Some MATLAB Functions

to accompany The Matrix Eigenvalue Problem: GR and Krylov Subspace Methods,

by David S. Watkins, professor of Mathematics at Washington State University .

Important Notice: These codes are for instructional and recreational use only. They were written with an eye to simplicity, so they lack certain safeguards that a foolproof code would possess. They have not been optimized for speed, nor are they guaranteed to be error free. If you need efficient, well-tested codes for your applications, use the built-in functions provided in MATLAB, or see LAPACK, ARPACK, HAPACK, etc.

Chapter 3

showmatrix.m makes a color plot of a matrix. This is used by many of the programs in this collection.
reflector.m makes a reflector such that Qx = alpha e_1 or Qx = alpha e_n.
hessgo.m is a simple driver program for downhess.m (or uphess.m).
downhess.m does a unitary similarity transformation to upper Hessenberg form from top to bottom.
uphess.m does a unitary similarity transformation to upper Hessenberg form from bottom to top.

tridiago.m is a simple driver program for pseudotridiag.m.
pseudotridiag.m transforms a pseudosymmetric matrix to tridiagonal form.
hyperbolic.m computes a partly hyperbolic matrix such that Hx = alpha e_{1}. (used by pseudotridiag.m)
hyp_apply.m applies the matrix computed by hyperbolic.m. (used by pseudotridiag.m)

Chapter 4

primitiveqr.m does basic unshifted QR iterations and displays them graphically.
dqd.m does one iteration of the unshifted differential qd algorithm.

qrgo.m is a simple driver program for downchase.m
downchase.m does one iteration of the implicit QR algorithm of arbitrary degree.
rqgo.m is a simple driver program for upchase.m.
upchase.m does one iteration of the implicit RQ algorithm of arbitrary degree.
qrexact.m is a simple demonstration of the effect of exact shifts.
downchase2.m is a variant of downchase.m that is used by qrexact.m.

hrgo.m is a simple driver program for hrchase.m
hrchase.m does one iteration of the implicit HR algorithm of arbitrary degree.

Chapter 7

blurgo.m is a simple driver program for blurtest.m.
blurtest.m does an iteration of the QR algorithm of degree m and checks for shift blurring.

swapgo.m is a driver program for bulgecrash.m.
bulgecrash.m does a QR and an RQ iteration simultaneously by passing bulges through each other.
swap.m is a bulge-swapping routine used by bulgecrash.m.

Chapter 9

arngo.m is a driver program for arnoldi.m.
arnoldi.m is the simple Arnoldi process with reorthogonalization.
arngo2.m is a driver program for arnoldi2.m
arnoldi2.m is the Arnoldi process with a shift-and-invert option.
afun.m does a matrix-vector multiply (possibly shift-and-invert)

symlango.m is a driver program for symlan.m.
symlan.m is the symmetric Lanczos process with shift-and-invert option and reorthogonalization.
relango.m is a driver program for relan.m
relan.m is the symmetric Lanczos process with shift-and-invert option and thick restarts. (second restart method)
afun.m does a matrix-vector multiply (possibly shift-and-invert)

hamlango.m is a driver program for hamlan.m.
hamlan.m is the Hamiltonian Lanczos process.
hfun.m multiplies a Hamiltonian matrix times a vector.
symplango.m is a driver program for symplan.m.
symplan.m is the symplectic Lanczos process.
sfun.m multiplies a symplectic matrix (Cayley transform of a Hamiltonian matrix) times a vector.
normalize.m symplectic normalization used by hamlan.m and symplan.m.

rehamlango.m is a driver program for rehamlan.m
rehamlan.m does Hamiltonian or symplectic Lanczos process with thick restarts.
hsort.m is an auxiliary sorting routine used by rehamlan.m.
hzeig.m is the double-shift HR (HZ) algorithm (used by rehamlan.m).
hz.m does one double-shift HR (HZ) iteration (used by hzeig.m).
transetup.m is an auxiliary routine used by hz.m.
transdown.m is an auxiliary routine used by hz.m.

blocklango.m is a driver program for blocklan.m.
blocklan.m is the block symmetric Lanczos process with reorthogonalization.
afun.m does a matrix-vector multiply (possibly shift-and-invert)