spbsvx (l) - Linux Manuals
spbsvx: 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,
NAME
SPBSVX - 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,SYNOPSIS
- SUBROUTINE SPBSVX(
- FACT, UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, IWORK, INFO )
- CHARACTER EQUED, FACT, UPLO
- INTEGER INFO, KD, LDAB, LDAFB, LDB, LDX, N, NRHS
- REAL RCOND
- INTEGER IWORK( * )
- REAL AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ), BERR( * ), FERR( * ), S( * ), WORK( * ), X( LDX, * )
PURPOSE
SPBSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to compute the solution to a real system of linear equationsA
Error bounds on the solution and a condition estimate are also provided.
DESCRIPTION
The following steps are performed:1. If FACT = aqEaq, real scaling factors are computed to equilibrate
the system:
diag(S)
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)
2. If FACT = aqNaq or aqEaq, the Cholesky decomposition is used to
factor the matrix A
A
A
where U is an upper triangular band matrix, and L is a lower
triangular band matrix.
3. If the leading i-by-i principal minor is not positive definite,
then the routine returns with INFO
form of A is used to estimate the condition number of the matrix
A.
precision, INFO
still goes on to solve for X and compute error bounds as
described below.
4. The system of equations is solved for X using the factored form
of A.
5. Iterative refinement is applied to improve the computed solution
matrix and calculate error bounds and backward error estimates
for it.
6. If equilibration was used, the matrix X is premultiplied by
diag(S)
equilibration.
ARGUMENTS
- 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.
= aqFaq: On entry, AFB contains the factored form of A.
If EQUED = aqYaq, the matrix A has been equilibrated
with scaling factors given by S. AB and AFB will not
be modified.
= aqNaq: The matrix A will be copied to AFB and factored.
= aqEaq: The matrix A will be equilibrated if necessary, then copied to AFB and factored. - UPLO (input) CHARACTER*1
-
= aqUaq: Upper triangle of A is stored;
= aqLaq: Lower triangle of A is stored. - N (input) INTEGER
- The number of linear equations, i.e., 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.
- NRHS (input) INTEGER
- The number of right-hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 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, except if FACT = aqFaq and EQUED = aqYaq, 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 = 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). See below for further details. On exit, if FACT = aqEaq and EQUED = aqYaq, A is overwritten by diag(S)*A*diag(S).
- LDAB (input) INTEGER
- The leading dimension of the array A. LDAB >= KD+1.
- AFB (input or output) REAL array, dimension (LDAFB,N)
- If FACT = aqFaq, 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 = aqYaq, then AFB is the factored form of the equilibrated matrix A. If FACT = aqNaq, 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 = aqEaq, 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).
- LDAFB (input) INTEGER
- The leading dimension of the array AFB. LDAFB >= KD+1.
- EQUED (input or output) CHARACTER*1
-
Specifies the form of equilibration that was done.
= aqNaq: No equilibration (always true if FACT = aqNaq).
= aqYaq: Equilibration was done, i.e., A has been replaced by diag(S) * A * diag(S). EQUED is an input argument if FACT = aqFaq; otherwise, it is an output argument. - S (input or output) REAL array, dimension (N)
- The scale factors for A; not accessed if EQUED = aqNaq. S is an input argument if FACT = aqFaq; otherwise, S is an output argument. If FACT = aqFaq and EQUED = aqYaq, each element of S must be positive.
- B (input/output) REAL array, dimension (LDB,NRHS)
- On entry, the N-by-NRHS right hand side matrix B. On exit, if EQUED = aqNaq, B is not modified; if EQUED = aqYaq, B is overwritten by diag(S) * B.
- LDB (input) INTEGER
- The leading dimension of the array B. LDB >= max(1,N).
- X (output) REAL 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 = aqYaq, A and B are modified on exit, and the solution to the equilibrated system is inv(diag(S))*X.
- LDX (input) INTEGER
- The leading dimension of the array X. LDX >= max(1,N).
- RCOND (output) REAL
- 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.
- FERR (output) REAL 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.
- BERR (output) REAL 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).
- WORK (workspace) REAL array, dimension (3*N)
- IWORK (workspace) INTEGER array, dimension (N)
- INFO (output) INTEGER
-
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, and i is
<= 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.
FURTHER DETAILS
The band storage scheme is illustrated by the following example, when N = 6, KD = 2, and UPLO = aqUaq:Two-dimensional storage of the symmetric matrix A:
a11
(aij=conjg(aji))
Band storage of the upper triangle of A:
a11
Similarly, if UPLO = aqLaq the format of A is as follows:
a11
a21
a31
Array elements marked * are not used by the routine.