.TH SPBTRF 1 "November 2006" " LAPACK routine (version 3.1) " " LAPACK routine (version 3.1) "
.SH NAME
SPBTRF - the Cholesky factorization of a real symmetric positive definite band matrix A
.SH SYNOPSIS
.TP 19
SUBROUTINE SPBTRF(
UPLO, N, KD, AB, LDAB, INFO )
.TP 19
.ti +4
CHARACTER
UPLO
.TP 19
.ti +4
INTEGER
INFO, KD, LDAB, N
.TP 19
.ti +4
REAL
AB( LDAB, * )
.SH PURPOSE
SPBTRF computes the Cholesky factorization of a real symmetric
positive definite band matrix A.
The factorization has the form
.br
A = U**T * U, if UPLO = \(aqU\(aq, or
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A = L * L**T, if UPLO = \(aqL\(aq,
.br
where U is an upper triangular matrix and L is lower triangular.
.SH ARGUMENTS
.TP 8
UPLO (input) CHARACTER*1
= \(aqU\(aq: Upper triangle of A is stored;
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= \(aqL\(aq: Lower triangle of A is stored.
.TP 8
N (input) INTEGER
The order of the matrix A. N >= 0.
.TP 8
KD (input) INTEGER
The number of superdiagonals of the matrix A if UPLO = \(aqU\(aq,
or the number of subdiagonals if UPLO = \(aqL\(aq. KD >= 0.
.TP 8
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 = \(aqU\(aq, AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
if UPLO = \(aqL\(aq, 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.
.TP 8
LDAB (input) INTEGER
The leading dimension of the array AB. LDAB >= KD+1.
.TP 8
INFO (output) INTEGER
= 0: successful exit
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< 0: if INFO = -i, the i-th argument had an illegal value
.br
> 0: if INFO = i, the leading minor of order i is not
positive definite, and the factorization could not be
completed.
.SH FURTHER DETAILS
The band storage scheme is illustrated by the following example, when
N = 6, KD = 2, and UPLO = \(aqU\(aq:
.br
On entry: On exit:
.br
* * 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 = \(aqL\(aq the format of A is as follows:
.br
On entry: On exit:
.br
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.
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Contributed by
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Peter Mayes and Giuseppe Radicati, IBM ECSEC, Rome, March 23, 1989