LFTQS
Computes the RTR Cholesky factorization of a real symmetric positive definite matrix in band symmetric storage mode.
Required Arguments
A — NCODA + 1 by N array containing the N by N positive definite band coefficient matrix in band symmetric storage mode to be factored. (Input)
NCODA — Number of upper codiagonals of A. (Input)
FACT — NCODA + 1 by N array containing the RT R factorization of the matrix A. (Output)
If A s not needed, A and FACT can share the same storage locations.
Optional Arguments
N — Order of the matrix. (Input)
Default: N = size (A,2).
LDA — Leading dimension of A exactly as specified in the dimension statement of the calling program. (Input)
Default: LDA = size (A,1).
LDFACT — Leading dimension of FACT exactly as specified in the dimension statement of the calling program. (Input)
Default: LDFACT = size (FACT,1).
FORTRAN 90 Interface
Generic: CALL LFTQS (A, NCODA, FACT [, …])
Specific: The specific interface names are S_LFTQS and D_LFTQS.
FORTRAN 77 Interface
Single: CALL LFTQS (N, A, LDA, NCODA, FACT, LDFACT)
Double: The double precision name is DLFTQS.
Description
Routine LFTQS computes an RT R Cholesky factorization of a real symmetric positive definite band coefficient matrix. R is an upper triangular band matrix.
LFTQS fails if any submatrix of R is not positive definite or if R has a zero diagonal element. These errors occur only if A is very close to a singular matrix or to a matrix which is not positive definite.
The
RT R factors are returned in a form that is compatible with routines
LFIQS,
LFSQS and
LFDQS. To solve systems of equations with multiple right hand-side vectors, use
LFTQS followed by either
LFIQS or
LFSQS called once for each right-hand side. The routine
LFDQS can be called to compute the determinant of the coefficient matrix after
LFTQS has performed the factorization.
LFTQS is based on the LINPACK routine CPBFA; see Dongarra et al. (1979).
Comments
Informational error
Type | Code | Description |
|---|
4 | 2 | The input matrix is not positive definite. |
Example
The inverse of a 3
× 3 matrix is computed.
LFTQS is called to factor the matrix and to check for nonpositive definiteness.
LFSQS is called to determine the columns of the inverse.
USE LFTQS_INT
USE WRRRN_INT
USE LFSQS_INT
! Declare variables
INTEGER LDA, LDFACT, N, NCODA
PARAMETER (LDA=2, LDFACT=2, N=4, NCODA=1)
REAL A(LDA,N), AINV(N,N), FACT(LDFACT,N), RJ(N)
!
! Set values for A in band symmetric form
!
! A = ( 0.0 1.0 1.0 1.0 )
! ( 2.0 2.5 2.5 2.0 )
!
DATA A/0.0, 2.0, 1.0, 2.5, 1.0, 2.5, 1.0, 2.0/
! Factor the matrix A
CALL LFTQS (A, NCODA, FACT)
! 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 LFSQS
! reference places the J-th column of
! the inverse of A in the J-th column
! of AINV
CALL LFSQS (FACT, NCODA, RJ, AINV(:,J))
RJ(J) = 0.0E0
10 CONTINUE
! Print the results
CALL WRRRN (’AINV’, AINV, ITRING=1)
END
Output
AINV
1 2 3 4
1 0.6667 -0.3333 0.1667 -0.0833
2 0.6667 -0.3333 0.1667
3 0.6667 -0.3333
4 0.6667
