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NAME
DSPSVX - use 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
SYNOPSIS
SUBROUTINE DSPSVX( FACT, UPLO, N, NRHS, AP, AFP, IPIV, B,
LDB, X, LDX, RCOND, FERR, BERR, WORK,
IWORK, INFO )
CHARACTER FACT, UPLO
INTEGER INFO, LDB, LDX, N, NRHS
DOUBLE PRECISION RCOND
INTEGER IPIV( * ), IWORK( * )
DOUBLE PRECISION AFP( * ), AP( * ), B( LDB, * ),
BERR( * ), FERR( * ), WORK( * ), X( LDX,
* )
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. 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 fac-
tored 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, 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 (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.
AP (input) 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)*(2n-j)/2) = A(i,j) for
j<=i<=n. See below for further details.
AFP (input or output) 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 fac-
tor 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.
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 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 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 DSPTRF.
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 (3*N)
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 and <= N: if INFO = i, D(i,i) is exactly zero.
The factorization has been completed, but the block
diagonal matrix D is exactly singular, so the solu-
tion and error bounds could not be computed. = N+1:
the block diagonal matrix D is nonsingular, but
RCOND is less than machine precision. The factori-
zation has been completed, but the matrix is singu-
lar to working precision, so the solution and error
bounds have not been computed.
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 ]