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# dsysvx

```
NAME
DSYSVX - use the diagonal pivoting factorization to compute
the solution to a real system of linear equations A * X = B,

SYNOPSIS
SUBROUTINE DSYSVX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF,
IPIV, B, LDB, X, LDX, RCOND, FERR, BERR,
WORK, LWORK, IWORK, INFO )

CHARACTER      FACT, UPLO

INTEGER        INFO, LDA, LDAF, LDB, LDX, LWORK, N, NRHS

DOUBLE         PRECISION RCOND

INTEGER        IPIV( * ), IWORK( * )

DOUBLE         PRECISION A( LDA, * ), AF( LDAF, * ), B(
LDB, * ), BERR( * ), FERR( * ), WORK( *
), X( LDX, * )

PURPOSE
DSYSVX uses the diagonal pivoting factorization to compute
the solution to a real system of linear equations A * X = B,
where A is an N-by-N symmetric matrix 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.
The form of the factorization is
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 diago-
nal with
1-by-1 and 2-by-2 diagonal blocks.

2. 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, steps 3 and 4 are skipped.

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.

ARGUMENTS
FACT    (input) CHARACTER*1
Specifies whether or not the factored form of A has
been supplied on entry.  = 'F':  On entry, AF and
IPIV contain the factored form of A.  AF and IPIV
will not be modified.  = 'N':  The matrix A will be
copied to AF and factored.

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

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

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

A       (input) DOUBLE PRECISION array, dimension (LDA,N)
The symmetric matrix A.  If UPLO = 'U', the leading
N-by-N upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly
lower triangular part of A is not referenced.  If
UPLO = 'L', the leading N-by-N lower triangular part
of A contains the lower triangular part of the
matrix A, and the strictly upper triangular part of
A is not referenced.

LDA     (input) INTEGER
The leading dimension of the array A.  LDA >=
max(1,N).

(LDAF,N)
AF      (input or output) DOUBLE PRECISION array, dimension
If FACT = 'F', then AF 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 DSYTRF.

If FACT = 'N', then AF is an output argument and on
exit returns 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.

LDAF    (input) INTEGER
The leading dimension of the array AF.  LDAF >=
max(1,N).

IPIV    (input or output) 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 DSYTRF.  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 inter-
changed 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 inter-
changed 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 DSYTRF.

B       (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
The N-by-NRHS right hand side matrix B.

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

X       (output) DOUBLE PRECISION array, dimension (LDX,NRHS)
If INFO = 0, the N-by-NRHS solution matrix X.

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

RCOND   (output) 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, and the
solution and error bounds are not computed.

FERR    (output) DOUBLE PRECISION array, dimension (NRHS)
The estimated forward error bounds for each solution
vector X(j) (the j-th column of the solution matrix

X).  If XTRUE is the true solution, FERR(j) bounds
the magnitude of the largest entry in (X(j) - XTRUE)
divided by the magnitude of the largest entry in
X(j).  The quality of the error bound depends on the
quality of the estimate of norm(inv(A)) computed in
the code; if the estimate of norm(inv(A)) is accu-
rate, the error bound is guaranteed.

BERR    (output) DOUBLE PRECISION array, dimension (NRHS)
The componentwise relative backward error of each
solution vector X(j) (i.e., the smallest relative
change in any entry of A or B that makes X(j) an
exact solution).

WORK    (workspace) DOUBLE PRECISION array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns the optimal
LWORK.

LWORK   (input) INTEGER
The length of WORK.  LWORK >= 3*N, and for best per-
formance LWORK >= N*NB, where NB is the optimal
blocksize for DSYTRF.

IWORK   (workspace) INTEGER array, dimension (N)

INFO    (output) 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 block diagonal matrix D is
exactly singular, so the solution and error bounds
could not be computed.  = N+1: the block diagonal
matrix D is nonsingular, but RCOND is less than
machine precision.  The factorization has been com-
pleted, but the matrix is singular to working preci-
sion, so the solution and error bounds have not been
computed.
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