spbtrf (l) - Linux Manuals

spbtrf: computes the Cholesky factorization of a real symmetric positive definite band matrix A

NAME

SPBTRF - computes the Cholesky factorization of a real symmetric positive definite band matrix A

SYNOPSIS

SUBROUTINE SPBTRF(
UPLO, N, KD, AB, LDAB, INFO )

    
CHARACTER UPLO

    
INTEGER INFO, KD, LDAB, N

    
REAL AB( LDAB, * )

PURPOSE

SPBTRF computes the Cholesky factorization of a real symmetric positive definite band matrix A. The factorization has the form

A = U**T * U,  if UPLO = aqUaq, or

A = L  * L**T,  if UPLO = aqLaq,
where U is an upper triangular matrix and L is lower triangular.

ARGUMENTS

UPLO (input) CHARACTER*1
= aqUaq: Upper triangle of A is stored;
= aqLaq: Lower triangle of A is stored.
N (input) INTEGER
The order of the matrix A. N >= 0.
KD (input) INTEGER
The number of superdiagonals of the matrix A if UPLO = aqUaq, or the number of subdiagonals if UPLO = aqLaq. KD >= 0.
AB (input/output) REAL array, dimension (LDAB,N)
On entry, the upper or lower triangle of the symmetric band matrix A, stored in the first KD+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows: if UPLO = aqUaq, AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; if UPLO = aqLaq, AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). On exit, if INFO = 0, 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.
LDAB (input) INTEGER
The leading dimension of the array AB. LDAB >= KD+1.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, the leading minor of order i is not positive definite, and the factorization could not be completed.

FURTHER DETAILS

The band storage scheme is illustrated by the following example, when N = 6, KD = 2, and UPLO = aqUaq:
On entry: On exit:

 *    *   a13  a24  a35  a46      *    *   u13  u24  u35  u46
 *   a12  a23  a34  a45  a56      *   u12  u23  u34  u45  u56
a11  a22  a33  a44  a55  a66     u11  u22  u33  u44  u55  u66 Similarly, if UPLO = aqLaq the format of A is as follows:
On entry: On exit:

a11  a22  a33  a44  a55  a66     l11  l22  l33  l44  l55  l66
a21  a32  a43  a54  a65   *      l21  l32  l43  l54  l65   *
a31  a42  a53  a64   *    *      l31  l42  l53  l64   *    * Array elements marked * are not used by the routine.
Contributed by
Peter Mayes and Giuseppe Radicati, IBM ECSEC, Rome, March 23, 1989

Pages related to spbtrf

  • spbtrf (3)
  • spbtrs (l) - solves a system of linear equations A*X = B with a symmetric positive definite band matrix A using the Cholesky factorization A = U**T*U or A = L*L**T computed by SPBTRF
  • spbtf2 (l) - computes the Cholesky factorization of a real symmetric positive definite band matrix A
  • spbcon (l) - estimates the reciprocal of the condition number (in the 1-norm) of a real symmetric positive definite band matrix using the Cholesky factorization A = U**T*U or A = L*L**T computed by SPBTRF
  • spbequ (l) - computes row and column scalings intended to equilibrate a symmetric positive definite band matrix A and reduce its condition number (with respect to the two-norm)
  • spbrfs (l) - improves the computed solution to a system of linear equations when the coefficient matrix is symmetric positive definite and banded, and provides error bounds and backward error estimates for the solution
  • spbstf (l) - computes a split Cholesky factorization of a real symmetric positive definite band matrix A
  • spbsv (l) - computes the solution to a real system of linear equations A * X = B,