POWER_METHOD
The Power Method for Eigenvalues and Eigenvectors

POWER_METHOD is a C++ library which carries out the power method.

The power method seeks to determine the eigenvalue of maximum modulus, and a corresponding eigenvector.

The basic power method will not perform as expected if, corresponding to the maximum modulus, there are complex eigenvalues, or a pair of real eigenvalues of opposite sign. The power method's behavior can break down or be very slow initially if the starting vector has a zero or very small component in the eigenspace corresponding to the maximal eigenvalue.

A second version of the power method is included which can handle the case of complex eigenvalues.

Licensing:

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

Languages:

POWER_METHOD 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:

ARPACK, a FORTRAN90 library which computes eigenvalues for large matrices;

EISPACK, a FORTRAN77 library which carries out eigenvalue computations; superseded by LAPACK;

TEST_EIGEN, a C++ library which implements test matrices for eigenvalue analysis.

TEST_MAT, a C++ library which defines test matrices.

TOMS343, a FORTRAN77 library which computes the eigenvalues and eigenvectors of a general real matrix;
this is a FORTRAN77 version of ACM TOMS algorithm 343.

TOMS384, a FORTRAN77 library which computes the eigenvalues and eigenvectors of a symmetric matrix;
this is a FORTRAN77 version of ACM TOMS algorithm 384.

Reference:

1. Richard Burden, Douglas Faires,
   Numerical Analysis,
   Thomson Brooks/Cole, 2004,
   ISBN13: 978-0534392000,
   LC: QA297.B84.
2. Ward Cheney, David Kincaid,
   Numerical Mathematics and Computing,
   Brooks-Cole Publishing, 2004,
   ISBN: 0534201121,
   LC: QA297.C426.
3. Gene Golub, Charles VanLoan,
   Matrix Computations, Third Edition,
   Johns Hopkins, 1996,
   ISBN: 0-8018-4513-X,
   LC: QA188.G65.
4. Eric VanDeVelde,
   Concurrent Scientific Programming,
   Springer, 1994,
   ISBN: 0-387-94195-9,
   LC: QA76.58.V35.

Source Code:

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

Examples and Tests:

• power_method_prb.cpp, the tests;
• power_method_prb.sh, commands to compile and run the sample program.
• power_method_prb_output.txt, the output file.

List of Routines:

• CPU_TIME reports the elapsed CPU time.
• FIBONACCI2 returns the FIBONACCI2 matrix.
• POWER_METHOD applies several steps of the power method.
• POWER_METHOD2 applies the power method for possibly complex eigenvalues.
• R8_ABS returns the absolute value of an R8.
• R8_EPSILON returns the R8 roundoff unit.
• R8MAT_MV multiplies a matrix times a vector.
• R8VEC_COPY copies an R8VEC.
• R8VEC_DIVIDE divides an R8VEC by a nonzero scalar.
• R8VEC_DOT computes the dot product of a pair of R8VEC's.
• R8VEC_NORM_L2 returns the L2 norm of an R8VEC.
• R8VEC_UNIFORM_01 returns a unit pseudorandom R8VEC.
• TIMESTAMP prints the current YMDHMS date as a time stamp.
• TRIS returns the TRIS matrix.
• TRIS_EIGENVALUES returns the eigenvalues of the TRIS matrix.

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