LAPACK  3.10.1
LAPACK: Linear Algebra PACKage

◆ dspsvx()

subroutine dspsvx ( character  FACT,
character  UPLO,
integer  N,
integer  NRHS,
double precision, dimension( * )  AP,
double precision, dimension( * )  AFP,
integer, dimension( * )  IPIV,
double precision, dimension( ldb, * )  B,
integer  LDB,
double precision, dimension( ldx, * )  X,
integer  LDX,
double precision  RCOND,
double precision, dimension( * )  FERR,
double precision, dimension( * )  BERR,
double precision, dimension( * )  WORK,
integer, dimension( * )  IWORK,
integer  INFO 
)

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

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

Purpose:
 DSPSVX 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DSPTRF, 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 DSPTRF, 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 DSPTRF.
          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 DSPTRF.
[in]B
          B is DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
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 dspsvx.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  DOUBLE PRECISION RCOND
285 * ..
286 * .. Array Arguments ..
287  INTEGER IPIV( * ), IWORK( * )
288  DOUBLE PRECISION AFP( * ), AP( * ), B( LDB, * ), BERR( * ),
289  $ FERR( * ), WORK( * ), X( LDX, * )
290 * ..
291 *
292 * =====================================================================
293 *
294 * .. Parameters ..
295  DOUBLE PRECISION ZERO
296  parameter( zero = 0.0d+0 )
297 * ..
298 * .. Local Scalars ..
299  LOGICAL NOFACT
300  DOUBLE PRECISION ANORM
301 * ..
302 * .. External Functions ..
303  LOGICAL LSAME
304  DOUBLE PRECISION DLAMCH, DLANSP
305  EXTERNAL lsame, dlamch, dlansp
306 * ..
307 * .. External Subroutines ..
308  EXTERNAL dcopy, dlacpy, dspcon, dsprfs, dsptrf, dsptrs,
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( 'DSPSVX', -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 dcopy( n*( n+1 ) / 2, ap, 1, afp, 1 )
344  CALL dsptrf( 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 = dlansp( 'I', uplo, n, ap, work )
357 *
358 * Compute the reciprocal of the condition number of A.
359 *
360  CALL dspcon( uplo, n, afp, ipiv, anorm, rcond, work, iwork, info )
361 *
362 * Compute the solution vectors X.
363 *
364  CALL dlacpy( 'Full', n, nrhs, b, ldb, x, ldx )
365  CALL dsptrs( 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 dsprfs( 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.dlamch( 'Epsilon' ) )
376  $ info = n + 1
377 *
378  RETURN
379 *
380 * End of DSPSVX
381 *
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:69
subroutine dlacpy(UPLO, M, N, A, LDA, B, LDB)
DLACPY copies all or part of one two-dimensional array to another.
Definition: dlacpy.f:103
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine dcopy(N, DX, INCX, DY, INCY)
DCOPY
Definition: dcopy.f:82
double precision function dlansp(NORM, UPLO, N, AP, WORK)
DLANSP returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,...
Definition: dlansp.f:114
subroutine dsptrf(UPLO, N, AP, IPIV, INFO)
DSPTRF
Definition: dsptrf.f:159
subroutine dsprfs(UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO)
DSPRFS
Definition: dsprfs.f:179
subroutine dsptrs(UPLO, N, NRHS, AP, IPIV, B, LDB, INFO)
DSPTRS
Definition: dsptrs.f:115
subroutine dspcon(UPLO, N, AP, IPIV, ANORM, RCOND, WORK, IWORK, INFO)
DSPCON
Definition: dspcon.f:125
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