# CYCLIC_REDUCTION A Direct Solution Method for Tridiagonal Linear Systems.

CYCLIC_REDUCTION is a MATLAB library which applies the cyclic reduction method to solve a tridiagonal system of linear equations A*x=b.

The matrix is assumed to be diagonally dominant - that is, for every row, we require that the magnitude of the diagonal entry be at least as great as the sum of the magnitudes of the two off-diagonal elements. This is (just barely) true for the "-1, 2, -1" matrix, for instance.

Other methods for solving tridiagonal linear systems include:

• Gauss elimination with pivoting;
• the Thomas algorithm, (Gauss elimination without pivoting);
• the Jacobi, Gauss-Seidel, and SOR iterative methods;

Cyclic reduction is a form of Gauss elimination. It proceeds by first eliminating half of the variables simultaneously, then half of the remainder, and so on. This amounts to more work, but the work in each elimination step can be done in parallel. Thus, unlike the Gauss and Thomas algorithms, cyclic reduction offers a procedure for the direct solution of a tridiagonal linear system that can take advantage of parallelism.

Cyclic reduction can also be adapted to the block tridiagonal linear systems that arise when Poisson's equation is discretized over a 2D region.

### Languages:

CYCLIC_REDUCTION is available in a C version and a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version.

### Related Data and Programs:

CSPARSE, a C library which carries out the direct solution of sparse linear systems.

DLAP, a FORTRAN90 library which carries out the iterative solution of sparse linear systems.

LAPACK_EXAMPLES, a FORTRAN90 program which demonstrates the use of the LAPACK linear algebra library.

LINPACK, a FORTRAN90 library which factors and solves systems of linear equations in a variety of formats and arithmetic types.

LINPLUS, a MATLAB library which carries out simple manipulations of matrices in a variety of formats.

MGMRES, a MATLAB library which applies the restarted GMRES algorithm to solve a sparse linear system.

SPARSEKIT, a FORTRAN90 library which carries out operations on sparse matrices, including conversion between various formats.

SUPER_LU, a C library which implements some very fast direct solvers for systems of sparse linear equations.

TEST_MAT, a MATLAB library which defines test matrices, some of which have known determinants, eigenvalues and eigenvectors, inverses and so on.

### Reference:

1. Gene Golub, Charles VanLoan,
Matrix Computations,
Third Edition,
Johns Hopkins, 1996,
ISBN: 0-8018-4513-X,
LC: QA188.G65.
2. Roger Hockney,
A fast direct solution of Poisson's equation using Fourier Analysis,
Journal of the ACM,
Volume 12, Number 1, pages 95-113, January 1965.

### Examples and Tests:

You can go up one level to the MATLAB source codes.

Last revised on 11 May 2010.