LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
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## ◆ 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

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.```
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 274 of file sspsvx.f.

276*
277* -- LAPACK driver routine --
278* -- LAPACK is a software package provided by Univ. of Tennessee, --
279* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
280*
281* .. Scalar Arguments ..
282 CHARACTER FACT, UPLO
283 INTEGER INFO, LDB, LDX, N, NRHS
284 REAL RCOND
285* ..
286* .. Array Arguments ..
287 INTEGER IPIV( * ), IWORK( * )
288 REAL AFP( * ), AP( * ), B( LDB, * ), BERR( * ),
289 \$ FERR( * ), WORK( * ), X( LDX, * )
290* ..
291*
292* =====================================================================
293*
294* .. Parameters ..
295 REAL ZERO
296 parameter( zero = 0.0e+0 )
297* ..
298* .. Local Scalars ..
299 LOGICAL NOFACT
300 REAL ANORM
301* ..
302* .. External Functions ..
303 LOGICAL LSAME
304 REAL SLAMCH, SLANSP
305 EXTERNAL lsame, slamch, slansp
306* ..
307* .. External Subroutines ..
308 EXTERNAL scopy, slacpy, sspcon, ssprfs, ssptrf, ssptrs,
309 \$ xerbla
310* ..
311* .. Intrinsic Functions ..
312 INTRINSIC max
313* ..
314* .. Executable Statements ..
315*
316* Test the input parameters.
317*
318 info = 0
319 nofact = lsame( fact, 'N' )
320 IF( .NOT.nofact .AND. .NOT.lsame( fact, 'F' ) ) THEN
321 info = -1
322 ELSE IF( .NOT.lsame( uplo, 'U' ) .AND. .NOT.lsame( uplo, 'L' ) )
323 \$ THEN
324 info = -2
325 ELSE IF( n.LT.0 ) THEN
326 info = -3
327 ELSE IF( nrhs.LT.0 ) THEN
328 info = -4
329 ELSE IF( ldb.LT.max( 1, n ) ) THEN
330 info = -9
331 ELSE IF( ldx.LT.max( 1, n ) ) THEN
332 info = -11
333 END IF
334 IF( info.NE.0 ) THEN
335 CALL xerbla( 'SSPSVX', -info )
336 RETURN
337 END IF
338*
339 IF( nofact ) THEN
340*
341* Compute the factorization A = U*D*U**T or A = L*D*L**T.
342*
343 CALL scopy( n*( n+1 ) / 2, ap, 1, afp, 1 )
344 CALL ssptrf( uplo, n, afp, ipiv, info )
345*
346* Return if INFO is non-zero.
347*
348 IF( info.GT.0 )THEN
349 rcond = zero
350 RETURN
351 END IF
352 END IF
353*
354* Compute the norm of the matrix A.
355*
356 anorm = slansp( 'I', uplo, n, ap, work )
357*
358* Compute the reciprocal of the condition number of A.
359*
360 CALL sspcon( uplo, n, afp, ipiv, anorm, rcond, work, iwork, info )
361*
362* Compute the solution vectors X.
363*
364 CALL slacpy( 'Full', n, nrhs, b, ldb, x, ldx )
365 CALL ssptrs( uplo, n, nrhs, afp, ipiv, x, ldx, info )
366*
367* Use iterative refinement to improve the computed solutions and
368* compute error bounds and backward error estimates for them.
369*
370 CALL ssprfs( uplo, n, nrhs, ap, afp, ipiv, b, ldb, x, ldx, ferr,
371 \$ berr, work, iwork, info )
372*
373* Set INFO = N+1 if the matrix is singular to working precision.
374*
375 IF( rcond.LT.slamch( 'Epsilon' ) )
376 \$ info = n + 1
377*
378 RETURN
379*
380* End of SSPSVX
381*
subroutine xerbla(srname, info)
Definition cblat2.f:3285
subroutine scopy(n, sx, incx, sy, incy)
SCOPY
Definition scopy.f:82
subroutine sspcon(uplo, n, ap, ipiv, anorm, rcond, work, iwork, info)
SSPCON
Definition sspcon.f:125
subroutine ssprfs(uplo, n, nrhs, ap, afp, ipiv, b, ldb, x, ldx, ferr, berr, work, iwork, info)
SSPRFS
Definition ssprfs.f:179
subroutine ssptrf(uplo, n, ap, ipiv, info)
SSPTRF
Definition ssptrf.f:157
subroutine ssptrs(uplo, n, nrhs, ap, ipiv, b, ldb, info)
SSPTRS
Definition ssptrs.f:115
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
real function slamch(cmach)
SLAMCH
Definition slamch.f:68
real function slansp(norm, uplo, n, ap, work)
SLANSP returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,...
Definition slansp.f:114
logical function lsame(ca, cb)
LSAME
Definition lsame.f:48
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