
DSYSVX(l) ) DSYSVX(l)
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 NbyN symmetric matrix and X and B are Nby
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 diagonal with
1by1 and 2by2 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.
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 NbyN 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 NbyN 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).
AF (input or output) DOUBLE PRECISION array, dimension (LDAF,N)
If FACT = 'F', then AF is an input argument and on entry contains the block diago
nal matrix D and the multipliers used to obtain the factor U or L from the factor
ization 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 diago
nal matrix D and the multipliers used to obtain the factor U or L from the factor
ization 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 1by1
diagonal block. If UPLO = 'U' and IPIV(k) = IPIV(k1) < 0, then rows and columns
k1 and IPIV(k) were interchanged and D(k1:k,k1:k) is a 2by2 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 2by2 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 NbyNRHS 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 or INFO = N+1, the NbyNRHS 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.
FERR (output) DOUBLE PRECISION array, dimension (NRHS)
The estimated forward error bound for each solution vector X(j) (the jth 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 esti
mate is as reliable as the estimate for RCOND, and is almost always a slight over
estimate of the true error.
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 element of A or B that makes X(j) an exact solu
tion).
WORK (workspace/output) 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 performance LWORK >= N*NB, where
NB is the optimal blocksize for DSYTRF.
If LWORK = 1, then a workspace query is assumed; the routine only calculates the
optimal size of the WORK array, returns this value as the first entry of the WORK
array, and no error message related to LWORK is issued by XERBLA.
IWORK (workspace) INTEGER array, dimension (N)
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = i, the ith 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 fac
tor 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. Neverthe
less, 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.
LAPACK version 3.0 15 June 2000 DSYSVX(l) 
