LSLQB

Computes the RH DR Cholesky factorization of a complex Hermitian positive-definite matrix A in codiagonal band Hermitian storage mode. Solve a system Ax = b.

Required Arguments

A — Array containing the N by N positive-definite band coefficient matrix and the right hand side in codiagonal band Hermitian storage mode. (Input/Output)

The number of array columns must be at least 2 * NCODA + 3. The number of columns is not an input to this subprogram.

The number of array columns must be at least 2 * NCODA + 3. The number of columns is not an input to this subprogram.

NCODA — Number of upper codiagonals of matrix A. (Input)

Must satisfy NCODA ≥ 0 and NCODA < N.

Must satisfy NCODA ≥ 0 and NCODA < N.

U — Array of flags that indicate any singularities of A, namely loss of positive-definiteness of a leading minor. (Output)

A value U(I) = 0. means that the leading minor of dimension I is not positive-definite. Otherwise, U(I) = 1.

A value U(I) = 0. means that the leading minor of dimension I is not positive-definite. Otherwise, U(I) = 1.

Optional Arguments

N — Order of the matrix. (Input)

Must satisfy N > 0.

Default: N = size (A,2).

Must satisfy N > 0.

Default: N = size (A,2).

LDA — Leading dimension of A exactly as specified in the dimension statement of the calling program. (Input)

Must satisfy LDA ≥ N + NCODA.

Default: LDA = size (A,1).

Must satisfy LDA ≥ N + NCODA.

Default: LDA = size (A,1).

IJOB — flag to direct the desired factorization or solving step. (Input)

Default: IJOB = 1.

Default: IJOB = 1.

IJOB Meaning

1 factor the matrix A and solve the system Ax = b; where the real part of b is stored in column 2 * NCODA + 2 and the imaginary part of b is stored in column 2 * NCODA + 3 of array A. The real and imaginary parts of b are overwritten by the real and imaginary parts of x.

2 solve step only. Use the real part of b as column 2 * NCODA + 2 and the imaginary part of b as column 2 * NCODA + 3 of A. (The factorization step has already been done.) The real and imaginary parts of b are overwritten by the real and imaginary parts of x.

3 factor the matrix A but do not solve a system.

4,5,6 same meaning as with the value IJOB ‑ 3. For efficiency, no error checking is done on values LDA, N, NCODA, and U(*).

FORTRAN 90 Interface

Generic: CALL LSLQB (A, NCODA, U [, …])

Specific: The specific interface names are S_LSLQB and D_LSLQB.

FORTRAN 77 Interface

Single: CALL LSLQB (N, A, LDA, NCODA, IJOB, U)

Double: The double precision name is DLSLQB.

Description

Routine LSLQB factors and solves the Hermitian positive definite banded linear system Ax = b. The matrix is factored so that A = RH DR, where R is unit upper triangular and D is diagonal and real. The reciprocals of the diagonal entries of D are computed and saved to make the solving step more efficient. Errors will occur if D has a nonpositive diagonal element. Such events occur only if A is very close to a singular matrix or is not positive definite.

LSLQB is efficient for problems with a small band width. The particular cases NCODA = 0, 1 are done with special loops within the code. These cases will give good performance. See Hanson (1989) for more on the algorithm. When solving tridiagonal systems, NCODA = 1, the cyclic reduction code LSLCQ should be considered as an alternative. The expectation is that LSLCQ will outperform LSLQB on vector or parallel computers. It may be inferior on scalar computers or even parallel computers with non-optimizing compilers.

Comments

1. Workspace may be explicitly provided, if desired, by use of L2LQB/DL2LQB The reference is:

CALL L2LQB (N, A, LDA, NCODA, IJOB, U, WK1, WK2)

The additional arguments are as follows:

WK1 — Work vector of length NCODA.

WK2 — Work vector of length NCODA.

2. Informational error

Type | Code | Description |
---|---|---|

4 | 2 | The input matrix is not positive definite. |

Example

A system of five linear equations is solved. The coefficient matrix has real positive definite codiagonal Hermitian band form and the right-hand-side vector b has five elements.

USE LSLQB_INT

USE WRRRN_INT

INTEGER LDA, N, NCODA

PARAMETER (N=5, NCODA=1, LDA=N+NCODA)

!

INTEGER I, IJOB, J

REAL A(LDA,2*NCODA+3), U(N)

!

! Set values for A and right hand side

! in codiagonal band Hermitian form:

!

! ( * * * * * )

! ( 2.0 * * 1.0 5.0)

! A = ( 4.0 -1.0 1.0 12.0 -6.0)

! (10.0 1.0 2.0 1.0 -16.0)

! ( 6.0 0.0 4.0 -3.0 -3.0)

! ( 9.0 1.0 1.0 25.0 16.0)

!

DATA ((A(I+NCODA,J),I=1,N),J=1,2*NCODA+3)/2.0, 4.0, 10.0, 6.0,&

9.0, 0.0, -1.0, 1.0, 0.0, 1.0, 0.0, 1.0, 2.0, 4.0, 1.0,&

1.0, 12.0, 1.0, -3.0, 25.0, 5.0, -6.0, -16.0, -3.0, 16.0/

!

! Factor and solve A*x = b.

!

IJOB = 1

CALL LSLQB (A, NCODA, U)

!

! Print results

!

CALL WRRRN (’REAL(X)’, A((NCODA+1):,(2*NCODA+2):), 1, N, 1)

CALL WRRRN (’IMAG(X)’, A((NCODA+1):,(2*NCODA+3):), 1, N, 1)

END

Output

REAL(X)

1 2 3 4 5

2.000 3.000 -1.000 0.000 3.000

IMAG(X)

1 2 3 4 5

1.000 0.000 -1.000 -2.000 2.000