 LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ zspsvx()

 subroutine zspsvx ( character FACT, character UPLO, integer N, integer NRHS, complex*16, dimension( * ) AP, complex*16, dimension( * ) AFP, integer, dimension( * ) IPIV, 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 )

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

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Purpose:
``` ZSPSVX uses the diagonal pivoting factorization A = U*D*U**T or
A = L*D*L**T to compute the solution to a complex system of linear
equations A * X = B, where A is an N-by-N symmetric 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**T,  if UPLO = 'U', or
A = L * D * L**T,  if UPLO = 'L',
where U (or L) is a product of permutation and unit upper (lower)
triangular matrices and D is symmetric 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. AP, 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*16 array, dimension (N*(N+1)/2) The upper or lower triangle of the symmetric 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*16 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**T or A = L*D*L**T as computed by ZSPTRF, 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**T or A = L*D*L**T as computed by ZSPTRF, 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 ZSPTRF. 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 ZSPTRF.``` [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 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 DOUBLE PRECISION 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 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: 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.```
Further Details:
```  The packed storage scheme is illustrated by the following example
when N = 4, UPLO = 'U':

Two-dimensional storage of the symmetric matrix A:

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

Packed storage of the upper triangle of A:

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

Definition at line 275 of file zspsvx.f.

277 *
278 * -- LAPACK driver routine --
279 * -- LAPACK is a software package provided by Univ. of Tennessee, --
280 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
281 *
282 * .. Scalar Arguments ..
283  CHARACTER FACT, UPLO
284  INTEGER INFO, LDB, LDX, N, NRHS
285  DOUBLE PRECISION RCOND
286 * ..
287 * .. Array Arguments ..
288  INTEGER IPIV( * )
289  DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
290  COMPLEX*16 AFP( * ), AP( * ), B( LDB, * ), WORK( * ),
291  \$ X( LDX, * )
292 * ..
293 *
294 * =====================================================================
295 *
296 * .. Parameters ..
297  DOUBLE PRECISION ZERO
298  parameter( zero = 0.0d+0 )
299 * ..
300 * .. Local Scalars ..
301  LOGICAL NOFACT
302  DOUBLE PRECISION ANORM
303 * ..
304 * .. External Functions ..
305  LOGICAL LSAME
306  DOUBLE PRECISION DLAMCH, ZLANSP
307  EXTERNAL lsame, dlamch, zlansp
308 * ..
309 * .. External Subroutines ..
310  EXTERNAL xerbla, zcopy, zlacpy, zspcon, zsprfs, zsptrf,
311  \$ zsptrs
312 * ..
313 * .. Intrinsic Functions ..
314  INTRINSIC max
315 * ..
316 * .. Executable Statements ..
317 *
318 * Test the input parameters.
319 *
320  info = 0
321  nofact = lsame( fact, 'N' )
322  IF( .NOT.nofact .AND. .NOT.lsame( fact, 'F' ) ) THEN
323  info = -1
324  ELSE IF( .NOT.lsame( uplo, 'U' ) .AND. .NOT.lsame( uplo, 'L' ) )
325  \$ THEN
326  info = -2
327  ELSE IF( n.LT.0 ) THEN
328  info = -3
329  ELSE IF( nrhs.LT.0 ) THEN
330  info = -4
331  ELSE IF( ldb.LT.max( 1, n ) ) THEN
332  info = -9
333  ELSE IF( ldx.LT.max( 1, n ) ) THEN
334  info = -11
335  END IF
336  IF( info.NE.0 ) THEN
337  CALL xerbla( 'ZSPSVX', -info )
338  RETURN
339  END IF
340 *
341  IF( nofact ) THEN
342 *
343 * Compute the factorization A = U*D*U**T or A = L*D*L**T.
344 *
345  CALL zcopy( n*( n+1 ) / 2, ap, 1, afp, 1 )
346  CALL zsptrf( uplo, n, afp, ipiv, info )
347 *
348 * Return if INFO is non-zero.
349 *
350  IF( info.GT.0 )THEN
351  rcond = zero
352  RETURN
353  END IF
354  END IF
355 *
356 * Compute the norm of the matrix A.
357 *
358  anorm = zlansp( 'I', uplo, n, ap, rwork )
359 *
360 * Compute the reciprocal of the condition number of A.
361 *
362  CALL zspcon( uplo, n, afp, ipiv, anorm, rcond, work, info )
363 *
364 * Compute the solution vectors X.
365 *
366  CALL zlacpy( 'Full', n, nrhs, b, ldb, x, ldx )
367  CALL zsptrs( uplo, n, nrhs, afp, ipiv, x, ldx, info )
368 *
369 * Use iterative refinement to improve the computed solutions and
370 * compute error bounds and backward error estimates for them.
371 *
372  CALL zsprfs( uplo, n, nrhs, ap, afp, ipiv, b, ldb, x, ldx, ferr,
373  \$ berr, work, rwork, info )
374 *
375 * Set INFO = N+1 if the matrix is singular to working precision.
376 *
377  IF( rcond.LT.dlamch( 'Epsilon' ) )
378  \$ info = n + 1
379 *
380  RETURN
381 *
382 * End of ZSPSVX
383 *
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:69
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine zcopy(N, ZX, INCX, ZY, INCY)
ZCOPY
Definition: zcopy.f:81
double precision function zlansp(NORM, UPLO, N, AP, WORK)
ZLANSP returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,...
Definition: zlansp.f:115
subroutine zlacpy(UPLO, M, N, A, LDA, B, LDB)
ZLACPY copies all or part of one two-dimensional array to another.
Definition: zlacpy.f:103
subroutine zsptrf(UPLO, N, AP, IPIV, INFO)
ZSPTRF
Definition: zsptrf.f:158
subroutine zsprfs(UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO)
ZSPRFS
Definition: zsprfs.f:180
subroutine zspcon(UPLO, N, AP, IPIV, ANORM, RCOND, WORK, INFO)
ZSPCON
Definition: zspcon.f:118
subroutine zsptrs(UPLO, N, NRHS, AP, IPIV, B, LDB, INFO)
ZSPTRS
Definition: zsptrs.f:115
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