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 posi
tive definite 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
matrix.
3. If the leading i-by-i principal minor is not positive
definite,
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.
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 premulti
plied 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 nec
essary, 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.
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 over
written 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 fac
tor 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 or 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 argument if FACT = 'F'; otherwise, it is an
output argument.
S (input or output) REAL array, dimension (N)
The scale factors for A; not accessed if EQUED =
'N'. S is an input argument if FACT = 'F'; other
wise, S is an output argument. If FACT = 'F' and
EQUED = 'Y', each element of S must be positive.
On entry, the N-by-NRHS right hand 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 or INFO = N+1, 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 par
ticular, if RCOND = 0), the matrix is singular to
working precision. This condition is indicated by
a return code of INFO > 0.
FERR (output) REAL array, dimension (NRHS)
The estimated forward error bound for each solu
tion vector X(j) (the j-th column of the solution
matrix X). If XTRUE is the true solution corre
sponding 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 reli
able as the estimate for RCOND, and is almost
always a slight overestimate of the true error.
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 element 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 ille
gal value
> 0: if INFO = i, and i is
positive definite, so the factorization could not
be completed, and the solution has not been com
puted. RCOND = 0 is returned. = N+1: U is nonsin
gular, 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.
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 ]
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