LAPACK  3.9.0 LAPACK: Linear Algebra PACKage

## ◆ chpsvx()

 subroutine chpsvx ( character FACT, character UPLO, integer N, integer NRHS, complex, dimension( * ) AP, complex, dimension( * ) AFP, integer, dimension( * ) IPIV, complex, dimension( ldb, * ) B, integer LDB, complex, dimension( ldx, * ) X, integer LDX, real RCOND, real, dimension( * ) FERR, real, dimension( * ) BERR, complex, dimension( * ) WORK, real, dimension( * ) RWORK, integer INFO )

CHPSVX computes the solution to system of linear equations A * X = B for OTHER matrices

Purpose:
CHPSVX uses the diagonal pivoting factorization A = U*D*U**H or
A = L*D*L**H to compute the solution to a complex system of linear
equations A * X = B, where A is an N-by-N Hermitian matrix stored
in packed format 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 = 'N', the diagonal pivoting method is used to factor A as
A = U * D * U**H,  if UPLO = 'U', or
A = L * D * L**H,  if UPLO = 'L',
where U (or L) is a product of permutation and unit upper (lower)
triangular matrices and D is Hermitian and block diagonal with
1-by-1 and 2-by-2 diagonal blocks.

2. If some D(i,i)=0, so that D is exactly singular, 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.

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

4. Iterative refinement is applied to improve the computed solution
matrix and calculate error bounds and backward error estimates
for it.
Parameters
 [in] FACT FACT is CHARACTER*1 Specifies whether or not the factored form of A has been supplied on entry. = 'F': On entry, AFP and IPIV contain the factored form of A. AFP and IPIV will not be modified. = 'N': The matrix A will be copied to AFP 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] AP AP is COMPLEX array, dimension (N*(N+1)/2) The upper or lower triangle of the Hermitian matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows: if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n. See below for further details. [in,out] AFP AFP is COMPLEX array, dimension (N*(N+1)/2) If FACT = 'F', then AFP is an input argument and on entry contains the block diagonal matrix D and the multipliers used to obtain the factor U or L from the factorization A = U*D*U**H or A = L*D*L**H as computed by CHPTRF, stored as a packed triangular matrix in the same storage format as A. If FACT = 'N', then AFP is an output argument and on exit contains the block diagonal matrix D and the multipliers used to obtain the factor U or L from the factorization A = U*D*U**H or A = L*D*L**H as computed by CHPTRF, stored as a packed triangular matrix in the same storage format as A. [in,out] IPIV IPIV is INTEGER array, dimension (N) If FACT = 'F', then IPIV is an input argument and on entry contains details of the interchanges and the block structure of D, as determined by CHPTRF. If IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal block. If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k) is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) = IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block. If FACT = 'N', then IPIV is an output argument and on exit contains details of the interchanges and the block structure of D, as determined by CHPTRF. [in] B B is COMPLEX 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 array, dimension (LDX,NRHS) If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix 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. 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 COMPLEX array, dimension (2*N) [out] RWORK RWORK is REAL 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: D(i,i) is exactly zero. The factorization has been completed but the factor D is exactly singular, so the solution and error bounds could not be computed. RCOND = 0 is returned. = N+1: D 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
Further Details:
The packed storage scheme is illustrated by the following example
when N = 4, UPLO = 'U':

Two-dimensional storage of the Hermitian matrix A:

a11 a12 a13 a14
a22 a23 a24
a33 a34     (aij = conjg(aji))
a44

Packed storage of the upper triangle of A:

AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]

Definition at line 279 of file chpsvx.f.

279 *
280 * -- LAPACK driver routine (version 3.7.0) --
281 * -- LAPACK is a software package provided by Univ. of Tennessee, --
282 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
283 * April 2012
284 *
285 * .. Scalar Arguments ..
286  CHARACTER FACT, UPLO
287  INTEGER INFO, LDB, LDX, N, NRHS
288  REAL RCOND
289 * ..
290 * .. Array Arguments ..
291  INTEGER IPIV( * )
292  REAL BERR( * ), FERR( * ), RWORK( * )
293  COMPLEX AFP( * ), AP( * ), B( LDB, * ), WORK( * ),
294  \$ X( LDX, * )
295 * ..
296 *
297 * =====================================================================
298 *
299 * .. Parameters ..
300  REAL ZERO
301  parameter( zero = 0.0e+0 )
302 * ..
303 * .. Local Scalars ..
304  LOGICAL NOFACT
305  REAL ANORM
306 * ..
307 * .. External Functions ..
308  LOGICAL LSAME
309  REAL CLANHP, SLAMCH
310  EXTERNAL lsame, clanhp, slamch
311 * ..
312 * .. External Subroutines ..
313  EXTERNAL ccopy, chpcon, chprfs, chptrf, chptrs, clacpy,
314  \$ xerbla
315 * ..
316 * .. Intrinsic Functions ..
317  INTRINSIC max
318 * ..
319 * .. Executable Statements ..
320 *
321 * Test the input parameters.
322 *
323  info = 0
324  nofact = lsame( fact, 'N' )
325  IF( .NOT.nofact .AND. .NOT.lsame( fact, 'F' ) ) THEN
326  info = -1
327  ELSE IF( .NOT.lsame( uplo, 'U' ) .AND. .NOT.lsame( uplo, 'L' ) )
328  \$ THEN
329  info = -2
330  ELSE IF( n.LT.0 ) THEN
331  info = -3
332  ELSE IF( nrhs.LT.0 ) THEN
333  info = -4
334  ELSE IF( ldb.LT.max( 1, n ) ) THEN
335  info = -9
336  ELSE IF( ldx.LT.max( 1, n ) ) THEN
337  info = -11
338  END IF
339  IF( info.NE.0 ) THEN
340  CALL xerbla( 'CHPSVX', -info )
341  RETURN
342  END IF
343 *
344  IF( nofact ) THEN
345 *
346 * Compute the factorization A = U*D*U**H or A = L*D*L**H.
347 *
348  CALL ccopy( n*( n+1 ) / 2, ap, 1, afp, 1 )
349  CALL chptrf( uplo, n, afp, ipiv, info )
350 *
351 * Return if INFO is non-zero.
352 *
353  IF( info.GT.0 )THEN
354  rcond = zero
355  RETURN
356  END IF
357  END IF
358 *
359 * Compute the norm of the matrix A.
360 *
361  anorm = clanhp( 'I', uplo, n, ap, rwork )
362 *
363 * Compute the reciprocal of the condition number of A.
364 *
365  CALL chpcon( uplo, n, afp, ipiv, anorm, rcond, work, info )
366 *
367 * Compute the solution vectors X.
368 *
369  CALL clacpy( 'Full', n, nrhs, b, ldb, x, ldx )
370  CALL chptrs( uplo, n, nrhs, afp, ipiv, x, ldx, info )
371 *
372 * Use iterative refinement to improve the computed solutions and
373 * compute error bounds and backward error estimates for them.
374 *
375  CALL chprfs( uplo, n, nrhs, ap, afp, ipiv, b, ldb, x, ldx, ferr,
376  \$ berr, work, rwork, info )
377 *
378 * Set INFO = N+1 if the matrix is singular to working precision.
379 *
380  IF( rcond.LT.slamch( 'Epsilon' ) )
381  \$ info = n + 1
382 *
383  RETURN
384 *
385 * End of CHPSVX
386 *
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chptrf
subroutine chptrf(UPLO, N, AP, IPIV, INFO)
CHPTRF
Definition: chptrf.f:161
clacpy
subroutine clacpy(UPLO, M, N, A, LDA, B, LDB)
CLACPY copies all or part of one two-dimensional array to another.
Definition: clacpy.f:105
clanhp
real function clanhp(NORM, UPLO, N, AP, WORK)
CLANHP returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,...
Definition: clanhp.f:119
xerbla
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
chprfs
subroutine chprfs(UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO)
CHPRFS
Definition: chprfs.f:182
lsame
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
slamch
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:70
chpcon
subroutine chpcon(UPLO, N, AP, IPIV, ANORM, RCOND, WORK, INFO)
CHPCON
Definition: chpcon.f:120
ccopy
subroutine ccopy(N, CX, INCX, CY, INCY)
CCOPY
Definition: ccopy.f:83
chptrs
subroutine chptrs(UPLO, N, NRHS, AP, IPIV, B, LDB, INFO)
CHPTRS
Definition: chptrs.f:117