CYCLIC_REDUCTION
A Direct Solution Method for Tridiagonal Linear Systems.

CYCLIC_REDUCTION is a C++ 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.

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

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 FORTRAN77 program which demonstrates the use of the LAPACK linear algebra library.

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

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

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

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

TEST_MAT, a C++ 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.

Source Code:

• cyclic_reduction.cpp, the source code.
• cyclic_reduction.hpp, the include file.
• cyclic_reduction.sh, commands to compile the source code.

Examples and Tests:

• cyclic_reduction_prb.cpp, a sample calling program.
• cyclic_reduction_prb.sh, commands to compile and run the sample program.
• cyclic_reduction_prb_output.txt, the output file.

List of Routines:

• I4_MAX returns the maximum of two I4's.
• I4_MIN returns the minimum of two I4's.
• R83_CR_FA decomposes a real tridiagonal matrix using cyclic reduction.
• R83_CR_SL solves a real linear system factored by R83_CR_FA.
• R83_GS_SL solves a R83 system using Gauss-Seidel iteration.
• R83_MXV_NEW multiplies a R83 matrix times a vector.
• R83_PRINT prints a R83 matrix.
• R83_PRINT_SOME prints some of a R83 matrix.
• R8MAT_PRINT prints an R8MAT.
• R8MAT_PRINT_SOME prints some of an R8MAT.
• R8VEC_INDICATOR_NEW sets an R8VEC to the indicator vector {1,2,3...}.
• R8VEC_PRINT prints an R8VEC.
• R8VEC_PRINT_SOME prints "some" of an R8VEC.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

You can go up one level to the C++ source codes.