UPLO, N, A, LDA, S, SCOND, AMAX, EQUED)
ZLAQHE scales a Hermitian matrix.
Definition: zlaqhe.f:136
subroutine zpocon(UPLO, N, A, LDA, ANORM, RCOND, WORK, RWORK, INFO)
ZPOCON
Definition: zpocon.f:123
subroutine zpoequ(N, A, LDA, S, SCOND, AMAX, INFO)
ZPOEQU
Definition: zpoequ.f:115

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 subroutine zposvxx ( character FACT, character UPLO, integer N, integer NRHS, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldaf, * ) AF, integer LDAF, character EQUED, double precision, dimension( * ) S, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension( ldx, * ) X, integer LDX, double precision RCOND, double precision RPVGRW, double precision, dimension( * ) BERR, integer N_ERR_BNDS, double precision, dimension( nrhs, * ) ERR_BNDS_NORM, double precision, dimension( nrhs, * ) ERR_BNDS_COMP, integer NPARAMS, double precision, dimension( * ) PARAMS, complex*16, dimension( * ) WORK, double precision, dimension( * ) RWORK, integer INFO )

ZPOSVXX computes the solution to system of linear equations A * X = B for PO matrices

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

Purpose:
```    ZPOSVXX uses the Cholesky factorization A = U**T*U or A = L*L**T
to compute the solution to a complex*16 system of linear equations
A * X = B, where A is an N-by-N symmetric positive definite matrix
and X and B are N-by-NRHS matrices.

If requested, both normwise and maximum componentwise error bounds
are returned. ZPOSVXX will return a solution with a tiny
guaranteed error (O(eps) where eps is the working machine
precision) unless the matrix is very ill-conditioned, in which
case a warning is returned. Relevant condition numbers also are
calculated and returned.

ZPOSVXX accepts user-provided factorizations and equilibration
factors; see the definitions of the FACT and EQUED options.
Solving with refinement and using a factorization from a previous
ZPOSVXX call will also produce a solution with either O(eps)
errors or warnings, but we cannot make that claim for general
user-provided factorizations and equilibration factors if they
differ from what ZPOSVXX would itself produce.```
Description:
```    The following steps are performed:

1. If FACT = 'E', double precision scaling factors are computed to equilibrate
the system:

diag(S)*A*diag(S)     *inv(diag(S))*X = diag(S)*B

Whether or not the system will be equilibrated depends on the
scaling of the matrix A, but if equilibration is used, A is
overwritten by diag(S)*A*diag(S) and B by diag(S)*B.

2. If FACT = 'N' or 'E', the Cholesky decomposition is used to
factor the matrix A (after equilibration if FACT = 'E') as
A = U**T* U,  if UPLO = 'U', or
A = L * L**T,  if UPLO = 'L',
where U is an upper triangular matrix and L is a lower triangular
matrix.

3. If the leading i-by-i principal minor is not positive definite,
then the routine returns with INFO = i. Otherwise, the factored
form of A is used to estimate the condition number of the matrix
A (see argument RCOND).  If the reciprocal of the condition number
is less than machine precision, the routine still goes on to solve
for X and compute error bounds as described below.

4. The system of equations is solved for X using the factored form
of A.

5. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero),
the routine will use iterative refinement to try to get a small
error and error bounds.  Refinement calculates the residual to at
least twice the working precision.

6. If equilibration was used, the matrix X is premultiplied by
diag(S) so that it solves the original system before
equilibration.```
```     Some optional parameters are bundled in the PARAMS array.  These
settings determine how refinement is performed, but often the
defaults are acceptable.  If the defaults are acceptable, users
can pass NPARAMS = 0 which prevents the source code from accessing
the PARAMS argument.```
Parameters
 [in] FACT ``` FACT is CHARACTER*1 Specifies whether or not the factored form of the matrix A is supplied on entry, and if not, whether the matrix A should be equilibrated before it is factored. = 'F': On entry, AF contains the factored form of A. If EQUED is not 'N', the matrix A has been equilibrated with scaling factors given by S. A and AF are not modified. = 'N': The matrix A will be copied to AF and factored. = 'E': The matrix A will be equilibrated if necessary, then copied to AF and factored.``` [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 number of linear equations, i.e., the order of the matrix A. N >= 0.``` [in] NRHS ``` NRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0.``` [in,out] A ``` A is COMPLEX*16 array, dimension (LDA,N) On entry, the symmetric matrix A, except if FACT = 'F' and EQUED = 'Y', then A must contain the equilibrated matrix diag(S)*A*diag(S). 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. A is not modified if FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit. On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by diag(S)*A*diag(S).``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).``` [in,out] AF ``` AF is COMPLEX*16 array, dimension (LDAF,N) If FACT = 'F', then AF is an input argument and on entry contains the triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T, in the same storage format as A. If EQUED .ne. 'N', then AF is the factored form of the equilibrated matrix diag(S)*A*diag(S). If FACT = 'N', then AF is an output argument and on exit returns the triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T of the original matrix A. If FACT = 'E', then AF is an output argument and on exit returns the triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T of the equilibrated matrix A (see the description of A for the form of the equilibrated matrix).``` [in] LDAF ``` LDAF is INTEGER The leading dimension of the array AF. LDAF >= max(1,N).``` [in,out] EQUED ``` EQUED is CHARACTER*1 Specifies the form of equilibration that was done. = 'N': No equilibration (always true if FACT = 'N'). = 'Y': Both row and column equilibration, i.e., A has been replaced by diag(S) * A * diag(S). EQUED is an input argument if FACT = 'F'; otherwise, it is an output argument.``` [in,out] S ``` S is DOUBLE PRECISION array, dimension (N) The row scale factors for A. If EQUED = 'Y', A is multiplied on the left and right by diag(S). S is an input argument if FACT = 'F'; otherwise, S is an output argument. If FACT = 'F' and EQUED = 'Y', each element of S must be positive. If S is output, each element of S is a power of the radix. If S is input, each element of S should be a power of the radix to ensure a reliable solution and error estimates. Scaling by powers of the radix does not cause rounding errors unless the result underflows or overflows. Rounding errors during scaling lead to refining with a matrix that is not equivalent to the input matrix, producing error estimates that may not be reliable.``` [in,out] B ``` B is COMPLEX*16 array, dimension (LDB,NRHS) On entry, the N-by-NRHS right hand side matrix B. On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y', B is overwritten by diag(S)*B;``` [in] LDB ``` LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).``` [out] X ``` X is COMPLEX*16 array, dimension (LDX,NRHS) If INFO = 0, the N-by-NRHS solution matrix X to the original system of equations. Note that A and B are modified on exit if EQUED .ne. 'N', and the solution to the equilibrated system is inv(diag(S))*X.``` [in] LDX ``` LDX is INTEGER The leading dimension of the array X. LDX >= max(1,N).``` [out] RCOND ``` RCOND is DOUBLE PRECISION Reciprocal scaled condition number. This is an estimate of the reciprocal Skeel condition number of the matrix A after equilibration (if done). If this is less than the machine precision (in particular, if it is zero), the matrix is singular to working precision. Note that the error may still be small even if this number is very small and the matrix appears ill- conditioned.``` [out] RPVGRW ``` RPVGRW is DOUBLE PRECISION Reciprocal pivot growth. On exit, this contains the reciprocal pivot growth factor norm(A)/norm(U). The "max absolute element" norm is used. If this is much less than 1, then the stability of the LU factorization of the (equilibrated) matrix A could be poor. This also means that the solution X, estimated condition numbers, and error bounds could be unreliable. If factorization fails with 0 0 and <= N: U(INFO,INFO) is exactly zero. The factorization has been completed, but the factor U is exactly singular, so the solution and error bounds could not be computed. RCOND = 0 is returned. = N+J: The solution corresponding to the Jth right-hand side is not guaranteed. The solutions corresponding to other right- hand sides K with K > J may not be guaranteed as well, but only the first such right-hand side is reported. If a small componentwise error is not requested (PARAMS(3) = 0.0) then the Jth right-hand side is the first with a normwise error bound that is not guaranteed (the smallest J such that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0) the Jth right-hand side is the first with either a normwise or componentwise error bound that is not guaranteed (the smallest J such that either ERR_BNDS_NORM(J,1) = 0.0 or ERR_BNDS_COMP(J,1) = 0.0). See the definition of ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information about all of the right-hand sides check ERR_BNDS_NORM or ERR_BNDS_COMP.```
Date
April 2012

Definition at line 495 of file zposvxx.f.

495 *
496 * -- LAPACK driver routine (version 3.4.1) --
497 * -- LAPACK is a software package provided by Univ. of Tennessee, --
498 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
499 * April 2012
500 *
501 * .. Scalar Arguments ..
502  CHARACTER equed, fact, uplo
503  INTEGER info, lda, ldaf, ldb, ldx, n, nrhs, nparams,
504  \$ n_err_bnds
505  DOUBLE PRECISION rcond, rpvgrw
506 * ..
507 * .. Array Arguments ..
508  COMPLEX*16 a( lda, * ), af( ldaf, * ), b( ldb, * ),
509  \$ work( * ), x( ldx, * )
510  DOUBLE PRECISION s( * ), params( * ), berr( * ), rwork( * ),
511  \$ err_bnds_norm( nrhs, * ),
512  \$ err_bnds_comp( nrhs, * )
513 * ..
514 *
515 * ==================================================================
516 *
517 * .. Parameters ..
518  DOUBLE PRECISION zero, one
519  parameter( zero = 0.0d+0, one = 1.0d+0 )
520  INTEGER final_nrm_err_i, final_cmp_err_i, berr_i
521  INTEGER rcond_i, nrm_rcond_i, nrm_err_i, cmp_rcond_i
522  INTEGER cmp_err_i, piv_growth_i
523  parameter( final_nrm_err_i = 1, final_cmp_err_i = 2,
524  \$ berr_i = 3 )
525  parameter( rcond_i = 4, nrm_rcond_i = 5, nrm_err_i = 6 )
526  parameter( cmp_rcond_i = 7, cmp_err_i = 8,
527  \$ piv_growth_i = 9 )
528 * ..
529 * .. Local Scalars ..
530  LOGICAL equil, nofact, rcequ
531  INTEGER infequ, j
532  DOUBLE PRECISION amax, bignum, smin, smax, scond, smlnum
533 * ..
534 * .. External Functions ..
535  EXTERNAL lsame, dlamch, zla_porpvgrw
536  LOGICAL lsame
537  DOUBLE PRECISION dlamch, zla_porpvgrw
538 * ..
539 * .. External Subroutines ..
540  EXTERNAL zpocon, zpoequb, zpotrf, zpotrs, zlacpy,
542 * ..
543 * .. Intrinsic Functions ..
544  INTRINSIC max, min
545 * ..
546 * .. Executable Statements ..
547 *
548  info = 0
549  nofact = lsame( fact, 'N' )
550  equil = lsame( fact, 'E' )
551  smlnum = dlamch( 'Safe minimum' )
552  bignum = one / smlnum
553  IF( nofact .OR. equil ) THEN
554  equed = 'N'
555  rcequ = .false.
556  ELSE
557  rcequ = lsame( equed, 'Y' )
558  ENDIF
559 *
560 * Default is failure. If an input parameter is wrong or
561 * factorization fails, make everything look horrible. Only the
562 * pivot growth is set here, the rest is initialized in ZPORFSX.
563 *
564  rpvgrw = zero
565 *
566 * Test the input parameters. PARAMS is not tested until ZPORFSX.
567 *
568  IF( .NOT.nofact .AND. .NOT.equil .AND. .NOT.
569  \$ lsame( fact, 'F' ) ) THEN
570  info = -1
571  ELSE IF( .NOT.lsame( uplo, 'U' ) .AND.
572  \$ .NOT.lsame( uplo, 'L' ) ) THEN
573  info = -2
574  ELSE IF( n.LT.0 ) THEN
575  info = -3
576  ELSE IF( nrhs.LT.0 ) THEN
577  info = -4
578  ELSE IF( lda.LT.max( 1, n ) ) THEN
579  info = -6
580  ELSE IF( ldaf.LT.max( 1, n ) ) THEN
581  info = -8
582  ELSE IF( lsame( fact, 'F' ) .AND. .NOT.
583  \$ ( rcequ .OR. lsame( equed, 'N' ) ) ) THEN
584  info = -9
585  ELSE
586  IF ( rcequ ) THEN
587  smin = bignum
588  smax = zero
589  DO 10 j = 1, n
590  smin = min( smin, s( j ) )
591  smax = max( smax, s( j ) )
592  10 CONTINUE
593  IF( smin.LE.zero ) THEN
594  info = -10
595  ELSE IF( n.GT.0 ) THEN
596  scond = max( smin, smlnum ) / min( smax, bignum )
597  ELSE
598  scond = one
599  END IF
600  END IF
601  IF( info.EQ.0 ) THEN
602  IF( ldb.LT.max( 1, n ) ) THEN
603  info = -12
604  ELSE IF( ldx.LT.max( 1, n ) ) THEN
605  info = -14
606  END IF
607  END IF
608  END IF
609 *
610  IF( info.NE.0 ) THEN
611  CALL xerbla( 'ZPOSVXX', -info )
612  RETURN
613  END IF
614 *
615  IF( equil ) THEN
616 *
617 * Compute row and column scalings to equilibrate the matrix A.
618 *
619  CALL zpoequb( n, a, lda, s, scond, amax, infequ )
620  IF( infequ.EQ.0 ) THEN
621 *
622 * Equilibrate the matrix.
623 *
624  CALL zlaqhe( uplo, n, a, lda, s, scond, amax, equed )
625  rcequ = lsame( equed, 'Y' )
626  END IF
627  END IF
628 *
629 * Scale the right-hand side.
630 *
631  IF( rcequ ) CALL zlascl2( n, nrhs, s, b, ldb )
632 *
633  IF( nofact .OR. equil ) THEN
634 *
635 * Compute the Cholesky factorization of A.
636 *
637  CALL zlacpy( uplo, n, n, a, lda, af, ldaf )
638  CALL zpotrf( uplo, n, af, ldaf, info )
639 *
640 * Return if INFO is non-zero.
641 *
642  IF( info.GT.0 ) THEN
643 *
644 * Pivot in column INFO is exactly 0
645 * Compute the reciprocal pivot growth factor of the
646 * leading rank-deficient INFO columns of A.
647 *
648  rpvgrw = zla_porpvgrw( uplo, n, a, lda, af, ldaf, rwork )
649  RETURN
650  END IF
651  END IF
652 *
653 * Compute the reciprocal pivot growth factor RPVGRW.
654 *
655  rpvgrw = zla_porpvgrw( uplo, n, a, lda, af, ldaf, rwork )
656 *
657 * Compute the solution matrix X.
658 *
659  CALL zlacpy( 'Full', n, nrhs, b, ldb, x, ldx )
660  CALL zpotrs( uplo, n, nrhs, af, ldaf, x, ldx, info )
661 *
662 * Use iterative refinement to improve the computed solution and
663 * compute error bounds and backward error estimates for it.
664 *
665  CALL zporfsx( uplo, equed, n, nrhs, a, lda, af, ldaf,
666  \$ s, b, ldb, x, ldx, rcond, berr, n_err_bnds, err_bnds_norm,
667  \$ err_bnds_comp, nparams, params, work, rwork, info )
668
669 *
670 * Scale solutions.
671 *
672  IF ( rcequ ) THEN
673  CALL zlascl2( n, nrhs, s, x, ldx )
674  END IF
675 *
676  RETURN
677 *
678 * End of ZPOSVXX
679 *
subroutine zpotrf(UPLO, N, A, LDA, INFO)
ZPOTRF VARIANT: right looking block version of the algorithm, calling Level 3 BLAS.
Definition: zpotrf.f:102
subroutine zlascl2(M, N, D, X, LDX)
ZLASCL2 performs diagonal scaling on a vector.
Definition: zlascl2.f:93
subroutine zlacpy(UPLO, M, N, A, LDA, B, LDB)
ZLACPY copies all or part of one two-dimensional array to another.
Definition: zlacpy.f:105
subroutine zpotrs(UPLO, N, NRHS, A, LDA, B, LDB, INFO)
ZPOTRS
Definition: zpotrs.f:112
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:65
double precision function zla_porpvgrw(UPLO, NCOLS, A, LDA, AF, LDAF, WORK)
ZLA_PORPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U) for a symmetric or Hermitian...
Definition: zla_porpvgrw.f:109
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
subroutine zlaqhe(UPLO, N, A, LDA, S, SCOND, AMAX, EQUED)
ZLAQHE scales a Hermitian matrix.
Definition: zlaqhe.f:136
subroutine zpoequb(N, A, LDA, S, SCOND, AMAX, INFO)
ZPOEQUB
Definition: zpoequb.f:115
subroutine zpocon(UPLO, N, A, LDA, ANORM, RCOND, WORK, RWORK, INFO)
ZPOCON
Definition: zpocon.f:123
subroutine zporfsx(UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF, S, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, RWORK, INFO)
ZPORFSX
Definition: zporfsx.f:395

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