LFSRB

Solves a real system of linear equations given the LU factorization of the coefficient matrix in band storage mode.

Required Arguments

FACT — (2 * NLCA + NUCA + 1) by N array containing the LU factorization of the coefficient matrix A as output from routine LFCRB/DLFCRB or LFTRB/DLFTRB. (Input)

NLCA — Number of lower codiagonals of A. (Input)

NUCA — Number of upper codiagonals of A. (Input)

IPVT — Vector of length N containing the pivoting information for the LU factorization of A as output from routine LFCRB/DLFCRB or LFTRB/DLFTRB. (Input)

B — Vector of length N containing the right-hand side of the linear system. (Input)

X — Vector of length N containing the solution to the linear system. (Output)
If B is not needed, B and X can share the same storage locations.

Optional Arguments

N — Number of equations. (Input)
Default: N = size (FACT,2).

LDFACT — Leading dimension of FACT exactly as specified in the dimension statement of the calling program. (Input)
Default: LDFACT = size (FACT,1).

IPATH — Path indicator. (Input)
IPATH = 1 means the system AX = B is solved.
IPATH = 2 means the system ATX = B is solved.
Default: IPATH = 1.

FORTRAN 90 Interface

Generic: CALL LFSRB (FACT, NLCA, NUCA, IPVT, B, X [, …])

Specific: The specific interface names are S_LFSRB and D_LFSRB.

FORTRAN 77 Interface

Single: CALL LFSRB (N, FACT, LDFACT, NLCA, NUCA, IPVT, B, IPATH, X)

Double: The double precision name is DLFSRB.

Description

Routine LFSRB computes the solution of a system of linear algebraic equations having a real banded coefficient matrix. To compute the solution, the coefficient matrix must first undergo an LU factorization. This may be done by calling either LFCRB or LFTRB. The solution to Ax = b is found by solving the banded triangular systems Ly = b and Ux = y. The forward elimination step consists of solving the system Ly = b by applying the same permutations and elimination operations to b that were applied to the columns of A in the factorization routine. The backward substitution step consists of solving the banded triangular system Ux = y for x.

LFSRB and LFIRB both solve a linear system given its LU factorization. LFIRB generally takes more time and produces a more accurate answer than LFSRB. Each iteration of the iterative refinement algorithm used by LFIRB calls LFSRB.

The underlying code is based on either LINPACK or LAPACK code depending upon which supporting libraries are used during linking. For a detailed explanation see Using ScaLAPACK, LAPACK, LINPACK, and EISPACK in the Introduction section of this manual.

Example

The inverse is computed for a real banded 4 × 4 matrix with one upper and one lower codiagonal. The input matrix is assumed to be well-conditioned, hence LFTRB is used rather than LFCRB.

USE LFSRB_INT

USE LFTRB_INT

USE WRRRN_INT

! Declare variables

INTEGER LDA, LDFACT, N, NLCA, NUCA

PARAMETER (LDA=3, LDFACT=4, N=4, NLCA=1, NUCA=1)

INTEGER IPVT(N)

REAL A(LDA,N), AINV(N,N), FACT(LDFACT,N), RJ(N)

! Set values for A in band form

! A = ( 0.0 -1.0 -2.0 2.0)

! ( 2.0 1.0 -1.0 1.0)

! ( -3.0 0.0 2.0 0.0)

DATA A/0.0, 2.0, -3.0, -1.0, 1.0, 0.0, -2.0, -1.0, 2.0,&

2.0, 1.0, 0.0/

CALL LFTRB (A, NLCA, NUCA, FACT, IPVT)

! Set up the columns of the identity

! matrix one at a time in RJ

RJ = 0.0E0

DO 10 J=1, N

RJ(J) = 1.0E0

! RJ is the J-th column of the identity

! matrix so the following LFSRB

! reference places the J-th column of

! the inverse of A in the J-th column

! of AINV

CALL LFSRB (FACT, NLCA, NUCA, IPVT, RJ, AINV(:,J))

RJ(J) = 0.0E0

10 CONTINUE

! Print results

CALL WRRRN (’AINV’, AINV)

END

Output

AINV

1 2 3 4

1 -1.000 -1.000 0.400 -0.800

2 -3.000 -2.000 0.800 -1.600

3 0.000 0.000 -0.200 0.400

4 0.000 0.000 0.400 0.200