.TH DPBSVX 1 "November 2006" " LAPACK driver routine (version 3.1) " " LAPACK driver routine (version 3.1) " .SH NAME DPBSVX - the Cholesky factorization A = U**T*U or A = L*L**T to compute the solution to a real system of linear equations A * X = B, .SH SYNOPSIS .TP 19 SUBROUTINE DPBSVX( FACT, UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, IWORK, INFO ) .TP 19 .ti +4 CHARACTER EQUED, FACT, UPLO .TP 19 .ti +4 INTEGER INFO, KD, LDAB, LDAFB, LDB, LDX, N, NRHS .TP 19 .ti +4 DOUBLE PRECISION RCOND .TP 19 .ti +4 INTEGER IWORK( * ) .TP 19 .ti +4 DOUBLE PRECISION AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ), BERR( * ), FERR( * ), S( * ), WORK( * ), X( LDX, * ) .SH PURPOSE DPBSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to compute 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 Error bounds on the solution and a condition estimate are also provided. .br .SH DESCRIPTION The following steps are performed: .br 1. If FACT = \(aqE\(aq, real scaling factors are computed to equilibrate the system: .br 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. .br 2. If FACT = \(aqN\(aq or \(aqE\(aq, the Cholesky decomposition is used to factor the matrix A (after equilibration if FACT = \(aqE\(aq) as 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. .br 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. .br 4. The system of equations is solved for X using the factored form of A. .br 5. Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it. .br 6. If equilibration was used, the matrix X is premultiplied by diag(S) so that it solves the original system before .br equilibration. .br .SH ARGUMENTS .TP 8 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. = \(aqF\(aq: On entry, AFB contains the factored form of A. If EQUED = \(aqY\(aq, the matrix A has been equilibrated with scaling factors given by S. AB and AFB will not be modified. = \(aqN\(aq: The matrix A will be copied to AFB and factored. .br = \(aqE\(aq: The matrix A will be equilibrated if necessary, then copied to AFB and factored. .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 matrices B and X. NRHS >= 0. .TP 8 AB (input/output) DOUBLE PRECISION 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, except if FACT = \(aqF\(aq and EQUED = \(aqY\(aq, 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 = \(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 FACT = \(aqE\(aq and EQUED = \(aqY\(aq, A is overwritten by diag(S)*A*diag(S). .TP 8 LDAB (input) INTEGER The leading dimension of the array A. LDAB >= KD+1. .TP 8 AFB (input or output) DOUBLE PRECISION array, dimension (LDAFB,N) If FACT = \(aqF\(aq, 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 = \(aqY\(aq, then AFB is the factored form of the equilibrated matrix A. If FACT = \(aqN\(aq, 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 = \(aqE\(aq, 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 matrix). .TP 8 LDAFB (input) INTEGER The leading dimension of the array AFB. LDAFB >= KD+1. .TP 8 EQUED (input or output) CHARACTER*1 Specifies the form of equilibration that was done. = \(aqN\(aq: No equilibration (always true if FACT = \(aqN\(aq). .br = \(aqY\(aq: Equilibration was done, i.e., A has been replaced by diag(S) * A * diag(S). EQUED is an input argument if FACT = \(aqF\(aq; otherwise, it is an output argument. .TP 8 S (input or output) DOUBLE PRECISION array, dimension (N) The scale factors for A; not accessed if EQUED = \(aqN\(aq. S is an input argument if FACT = \(aqF\(aq; otherwise, S is an output argument. If FACT = \(aqF\(aq and EQUED = \(aqY\(aq, each element of S must be positive. .TP 8 B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS) On entry, the N-by-NRHS right hand side matrix B. On exit, if EQUED = \(aqN\(aq, B is not modified; if EQUED = \(aqY\(aq, B is overwritten by diag(S) * B. .TP 8 LDB (input) INTEGER The leading dimension of the array B. LDB >= max(1,N). .TP 8 X (output) DOUBLE PRECISION 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 = \(aqY\(aq, A and B are modified on exit, and the solution to the equilibrated system is inv(diag(S))*X. .TP 8 LDX (input) INTEGER The leading dimension of the array X. LDX >= max(1,N). .TP 8 RCOND (output) DOUBLE PRECISION 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 matrix is singular to working precision. This condition is indicated by a return code of INFO > 0. .TP 8 FERR (output) DOUBLE PRECISION array, dimension (NRHS) The estimated forward error bound for each solution 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. .TP 8 BERR (output) DOUBLE PRECISION 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). .TP 8 WORK (workspace) DOUBLE PRECISION array, dimension (3*N) .TP 8 IWORK (workspace) INTEGER array, dimension (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, and i is .br <= 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. Nevertheless, 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. .SH FURTHER DETAILS The band storage scheme is illustrated by the following example, when N = 6, KD = 2, and UPLO = \(aqU\(aq: .br Two-dimensional storage of the symmetric matrix A: .br a11 a12 a13 .br a22 a23 a24 .br a33 a34 a35 .br a44 a45 a46 .br a55 a56 .br (aij=conjg(aji)) a66 .br Band storage of the upper triangle of A: .br * * a13 a24 a35 a46 .br * a12 a23 a34 a45 a56 .br a11 a22 a33 a44 a55 a66 .br Similarly, if UPLO = \(aqL\(aq the format of A is as follows: .br a11 a22 a33 a44 a55 a66 .br a21 a32 a43 a54 a65 * .br a31 a42 a53 a64 * * .br Array elements marked * are not used by the routine. .br