LSLTO

Solves a complex sparse Hermitian positive definite system of linear equations, given the Cholesky factorization of the coefficient matrix.

Required Arguments

A — Real vector of length 2N − 1 containing the first row of the coefficient matrix followed by its first column beginning with the second element. (Input)
See Comment 2.

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

X — Real vector of length N containing the solution of the linear system. (Output)
If B is not needed then B and X may share the same storage locations.

Optional Arguments

N — Order of the matrix represented by A. (Input)
Default: N = (size (A,1) + 1)/2

IPATH — Integer flag. (Input)
IPATH = 1 means the system Ax = B is solved.
IPATH = 2 means the system AT x = B is solved.
Default: IPATH =1.

FORTRAN 90 Interface

Generic: CALL LSLTO (A, B, X [, …])

Specific: The specific interface names are S_LSLTO and D_LSLTO.

FORTRAN 77 Interface

Single: CALL LSLTO (N, A, B, IPATH, X)

Double: The double precision name is DLSLTO.

Description

Toeplitz matrices have entries that are constant along each diagonal, for example,

 

The routine LSLTO is based on the routine TSLS in the TOEPLITZ package, see Arushanian et al. (1983). It is based on an algorithm of Trench (1964). This algorithm is also described by Golub and van Loan (1983), pages 125−133.

Comments

1. Workspace may be explicitly provided, if desired, by use of L2LTO/DL2LTO. The reference is:

CALL L2LTO (N, A, B, IPATH, X, WK)

The additional argument is:

WK — Work vector of length 2N − 2.

2. Because of the special structure of Toeplitz matrices, the first row and the first column of a Toeplitz matrix completely characterize the matrix. Hence, only the elements
A(1, 1), …, A(1, N), A(2, 1), …, A(N, 1) need to be stored.

Example

A system of four linear equations is solved. Note that only the first row and column of the matrix A are entered.

 

USE LSLTO_INT

USE WRRRN_INT

! Declare variables

INTEGER N

PARAMETER (N=4)

REAL A(2*N-1), B(N), X(N)

! Set values for A, and B

!

! A = ( 2 -3 -1 6 )

! ( 1 2 -3 -1 )

! ( 4 1 2 -3 )

! ( 3 4 1 2 )

!

! B = ( 16 -29 -7 5 )

!

DATA A/2.0, -3.0, -1.0, 6.0, 1.0, 4.0, 3.0/

DATA B/16.0, -29.0, -7.0, 5.0/

! Solve AX = B

CALL LSLTO (A, B, X)

! Print results

CALL WRRRN (’X’, X, 1, N, 1)

END

Output

 

X

1 2 3 4

-2.000 -1.000 7.000 4.000