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NAME
SPPSVX - use the Cholesky factorization A = U**T*U or A =
L*L**T to compute the solution to a real system of linear
equations A * X = B,
SYNOPSIS
SUBROUTINE SPPSVX( FACT, UPLO, N, NRHS, AP, AFP, EQUED, S,
B, LDB, X, LDX, RCOND, FERR, BERR, WORK,
IWORK, INFO )
CHARACTER EQUED, FACT, UPLO
INTEGER INFO, LDB, LDX, N, NRHS
REAL RCOND
INTEGER IWORK( * )
REAL AFP( * ), AP( * ), B( LDB, * ), BERR( *
), FERR( * ), S( * ), WORK( * ), X( LDX,
* )
PURPOSE
SPPSVX uses the Cholesky factorization A = U**T*U or A =
L*L**T to compute the solution to a real system of linear
equations
A * X = B, where A is an N-by-N symmetric positive defin-
ite 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 = 'E', real scaling factors are computed to
equilibrate
the system:
diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B
Whether or not the system will be equilibrated depends on
the
scaling of the matrix A, but if equilibration is used, A
is
overwritten by diag(S)*A*diag(S) and B by diag(S)*B.
2. If FACT = 'N' or 'E', the Cholesky decomposition is used
to
factor the matrix A (after equilibration if FACT = 'E')
as
A = U**T* U, if UPLO = 'U', or
A = L * L**T, if UPLO = 'L',
where U is an upper triangular matrix and L is a lower
triangular
matrix.
3. 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 4-6 are skipped.
4. The system of equations is solved for X using the fac-
tored form
of A.
5. Iterative refinement is applied to improve the computed
solution
matrix and calculate error bounds and backward error
estimates
for it.
6. If equilibration was used, the matrix X is premultiplied
by
diag(S) so that it solves the original system before
equilibration.
ARGUMENTS
FACT (input) CHARACTER*1
Specifies whether or not the factored form of the
matrix A is supplied on entry, and if not, whether
the matrix A should be equilibrated before it is
factored. = 'F': On entry, AFP contains the fac-
tored form of A. If EQUED = 'Y', the matrix A has
been equilibrated with scaling factors given by S.
AP and AFP will not be modified. = 'N': The matrix
A will be copied to AFP and factored.
= 'E': The matrix A will be equilibrated if neces-
sary, then 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/output) REAL array, dimension (N*(N+1)/2)
On entry, the upper or lower triangle of the sym-
metric matrix A, packed columnwise in a linear
array, except if FACT = 'F' and EQUED = 'Y', then A
must contain the equilibrated matrix
diag(S)*A*diag(S). 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. A is not modified if FACT =
'F' or 'N', or if FACT = 'E' and EQUED = 'N' on
exit.
On exit, if FACT = 'E' and EQUED = 'Y', A is
overwritten by diag(S)*A*diag(S).
AFP (input or output) REAL array, dimension
(N*(N+1)/2) If FACT = 'F', then AFP is an input
argument and on entry contains the triangular factor
U or L from the Cholesky factorization A = U'*U or A
= L*L', in the same storage format as A. If EQUED
.ne. 'N', then AFP is the factored form of the
equilibrated matrix A.
If FACT = 'N', then AFP is an output argument and on
exit returns the triangular factor U or L from the
Cholesky factorization A = U'*U or A = L*L' of the
original matrix A.
If FACT = 'E', then AFP is an output argument and on
exit returns the triangular factor U or L from the
Cholesky factorization A = U'*U or A = L*L' of the
equilibrated matrix A (see the description of AP for
the form of the equilibrated matrix).
EQUED (input/output) CHARACTER*1
Specifies the form of equilibration that was done.
= 'N': No equilibration (always true if FACT =
'N').
= 'Y': Equilibration was done, i.e., A has been
replaced by diag(S) * A * diag(S). EQUED is an
input variable if FACT = 'F'; otherwise, it is an
output variable.
S (input/output) REAL array, dimension (N)
The scale factors for A; not accessed if EQUED =
'N'. S is an input variable if FACT = 'F'; other-
wise, S is an output variable. If FACT = 'F' and
EQUED = 'Y', each element of S must be positive.
B (input/output) REAL array, dimension (LDB,NRHS)
On entry, the N-by-NRHS righthand side matrix B. On
exit, if EQUED = 'N', B is not modified; if EQUED =
'Y', B is overwritten by diag(S) * B.
LDB (input) INTEGER
The leading dimension of the array B. LDB >=
max(1,N).
X (output) REAL array, dimension (LDX,NRHS)
If INFO = 0, the N-by-NRHS solution matrix X to the
original system of equations. Note that if EQUED =
'Y', A and B are modified on exit, and the solution
to the equilibrated system is inv(diag(S))*X.
LDX (input) INTEGER
The leading dimension of the array X. LDX >=
max(1,N).
RCOND (output) REAL
The estimate of the reciprocal condition number of
the matrix A after equilibration (if done). If
RCOND is less than the machine precision (in partic-
ular, if RCOND = 0), the matrix is singular to work-
ing precision. This condition is indicated by a
return code of INFO > 0, and the solution and error
bounds are not computed.
FERR (output) REAL 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) REAL 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) REAL 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: if INFO = i, and i is
<= N: if INFO = i, the leading minor of order i of A
is not positive definite, so the factorization could
not be completed, and the solution and error bounds
could not be computed. = N+1: RCOND is less than
machine precision. The factorization has been com-
pleted, but the matrix is singular to working preci-
sion, and 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 = conjg(aji))
a44
Packed storage of the upper triangle of A:
AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]