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

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

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