spbsvx(3)

NAME

SPBSVX - 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 SPBSVX( FACT, UPLO, N, KD, NRHS, AB, LDAB, AFB,
LDAFB,  EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, IWORK,
INFO )
    CHARACTER      EQUED, FACT, UPLO
    INTEGER        INFO, KD, LDAB,  LDAFB,  LDB,  LDX,  N,
NRHS
    REAL           RCOND
    INTEGER        IWORK( * )
    REAL           AB( LDAB, * ), AFB( LDAFB, * ), B( LDB,
* ), BERR( * ), FERR( * ), S( * ), WORK( * ), X( LDX, * )

PURPOSE

SPBSVX 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

band matrix 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 band matrix, and L is a

lower
triangular band 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,

AFB contains the factored form of A. If EQUED = 'Y', the matrix

A has been equilibrated with scaling factors given by S. AB and

AFB will not be modified. = 'N': The matrix A will be copied to

AFB and factored.
= 'E': The matrix A will be equilibrated if nec

essary, then copied to AFB 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.

KD (input) INTEGER
The number of superdiagonals of the matrix A if

UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >=

0.

NRHS (input) INTEGER
The number of right-hand sides, i.e., the number

of columns of the matrices B and X. NRHS >= 0.

AB (input/output) REAL array, dimension (LDAB,N)On entry, the upper or lower triangle of the sym

metric band matrix A, stored in the first KD+1 rows of the 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 j-th column of the array AB as follows: if UPLO =

'U', AB(KD+1+i-j,j) = A(i,j) for max(1,j-KD)<=i<=j; if UPLO =

'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(N,j+KD). See below

for further details.

On exit, if FACT = 'E' and EQUED = 'Y', A is over

written by diag(S)*A*diag(S).

LDAB (input) INTEGER
The leading dimension of the array A. LDAB >=

KD+1.

AFB (input or output) REAL array, dimension (LDAFB,N)
If FACT = 'F', then AFB is an input argument and

on entry contains the triangular factor U or L from the Cholesky

factorization A = U**T*U or A = L*L**T of the band matrix A, in

the same storage format as A (see AB). If EQUED = 'Y', then AFB

is the factored form of the equilibrated matrix A.

If FACT = 'N', then AFB is an output argument and

on exit returns the triangular factor U or L from the Cholesky

factorization A = U**T*U or A = L*L**T.

If FACT = 'E', then AFB is an output argument and

on exit returns the triangular factor U or L from the Cholesky

factorization A = U**T*U or A = L*L**T of the equilibrated matrix

A (see the description of A for the form of the equilibrated ma

trix).

LDAFB (input) INTEGER
The leading dimension of the array AFB. LDAFB >=

KD+1.

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 band storage scheme is illustrated by the following

example, when N = 6, KD = 2, and UPLO = 'U':

Two-dimensional storage of the symmetric matrix A:

a11 a12 a13
a22 a23 a24
a33 a34 a35
a44 a45 a46
a55 a56

(aij=conjg(aji)) a66

Band storage of the upper triangle of A:

* * a13 a24 a35 a46
* a12 a23 a34 a45 a56

a11 a22 a33 a44 a55 a66

Similarly, if UPLO = 'L' the format of A is as follows:

a11 a22 a33 a44 a55 a66
a21 a32 a43 a54 a65 *
a31 a42 a53 a64 * *

Array elements marked * are not used by the routine.

LAPACK version 3.0 15 June 2000

SPBSVX(3)