
SPPSVX(l) ) SPPSVX(l)
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 NbyN symmetric positive defi
nite matrix stored in packed format and X and B are NbyNRHS 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. If the leading ibyi 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 factored 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 factored 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 necessary, 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 symmetric 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 jth column of A is stored in the
array AP as follows: if UPLO = 'U', AP(i + (j1)*j/2) = A(i,j) for 1<=i<=j; if
UPLO = 'L', AP(i + (j1)*(2nj)/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 equili
brated 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'; otherwise, S is an output argument. If FACT = 'F' and EQUED = 'Y',
each element of S must be positive.
B (input/output) REAL array, dimension (LDB,NRHS)
On entry, the NbyNRHS 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 NbyNRHS 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 equilibra
tion (if done). If RCOND is less than the machine precision (in particular, if
RCOND = 0), the matrix is singular to working precision. This condition is indi
cated by a return code of INFO > 0.
FERR (output) REAL 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) 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 solu
tion).
WORK (workspace) REAL array, dimension (3*N)
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: the leading minor of order i of A is not positive definite, so the factor
ization could not be completed, and the solution has not been computed. RCOND = 0
is returned. = N+1: U is nonsingular, but RCOND is less than machine precision,
meaning that the matrix is singular to working precision. Nevertheless, the solu
tion 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':
Twodimensional 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 ]
LAPACK version 3.0 15 June 2000 SPPSVX(l) 
