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LAPACK » Positive Definite Matrix » Linear Solve

Functions

subroutine sposv (uplo, n, nrhs, a, lda, b, ldb, info)
  SPOSV computes the solution to system of linear equations A * X = B for PO matrices
subroutine sposvx (fact, uplo, n, nrhs, a, lda, af, ldaf, equed, s, b, ldb, x, ldx, rcond, ferr, berr, work, iwork, info)
  SPOSVX computes the solution to system of linear equations A * X = B for PO matrices
subroutine sposvxx (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, iwork, info)
  SPOSVXX computes the solution to system of linear equations A * X = B for PO matrices

Detailed Description

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

Function Documentation

◆ sposv()

subroutine sposv ( character uplo,
integer n,
integer nrhs,
real, dimension( lda, * ) a,
integer lda,
real, dimension( ldb, * ) b,
integer ldb,
integer info )

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

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

Purpose:
!>
!> SPOSV computes the solution to a real 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.
!>
!> The Cholesky decomposition is used to factor A 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.  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 REAL 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, if INFO = 0, the factor U or L from the Cholesky
!>          factorization A = U**T*U or A = L*L**T.
!>
[in]LDA
!>          LDA is INTEGER
!>          The leading dimension of the array A.  LDA >= max(1,N).
!>
[in,out]B
!>          B is REAL 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.
!>
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

Definition at line 129 of file sposv.f.

130*
131* -- LAPACK driver routine --
132* -- LAPACK is a software package provided by Univ. of Tennessee, --
133* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
134*
135* .. Scalar Arguments ..
136 CHARACTER UPLO
137 INTEGER INFO, LDA, LDB, N, NRHS
138* ..
139* .. Array Arguments ..
140 REAL A( LDA, * ), B( LDB, * )
141* ..
142*
143* =====================================================================
144*
145* .. External Functions ..
146 LOGICAL LSAME
147 EXTERNAL lsame
148* ..
149* .. External Subroutines ..
150 EXTERNAL spotrf, spotrs, xerbla
151* ..
152* .. Intrinsic Functions ..
153 INTRINSIC max
154* ..
155* .. Executable Statements ..
156*
157* Test the input parameters.
158*
159 info = 0
160 IF( .NOT.lsame( uplo, 'U' ) .AND. .NOT.lsame( uplo, 'L' ) ) THEN
161 info = -1
162 ELSE IF( n.LT.0 ) THEN
163 info = -2
164 ELSE IF( nrhs.LT.0 ) THEN
165 info = -3
166 ELSE IF( lda.LT.max( 1, n ) ) THEN
167 info = -5
168 ELSE IF( ldb.LT.max( 1, n ) ) THEN
169 info = -7
170 END IF
171 IF( info.NE.0 ) THEN
172 CALL xerbla( 'SPOSV ', -info )
173 RETURN
174 END IF
175*
176* Compute the Cholesky factorization A = U**T*U or A = L*L**T.
177*
178 CALL spotrf( uplo, n, a, lda, info )
179 IF( info.EQ.0 ) THEN
180*
181* Solve the system A*X = B, overwriting B with X.
182*
183 CALL spotrs( uplo, n, nrhs, a, lda, b, ldb, info )
184*
185 END IF
186 RETURN
187*
188* End of SPOSV
189*
xerbla
subroutine xerbla(srname, info)
XERBLA
Definition xerbla.f:60
lsame
logical function lsame(ca, cb)
LSAME
Definition lsame.f:53
spotrs
subroutine spotrs(uplo, n, nrhs, a, lda, b, ldb, info)
SPOTRS
Definition spotrs.f:110
spotrf
subroutine spotrf(uplo, n, a, lda, info)
SPOTRF
Definition spotrf.f:107
max
#define max(a, b)
Definition macros.h:21

◆ sposvx()

subroutine sposvx ( character fact,
character uplo,
integer n,
integer nrhs,
real, dimension( lda, * ) a,
integer lda,
real, dimension( ldaf, * ) af,
integer ldaf,
character equed,
real, dimension( * ) s,
real, dimension( ldb, * ) b,
integer ldb,
real, dimension( ldx, * ) x,
integer ldx,
real rcond,
real, dimension( * ) ferr,
real, dimension( * ) berr,
real, dimension( * ) work,
integer, dimension( * ) iwork,
integer info )

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

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

Purpose:
!>
!> SPOSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to
!> compute the solution to a real 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.
!>
!> 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**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.  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 REAL 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 REAL 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':  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 REAL 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 REAL 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 REAL 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 REAL
!>          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 REAL 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 REAL 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 REAL array, dimension (3*N)
!>
[out]IWORK
!>          IWORK is INTEGER 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.
!>
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

Definition at line 304 of file sposvx.f.

307*
308* -- LAPACK driver routine --
309* -- LAPACK is a software package provided by Univ. of Tennessee, --
310* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
311*
312* .. Scalar Arguments ..
313 CHARACTER EQUED, FACT, UPLO
314 INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
315 REAL RCOND
316* ..
317* .. Array Arguments ..
318 INTEGER IWORK( * )
319 REAL A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
320 $ BERR( * ), FERR( * ), S( * ), WORK( * ),
321 $ X( LDX, * )
322* ..
323*
324* =====================================================================
325*
326* .. Parameters ..
327 REAL ZERO, ONE
328 parameter( zero = 0.0e+0, one = 1.0e+0 )
329* ..
330* .. Local Scalars ..
331 LOGICAL EQUIL, NOFACT, RCEQU
332 INTEGER I, INFEQU, J
333 REAL AMAX, ANORM, BIGNUM, SCOND, SMAX, SMIN, SMLNUM
334* ..
335* .. External Functions ..
336 LOGICAL LSAME
337 REAL SLAMCH, SLANSY
338 EXTERNAL lsame, slamch, slansy
339* ..
340* .. External Subroutines ..
341 EXTERNAL slacpy, slaqsy, spocon, spoequ, sporfs, spotrf,
342 $ spotrs, xerbla
343* ..
344* .. Intrinsic Functions ..
345 INTRINSIC max, min
346* ..
347* .. Executable Statements ..
348*
349 info = 0
350 nofact = lsame( fact, 'N' )
351 equil = lsame( fact, 'E' )
352 IF( nofact .OR. equil ) THEN
353 equed = 'N'
354 rcequ = .false.
355 ELSE
356 rcequ = lsame( equed, 'Y' )
357 smlnum = slamch( 'Safe minimum' )
358 bignum = one / smlnum
359 END IF
360*
361* Test the input parameters.
362*
363 IF( .NOT.nofact .AND. .NOT.equil .AND. .NOT.lsame( fact, 'F' ) )
364 $ THEN
365 info = -1
366 ELSE IF( .NOT.lsame( uplo, 'U' ) .AND. .NOT.lsame( uplo, 'L' ) )
367 $ THEN
368 info = -2
369 ELSE IF( n.LT.0 ) THEN
370 info = -3
371 ELSE IF( nrhs.LT.0 ) THEN
372 info = -4
373 ELSE IF( lda.LT.max( 1, n ) ) THEN
374 info = -6
375 ELSE IF( ldaf.LT.max( 1, n ) ) THEN
376 info = -8
377 ELSE IF( lsame( fact, 'F' ) .AND. .NOT.
378 $ ( rcequ .OR. lsame( equed, 'N' ) ) ) THEN
379 info = -9
380 ELSE
381 IF( rcequ ) THEN
382 smin = bignum
383 smax = zero
384 DO 10 j = 1, n
385 smin = min( smin, s( j ) )
386 smax = max( smax, s( j ) )
387 10 CONTINUE
388 IF( smin.LE.zero ) THEN
389 info = -10
390 ELSE IF( n.GT.0 ) THEN
391 scond = max( smin, smlnum ) / min( smax, bignum )
392 ELSE
393 scond = one
394 END IF
395 END IF
396 IF( info.EQ.0 ) THEN
397 IF( ldb.LT.max( 1, n ) ) THEN
398 info = -12
399 ELSE IF( ldx.LT.max( 1, n ) ) THEN
400 info = -14
401 END IF
402 END IF
403 END IF
404*
405 IF( info.NE.0 ) THEN
406 CALL xerbla( 'SPOSVX', -info )
407 RETURN
408 END IF
409*
410 IF( equil ) THEN
411*
412* Compute row and column scalings to equilibrate the matrix A.
413*
414 CALL spoequ( n, a, lda, s, scond, amax, infequ )
415 IF( infequ.EQ.0 ) THEN
416*
417* Equilibrate the matrix.
418*
419 CALL slaqsy( uplo, n, a, lda, s, scond, amax, equed )
420 rcequ = lsame( equed, 'Y' )
421 END IF
422 END IF
423*
424* Scale the right hand side.
425*
426 IF( rcequ ) THEN
427 DO 30 j = 1, nrhs
428 DO 20 i = 1, n
429 b( i, j ) = s( i )*b( i, j )
430 20 CONTINUE
431 30 CONTINUE
432 END IF
433*
434 IF( nofact .OR. equil ) THEN
435*
436* Compute the Cholesky factorization A = U**T *U or A = L*L**T.
437*
438 CALL slacpy( uplo, n, n, a, lda, af, ldaf )
439 CALL spotrf( uplo, n, af, ldaf, info )
440*
441* Return if INFO is non-zero.
442*
443 IF( info.GT.0 )THEN
444 rcond = zero
445 RETURN
446 END IF
447 END IF
448*
449* Compute the norm of the matrix A.
450*
451 anorm = slansy( '1', uplo, n, a, lda, work )
452*
453* Compute the reciprocal of the condition number of A.
454*
455 CALL spocon( uplo, n, af, ldaf, anorm, rcond, work, iwork, info )
456*
457* Compute the solution matrix X.
458*
459 CALL slacpy( 'Full', n, nrhs, b, ldb, x, ldx )
460 CALL spotrs( uplo, n, nrhs, af, ldaf, x, ldx, info )
461*
462* Use iterative refinement to improve the computed solution and
463* compute error bounds and backward error estimates for it.
464*
465 CALL sporfs( uplo, n, nrhs, a, lda, af, ldaf, b, ldb, x, ldx,
466 $ ferr, berr, work, iwork, info )
467*
468* Transform the solution matrix X to a solution of the original
469* system.
470*
471 IF( rcequ ) THEN
472 DO 50 j = 1, nrhs
473 DO 40 i = 1, n
474 x( i, j ) = s( i )*x( i, j )
475 40 CONTINUE
476 50 CONTINUE
477 DO 60 j = 1, nrhs
478 ferr( j ) = ferr( j ) / scond
479 60 CONTINUE
480 END IF
481*
482* Set INFO = N+1 if the matrix is singular to working precision.
483*
484 IF( rcond.LT.slamch( 'Epsilon' ) )
485 $ info = n + 1
486*
487 RETURN
488*
489* End of SPOSVX
490*
slacpy
subroutine slacpy(uplo, m, n, a, lda, b, ldb)
SLACPY copies all or part of one two-dimensional array to another.
Definition slacpy.f:103
sporfs
subroutine sporfs(uplo, n, nrhs, a, lda, af, ldaf, b, ldb, x, ldx, ferr, berr, work, iwork, info)
SPORFS
Definition sporfs.f:183
spocon
subroutine spocon(uplo, n, a, lda, anorm, rcond, work, iwork, info)
SPOCON
Definition spocon.f:121
spoequ
subroutine spoequ(n, a, lda, s, scond, amax, info)
SPOEQU
Definition spoequ.f:112
slansy
real function slansy(norm, uplo, n, a, lda, work)
SLANSY returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,...
Definition slansy.f:122
slaqsy
subroutine slaqsy(uplo, n, a, lda, s, scond, amax, equed)
SLAQSY scales a symmetric/Hermitian matrix, using scaling factors computed by spoequ.
Definition slaqsy.f:133
slamch
real function slamch(cmach)
SLAMCH
Definition slamch.f:68
min
#define min(a, b)
Definition macros.h:20

◆ sposvxx()

subroutine sposvxx ( character fact,
character uplo,
integer n,
integer nrhs,
real, dimension( lda, * ) a,
integer lda,
real, dimension( ldaf, * ) af,
integer ldaf,
character equed,
real, dimension( * ) s,
real, dimension( ldb, * ) b,
integer ldb,
real, dimension( ldx, * ) x,
integer ldx,
real rcond,
real rpvgrw,
real, dimension( * ) berr,
integer n_err_bnds,
real, dimension( nrhs, * ) err_bnds_norm,
real, dimension( nrhs, * ) err_bnds_comp,
integer nparams,
real, dimension( * ) params,
real, dimension( * ) work,
integer, dimension( * ) iwork,
integer info )

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

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

Purpose:
!>
!>    SPOSVXX uses the Cholesky factorization A = U**T*U or A = L*L**T
!>    to compute the solution to a real 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. SPOSVXX 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.
!>
!>    SPOSVXX 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
!>    SPOSVXX 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 SPOSVXX would itself produce.
!>
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**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 REAL 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL
!>     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 REAL
!>     Reciprocal pivot growth.  On exit, this contains the reciprocal
!>     pivot growth factor norm(A)/norm(U). The 
!>     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<INFO<=N, then this contains the reciprocal pivot growth factor
!>     for the leading INFO columns of A.
!>
[out]BERR
!>          BERR is REAL array, dimension (NRHS)
!>     Componentwise relative backward error.  This is 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).
!>
[in]N_ERR_BNDS
!>          N_ERR_BNDS is INTEGER
!>     Number of error bounds to return for each right hand side
!>     and each type (normwise or componentwise).  See ERR_BNDS_NORM and
!>     ERR_BNDS_COMP below.
!>
[out]ERR_BNDS_NORM
!>          ERR_BNDS_NORM is REAL array, dimension (NRHS, N_ERR_BNDS)
!>     For each right-hand side, this array contains information about
!>     various error bounds and condition numbers corresponding to the
!>     normwise relative error, which is defined as follows:
!>
!>     Normwise relative error in the ith solution vector:
!>             max_j (abs(XTRUE(j,i) - X(j,i)))
!>            ------------------------------
!>                  max_j abs(X(j,i))
!>
!>     The array is indexed by the type of error information as described
!>     below. There currently are up to three pieces of information
!>     returned.
!>
!>     The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
!>     right-hand side.
!>
!>     The second index in ERR_BNDS_NORM(:,err) contains the following
!>     three fields:
!>     err = 1  boolean. Trust the answer if the
!>              reciprocal condition number is less than the threshold
!>              sqrt(n) * slamch('Epsilon').
!>
!>     err = 2  error bound: The estimated forward error,
!>              almost certainly within a factor of 10 of the true error
!>              so long as the next entry is greater than the threshold
!>              sqrt(n) * slamch('Epsilon'). This error bound should only
!>              be trusted if the previous boolean is true.
!>
!>     err = 3  Reciprocal condition number: Estimated normwise
!>              reciprocal condition number.  Compared with the threshold
!>              sqrt(n) * slamch('Epsilon') to determine if the error
!>              estimate is . These reciprocal condition
!>              numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
!>              appropriately scaled matrix Z.
!>              Let Z = S*A, where S scales each row by a power of the
!>              radix so all absolute row sums of Z are approximately 1.
!>
!>     See Lapack Working Note 165 for further details and extra
!>     cautions.
!>
[out]ERR_BNDS_COMP
!>          ERR_BNDS_COMP is REAL array, dimension (NRHS, N_ERR_BNDS)
!>     For each right-hand side, this array contains information about
!>     various error bounds and condition numbers corresponding to the
!>     componentwise relative error, which is defined as follows:
!>
!>     Componentwise relative error in the ith solution vector:
!>                    abs(XTRUE(j,i) - X(j,i))
!>             max_j ----------------------
!>                         abs(X(j,i))
!>
!>     The array is indexed by the right-hand side i (on which the
!>     componentwise relative error depends), and the type of error
!>     information as described below. There currently are up to three
!>     pieces of information returned for each right-hand side. If
!>     componentwise accuracy is not requested (PARAMS(3) = 0.0), then
!>     ERR_BNDS_COMP is not accessed.  If N_ERR_BNDS < 3, then at most
!>     the first (:,N_ERR_BNDS) entries are returned.
!>
!>     The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
!>     right-hand side.
!>
!>     The second index in ERR_BNDS_COMP(:,err) contains the following
!>     three fields:
!>     err = 1  boolean. Trust the answer if the
!>              reciprocal condition number is less than the threshold
!>              sqrt(n) * slamch('Epsilon').
!>
!>     err = 2  error bound: The estimated forward error,
!>              almost certainly within a factor of 10 of the true error
!>              so long as the next entry is greater than the threshold
!>              sqrt(n) * slamch('Epsilon'). This error bound should only
!>              be trusted if the previous boolean is true.
!>
!>     err = 3  Reciprocal condition number: Estimated componentwise
!>              reciprocal condition number.  Compared with the threshold
!>              sqrt(n) * slamch('Epsilon') to determine if the error
!>              estimate is . These reciprocal condition
!>              numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
!>              appropriately scaled matrix Z.
!>              Let Z = S*(A*diag(x)), where x is the solution for the
!>              current right-hand side and S scales each row of
!>              A*diag(x) by a power of the radix so all absolute row
!>              sums of Z are approximately 1.
!>
!>     See Lapack Working Note 165 for further details and extra
!>     cautions.
!>
[in]NPARAMS
!>          NPARAMS is INTEGER
!>     Specifies the number of parameters set in PARAMS.  If <= 0, the
!>     PARAMS array is never referenced and default values are used.
!>
[in,out]PARAMS
!>          PARAMS is REAL array, dimension NPARAMS
!>     Specifies algorithm parameters.  If an entry is < 0.0, then
!>     that entry will be filled with default value used for that
!>     parameter.  Only positions up to NPARAMS are accessed; defaults
!>     are used for higher-numbered parameters.
!>
!>       PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
!>            refinement or not.
!>         Default: 1.0
!>            = 0.0:  No refinement is performed, and no error bounds are
!>                    computed.
!>            = 1.0:  Use the double-precision refinement algorithm,
!>                    possibly with doubled-single computations if the
!>                    compilation environment does not support DOUBLE
!>                    PRECISION.
!>              (other values are reserved for future use)
!>
!>       PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
!>            computations allowed for refinement.
!>         Default: 10
!>         Aggressive: Set to 100 to permit convergence using approximate
!>                     factorizations or factorizations other than LU. If
!>                     the factorization uses a technique other than
!>                     Gaussian elimination, the guarantees in
!>                     err_bnds_norm and err_bnds_comp may no longer be
!>                     trustworthy.
!>
!>       PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code
!>            will attempt to find a solution with small componentwise
!>            relative error in the double-precision algorithm.  Positive
!>            is true, 0.0 is false.
!>         Default: 1.0 (attempt componentwise convergence)
!>
[out]WORK
!>          WORK is REAL array, dimension (4*N)
!>
[out]IWORK
!>          IWORK is INTEGER array, dimension (N)
!>
[out]INFO
!>          INFO is INTEGER
!>       = 0:  Successful exit. The solution to every right-hand side is
!>         guaranteed.
!>       < 0:  If INFO = -i, the i-th argument had an illegal value
!>       > 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.
!>
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

Definition at line 493 of file sposvxx.f.

497*
498* -- LAPACK driver routine --
499* -- LAPACK is a software package provided by Univ. of Tennessee, --
500* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
501*
502* .. Scalar Arguments ..
503 CHARACTER EQUED, FACT, UPLO
504 INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS,
505 $ N_ERR_BNDS
506 REAL RCOND, RPVGRW
507* ..
508* .. Array Arguments ..
509 INTEGER IWORK( * )
510 REAL A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
511 $ X( LDX, * ), WORK( * )
512 REAL S( * ), PARAMS( * ), BERR( * ),
513 $ ERR_BNDS_NORM( NRHS, * ),
514 $ ERR_BNDS_COMP( NRHS, * )
515* ..
516*
517* ==================================================================
518*
519* .. Parameters ..
520 REAL ZERO, ONE
521 parameter( zero = 0.0e+0, one = 1.0e+0 )
522 INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I
523 INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I
524 INTEGER CMP_ERR_I, PIV_GROWTH_I
525 parameter( final_nrm_err_i = 1, final_cmp_err_i = 2,
526 $ berr_i = 3 )
527 parameter( rcond_i = 4, nrm_rcond_i = 5, nrm_err_i = 6 )
528 parameter( cmp_rcond_i = 7, cmp_err_i = 8,
529 $ piv_growth_i = 9 )
530* ..
531* .. Local Scalars ..
532 LOGICAL EQUIL, NOFACT, RCEQU
533 INTEGER INFEQU, J
534 REAL AMAX, BIGNUM, SMIN, SMAX,
535 $ SCOND, SMLNUM
536* ..
537* .. External Functions ..
538 EXTERNAL lsame, slamch, sla_porpvgrw
539 LOGICAL LSAME
540 REAL SLAMCH, SLA_PORPVGRW
541* ..
542* .. External Subroutines ..
543 EXTERNAL spoequb, spotrf, spotrs, slacpy, slaqsy,
544 $ xerbla, slascl2, sporfsx
545* ..
546* .. Intrinsic Functions ..
547 INTRINSIC max, min
548* ..
549* .. Executable Statements ..
550*
551 info = 0
552 nofact = lsame( fact, 'N' )
553 equil = lsame( fact, 'E' )
554 smlnum = slamch( 'Safe minimum' )
555 bignum = one / smlnum
556 IF( nofact .OR. equil ) THEN
557 equed = 'N'
558 rcequ = .false.
559 ELSE
560 rcequ = lsame( equed, 'Y' )
561 ENDIF
562*
563* Default is failure. If an input parameter is wrong or
564* factorization fails, make everything look horrible. Only the
565* pivot growth is set here, the rest is initialized in SPORFSX.
566*
567 rpvgrw = zero
568*
569* Test the input parameters. PARAMS is not tested until SPORFSX.
570*
571 IF( .NOT.nofact .AND. .NOT.equil .AND. .NOT.
572 $ lsame( fact, 'F' ) ) THEN
573 info = -1
574 ELSE IF( .NOT.lsame( uplo, 'U' ) .AND.
575 $ .NOT.lsame( uplo, 'L' ) ) THEN
576 info = -2
577 ELSE IF( n.LT.0 ) THEN
578 info = -3
579 ELSE IF( nrhs.LT.0 ) THEN
580 info = -4
581 ELSE IF( lda.LT.max( 1, n ) ) THEN
582 info = -6
583 ELSE IF( ldaf.LT.max( 1, n ) ) THEN
584 info = -8
585 ELSE IF( lsame( fact, 'F' ) .AND. .NOT.
586 $ ( rcequ .OR. lsame( equed, 'N' ) ) ) THEN
587 info = -9
588 ELSE
589 IF ( rcequ ) THEN
590 smin = bignum
591 smax = zero
592 DO 10 j = 1, n
593 smin = min( smin, s( j ) )
594 smax = max( smax, s( j ) )
595 10 CONTINUE
596 IF( smin.LE.zero ) THEN
597 info = -10
598 ELSE IF( n.GT.0 ) THEN
599 scond = max( smin, smlnum ) / min( smax, bignum )
600 ELSE
601 scond = one
602 END IF
603 END IF
604 IF( info.EQ.0 ) THEN
605 IF( ldb.LT.max( 1, n ) ) THEN
606 info = -12
607 ELSE IF( ldx.LT.max( 1, n ) ) THEN
608 info = -14
609 END IF
610 END IF
611 END IF
612*
613 IF( info.NE.0 ) THEN
614 CALL xerbla( 'SPOSVXX', -info )
615 RETURN
616 END IF
617*
618 IF( equil ) THEN
619*
620* Compute row and column scalings to equilibrate the matrix A.
621*
622 CALL spoequb( n, a, lda, s, scond, amax, infequ )
623 IF( infequ.EQ.0 ) THEN
624*
625* Equilibrate the matrix.
626*
627 CALL slaqsy( uplo, n, a, lda, s, scond, amax, equed )
628 rcequ = lsame( equed, 'Y' )
629 END IF
630 END IF
631*
632* Scale the right-hand side.
633*
634 IF( rcequ ) CALL slascl2( n, nrhs, s, b, ldb )
635*
636 IF( nofact .OR. equil ) THEN
637*
638* Compute the Cholesky factorization of A.
639*
640 CALL slacpy( uplo, n, n, a, lda, af, ldaf )
641 CALL spotrf( uplo, n, af, ldaf, info )
642*
643* Return if INFO is non-zero.
644*
645 IF( info.NE.0 ) THEN
646*
647* Pivot in column INFO is exactly 0
648* Compute the reciprocal pivot growth factor of the
649* leading rank-deficient INFO columns of A.
650*
651 rpvgrw = sla_porpvgrw( uplo, info, a, lda, af, ldaf, work )
652 RETURN
653 ENDIF
654 END IF
655*
656* Compute the reciprocal growth factor RPVGRW.
657*
658 rpvgrw = sla_porpvgrw( uplo, n, a, lda, af, ldaf, work )
659*
660* Compute the solution matrix X.
661*
662 CALL slacpy( 'Full', n, nrhs, b, ldb, x, ldx )
663 CALL spotrs( uplo, n, nrhs, af, ldaf, x, ldx, info )
664*
665* Use iterative refinement to improve the computed solution and
666* compute error bounds and backward error estimates for it.
667*
668 CALL sporfsx( uplo, equed, n, nrhs, a, lda, af, ldaf,
669 $ s, b, ldb, x, ldx, rcond, berr, n_err_bnds, err_bnds_norm,
670 $ err_bnds_comp, nparams, params, work, iwork, info )
671
672*
673* Scale solutions.
674*
675 IF ( rcequ ) THEN
676 CALL slascl2 ( n, nrhs, s, x, ldx )
677 END IF
678*
679 RETURN
680*
681* End of SPOSVXX
682*
slascl2
subroutine slascl2(m, n, d, x, ldx)
SLASCL2 performs diagonal scaling on a vector.
Definition slascl2.f:90
spoequb
subroutine spoequb(n, a, lda, s, scond, amax, info)
SPOEQUB
Definition spoequb.f:118
sla_porpvgrw
real function sla_porpvgrw(uplo, ncols, a, lda, af, ldaf, work)
SLA_PORPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U) for a symmetric or Hermitian...
Definition sla_porpvgrw.f:104
sporfsx
subroutine sporfsx(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, iwork, info)
SPORFSX
Definition sporfsx.f:394