Block Lanczos Tridiagonalization

Block Lanczos Tridiagonalization of
Complex Symmetric Matrices Matlab Package

1. Introduction

For any symmetric matrix A, there exists a special form of SVD, called Takagi factorization:

where Q is unitary and is the diagonal singular value matrix.

This package tridiagonalizes a complex symmetric matrix using block Lanczos algorithm. It is more efficient than the vector version. It can be followed by the divide-and-conquer method or the QR method for the symmetric SVD of a complex symmetric tridiagonal matrix to compute the Takagi factorization or symmetric SVD of a complex symmetric matrix.

2. Dependency

                            compare                            csgen
            ___________ _______|_______ ___________              |
           |           |               |           |         unitarand
         LanMPO    BlkLanCom        testinfo     errchk          |
                   BlkLanNorm                                  house
                       |
                    LanTri           
                       |        
                    sbmvmul

3. Functions

BlkLanCom.m Block Lanczos tridiagonalization of a complex-symmetric matrix using the componentwise orthogonalization.

BlkLanNorm.m Block Lanczos tridiagonalization of a complex-symmetric matrix using the normwise orthogonalization.

compare.m Compare the performance between componentwise and normwise orthogonalization scheme.

csgen.m Generate a random complex symmetric matrix with specified singular values.

errchk.m Check the errors of orthogonality and factorization of the block Lanczos algorithm.

house.m Householder transformation given a vector.

LanMPO.m Lanczos tridiagonalization of a complex-symmetric matrix using the modified partial orthogonalization.

LanTri.m Lanczos tridiagonalization of a complex-symmetric and block tridiagonal matrix.

sbmvmul.m Complex-symmetric and block tridiagonal matrix-vector multiplication.

testinfo.m Collect experiment results into a file.

unitarand.m Generate a random unitary matrix.

4. References

Sanzheng Qiao, Guohong Liu, and Wei Xu. in Advanced Signal Processing Algorithms, Architectures, and Implementations XV, edited by Franklin T. Luk, Proc. of SPIE Vol. 5910, 591010 (2005). (ps, pdf)

5. Download

Download whole package: package.zip

Download source code of functions:

BlkLanCom.m BlkLanNorm.m compare.m csgen.m
errchk.m house.m LanMPO.m LanTri.m
sbmvmul.m testinfo.m unitarand.m

6. Contact

Sanzheng Qiao

Professor, Department of Computing and Software
McMaster University, Hamilton, Ontario L8S 4K1, Canada
(905)-5259140 ext 27234

Developed by Guohong Liu. Last updated on May, 2005

BlkLanCom.m	Block Lanczos tridiagonalization of a complex-symmetric matrix using the componentwise orthogonalization.
BlkLanNorm.m	Block Lanczos tridiagonalization of a complex-symmetric matrix using the normwise orthogonalization.
compare.m	Compare the performance between componentwise and normwise orthogonalization scheme.
csgen.m	Generate a random complex symmetric matrix with specified singular values.
errchk.m	Check the errors of orthogonality and factorization of the block Lanczos algorithm.
house.m	Householder transformation given a vector.
LanMPO.m	Lanczos tridiagonalization of a complex-symmetric matrix using the modified partial orthogonalization.
LanTri.m	Lanczos tridiagonalization of a complex-symmetric and block tridiagonal matrix.
sbmvmul.m	Complex-symmetric and block tridiagonal matrix-vector multiplication.
testinfo.m	Collect experiment results into a file.
unitarand.m	Generate a random unitary matrix.