.TH SPBSV 1 "November 2006" " LAPACK driver routine (version 3.1) " " LAPACK driver routine (version 3.1) " .SH NAME SPBSV - the solution to a real system of linear equations A * X = B, .SH SYNOPSIS .TP 18 SUBROUTINE SPBSV( UPLO, N, KD, NRHS, AB, LDAB, B, LDB, INFO ) .TP 18 .ti +4 CHARACTER UPLO .TP 18 .ti +4 INTEGER INFO, KD, LDAB, LDB, N, NRHS .TP 18 .ti +4 REAL AB( LDAB, * ), B( LDB, * ) .SH PURPOSE SPBSV computes the solution to a real system of linear equations A * X = B, where A is an N-by-N symmetric positive definite band matrix and X and B are N-by-NRHS matrices. .br The Cholesky decomposition is used to factor A as .br A = U**T * U, if UPLO = \(aqU\(aq, or .br A = L * L**T, if UPLO = \(aqL\(aq, .br where U is an upper triangular band matrix, and L is a lower triangular band matrix, with the same number of superdiagonals or subdiagonals as A. The factored form of A is then used to solve the system of equations A * X = B. .br .SH ARGUMENTS .TP 8 UPLO (input) CHARACTER*1 = \(aqU\(aq: Upper triangle of A is stored; .br = \(aqL\(aq: Lower triangle of A is stored. .TP 8 N (input) INTEGER The number of linear equations, i.e., 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 NRHS (input) INTEGER The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 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). See below for further details. 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 B (input/output) REAL array, dimension (LDB,NRHS) On entry, the N-by-NRHS right hand side matrix B. On exit, if INFO = 0, the N-by-NRHS solution matrix X. .TP 8 LDB (input) INTEGER The leading dimension of the array B. LDB >= max(1,N). .TP 8 INFO (output) INTEGER = 0: successful exit .br < 0: if INFO = -i, the i-th argument had an illegal value .br > 0: if INFO = i, 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. .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. .br