sppsvx(3)

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 equa

tions 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 equa

tions A * X = B, where A is an N-by-N symmetric positive definite

matrix stored in packed format and X and B are N-by-NRHS matri

ces.

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 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 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 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.

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 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 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 ma

trix).

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 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 particular, if RCOND = 0), the ma

trix 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 solu

tion vector X(j) (the j-th 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 estimate is as reliable 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
<= N: the leading minor of order i of A is not

positive definite, so the factorization 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. Never

theless, 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 ]

LAPACK version 3.0 15 June 2000

SPPSVX(3)