zspsvx.f man page

zspsvx.f —

Synopsis

Functions/Subroutines

subroutine zspsvx (FACT, UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, RWORK, INFO)
ZSPSVX computes the solution to system of linear equations A * X = B for OTHER matrices

Function/Subroutine Documentation

subroutine zspsvx (characterFACT, characterUPLO, integerN, integerNRHS, complex*16, dimension( * )AP, complex*16, dimension( * )AFP, integer, dimension( * )IPIV, complex*16, dimension( ldb, * )B, integerLDB, complex*16, dimension( ldx, * )X, integerLDX, double precisionRCOND, double precision, dimension( * )FERR, double precision, dimension( * )BERR, complex*16, dimension( * )WORK, double precision, dimension( * )RWORK, integerINFO)

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

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:

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.

UPLO

UPLO is CHARACTER*1
= 'U':  Upper triangle of A is stored;
= 'L':  Lower triangle of A is stored.

N

N is INTEGER
The number of linear equations, i.e., the order of the
matrix A.  N >= 0.

NRHS

NRHS is INTEGER
The number of right hand sides, i.e., the number of columns
of the matrices B and X.  NRHS >= 0.

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.

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.

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.

B

B is COMPLEX*16 array, dimension (LDB,NRHS)
The N-by-NRHS right hand side matrix B.

LDB

LDB is INTEGER
The leading dimension of the array B.  LDB >= max(1,N).

X

X is COMPLEX*16 array, dimension (LDX,NRHS)
If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.

LDX

LDX is INTEGER
The leading dimension of the array X.  LDX >= max(1,N).

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.

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.

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).

WORK

WORK is COMPLEX*16 array, dimension (2*N)

RWORK

RWORK is DOUBLE PRECISION array, dimension (N)

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.

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 277 of file zspsvx.f.

Author

Generated automatically by Doxygen for LAPACK from the source code.

Referenced By

zspsvx(3) is an alias of zspsvx.f(3).

Sat Nov 16 2013 Version 3.4.2 LAPACK