dsptrf(3)

NAME

DSPTRF - compute the factorization of a real symmetric ma

trix A stored in packed format using the Bunch-Kaufman diagonal

pivoting method

SYNOPSIS

SUBROUTINE DSPTRF( UPLO, N, AP, IPIV, INFO )
    CHARACTER      UPLO
    INTEGER        INFO, N
    INTEGER        IPIV( * )
    DOUBLE         PRECISION AP( * )

PURPOSE

DSPTRF computes the factorization of a real symmetric ma

trix A stored in packed format using the Bunch-Kaufman diagonal

pivoting method:

A = U*D*U**T or A = L*D*L**T

where U (or L) is a product of permutation and unit upper

(lower) triangular matrices, and D is symmetric and block diago

nal with 1-by-1 and 2-by-2 diagonal blocks.

ARGUMENTS

UPLO (input) CHARACTER*1

= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.

N (input) INTEGER

The order of the matrix A. N >= 0.

AP (input/output) DOUBLE PRECISION 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. 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.

On exit, the block diagonal matrix D and the mul

tipliers used to obtain the factor U or L, stored as a packed

triangular matrix overwriting A (see below for further details).

IPIV (output) INTEGER array, dimension (N)

Details of the interchanges and the block struc

ture of D. If IPIV(k) > 0, then rows and columns k and IPIV(k)

were interchanged and D(k,k) is a 1-by-1 diagonal block. If UPLO

= 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and columns k-1 and

-IPIV(k) were interchanged and D(k-1:k,k-1:k) is a 2-by-2 diago

nal block. If UPLO = 'L' and IPIV(k) = IPIV(k+1) < 0, then rows

and columns k+1 and -IPIV(k) were interchanged and D(k:k+1,k:k+1)

is a 2-by-2 diagonal block.

INFO (output) INTEGER

= 0: successful exit
< 0: if INFO = -i, the i-th argument had an ille

gal value
> 0: if INFO = i, D(i,i) is exactly zero. The

factorization has been completed, but the block diagonal matrix D

is exactly singular, and division by zero will occur if it is

used to solve a system of equations.

FURTHER DETAILS

5-96 - Based on modifications by J. Lewis, Boeing Computer

Services

Company

If UPLO = 'U', then A = U*D*U', where

U = P(n)*U(n)* ... *P(k)U(k)* ...,

i.e., U is a product of terms P(k)*U(k), where k decreases

from n to 1 in steps of 1 or 2, and D is a block diagonal matrix

with 1-by-1 and 2-by-2 diagonal blocks D(k). P(k) is a permuta

tion matrix as defined by IPIV(k), and U(k) is a unit upper tri

angular matrix, such that if the diagonal block D(k) is of order

s (s = 1 or 2), then

( I v 0 ) k-s

U(k) = ( 0 I 0 ) s

( 0 0 I ) n-k

k-s s n-k

If s = 1, D(k) overwrites A(k,k), and v overwrites

A(1:k-1,k). If s = 2, the upper triangle of D(k) overwrites

A(k-1,k-1), A(k-1,k), and A(k,k), and v overwrites

A(1:k-2,k-1:k).

If UPLO = 'L', then A = L*D*L', where

L = P(1)*L(1)* ... *P(k)*L(k)* ...,

i.e., L is a product of terms P(k)*L(k), where k increases

from 1 to n in steps of 1 or 2, and D is a block diagonal matrix

with 1-by-1 and 2-by-2 diagonal blocks D(k). P(k) is a permuta

tion matrix as defined by IPIV(k), and L(k) is a unit lower tri

angular matrix, such that if the diagonal block D(k) is of order

s (s = 1 or 2), then

( I 0 0 ) k-1

L(k) = ( 0 I 0 ) s

( 0 v I ) n-k-s+1

k-1 s n-k-s+1

If s = 1, D(k) overwrites A(k,k), and v overwrites

A(k+1:n,k). If s = 2, the lower triangle of D(k) overwrites

A(k,k), A(k+1,k), and A(k+1,k+1), and v overwrites

A(k+2:n,k:k+1).

LAPACK version 3.0 15 June 2000

DSPTRF(3)