 LAPACK  3.8.0 LAPACK: Linear Algebra PACKage

## ◆ sspsvx()

 subroutine sspsvx ( character FACT, character UPLO, integer N, integer NRHS, real, dimension( * ) AP, real, dimension( * ) AFP, integer, dimension( * ) IPIV, 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 )

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

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

Purpose:
``` SSPSVX uses the diagonal pivoting factorization A = U*D*U**T or
A = L*D*L**T to compute the solution to a real 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 REAL 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 REAL 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 SSPTRF, 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 SSPTRF, 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 SSPTRF. 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 SSPTRF.``` [in] B ``` B is REAL 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 REAL 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 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: 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 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 278 of file sspsvx.f.

278 *
279 * -- LAPACK driver routine (version 3.7.1) --
280 * -- LAPACK is a software package provided by Univ. of Tennessee, --
281 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
282 * April 2012
283 *
284 * .. Scalar Arguments ..
285  CHARACTER fact, uplo
286  INTEGER info, ldb, ldx, n, nrhs
287  REAL rcond
288 * ..
289 * .. Array Arguments ..
290  INTEGER ipiv( * ), iwork( * )
291  REAL afp( * ), ap( * ), b( ldb, * ), berr( * ),
292  \$ ferr( * ), work( * ), x( ldx, * )
293 * ..
294 *
295 * =====================================================================
296 *
297 * .. Parameters ..
298  REAL zero
299  parameter( zero = 0.0e+0 )
300 * ..
301 * .. Local Scalars ..
302  LOGICAL nofact
303  REAL anorm
304 * ..
305 * .. External Functions ..
306  LOGICAL lsame
307  REAL slamch, slansp
308  EXTERNAL lsame, slamch, slansp
309 * ..
310 * .. External Subroutines ..
311  EXTERNAL scopy, slacpy, sspcon, ssprfs, ssptrf, ssptrs,
312  \$ xerbla
313 * ..
314 * .. Intrinsic Functions ..
315  INTRINSIC max
316 * ..
317 * .. Executable Statements ..
318 *
319 * Test the input parameters.
320 *
321  info = 0
322  nofact = lsame( fact, 'N' )
323  IF( .NOT.nofact .AND. .NOT.lsame( fact, 'F' ) ) THEN
324  info = -1
325  ELSE IF( .NOT.lsame( uplo, 'U' ) .AND. .NOT.lsame( uplo, 'L' ) )
326  \$ THEN
327  info = -2
328  ELSE IF( n.LT.0 ) THEN
329  info = -3
330  ELSE IF( nrhs.LT.0 ) THEN
331  info = -4
332  ELSE IF( ldb.LT.max( 1, n ) ) THEN
333  info = -9
334  ELSE IF( ldx.LT.max( 1, n ) ) THEN
335  info = -11
336  END IF
337  IF( info.NE.0 ) THEN
338  CALL xerbla( 'SSPSVX', -info )
339  RETURN
340  END IF
341 *
342  IF( nofact ) THEN
343 *
344 * Compute the factorization A = U*D*U**T or A = L*D*L**T.
345 *
346  CALL scopy( n*( n+1 ) / 2, ap, 1, afp, 1 )
347  CALL ssptrf( uplo, n, afp, ipiv, info )
348 *
349 * Return if INFO is non-zero.
350 *
351  IF( info.GT.0 )THEN
352  rcond = zero
353  RETURN
354  END IF
355  END IF
356 *
357 * Compute the norm of the matrix A.
358 *
359  anorm = slansp( 'I', uplo, n, ap, work )
360 *
361 * Compute the reciprocal of the condition number of A.
362 *
363  CALL sspcon( uplo, n, afp, ipiv, anorm, rcond, work, iwork, info )
364 *
365 * Compute the solution vectors X.
366 *
367  CALL slacpy( 'Full', n, nrhs, b, ldb, x, ldx )
368  CALL ssptrs( uplo, n, nrhs, afp, ipiv, x, ldx, info )
369 *
370 * Use iterative refinement to improve the computed solutions and
371 * compute error bounds and backward error estimates for them.
372 *
373  CALL ssprfs( uplo, n, nrhs, ap, afp, ipiv, b, ldb, x, ldx, ferr,
374  \$ berr, work, iwork, info )
375 *
376 * Set INFO = N+1 if the matrix is singular to working precision.
377 *
378  IF( rcond.LT.slamch( 'Epsilon' ) )
379  \$ info = n + 1
380 *
381  RETURN
382 *
383 * End of SSPSVX
384 *
subroutine sspcon(UPLO, N, AP, IPIV, ANORM, RCOND, WORK, IWORK, INFO)
SSPCON
Definition: sspcon.f:127
real function slansp(NORM, UPLO, N, AP, WORK)
SLANSP returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a symmetric matrix supplied in packed form.
Definition: slansp.f:116
subroutine ssprfs(UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO)
SSPRFS
Definition: ssprfs.f:181
subroutine ssptrs(UPLO, N, NRHS, AP, IPIV, B, LDB, INFO)
SSPTRS
Definition: ssptrs.f:117
subroutine ssptrf(UPLO, N, AP, IPIV, INFO)
SSPTRF
Definition: ssptrf.f:159
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:69
subroutine slacpy(UPLO, M, N, A, LDA, B, LDB)
SLACPY copies all or part of one two-dimensional array to another.
Definition: slacpy.f:105
subroutine scopy(N, SX, INCX, SY, INCY)
SCOPY
Definition: scopy.f:84
Here is the call graph for this function:
Here is the caller graph for this function: