readme.txt

Mathematical Methods by Moor n Stiling.

% thetac = matrix of c values 
% thetas = matrix of s values (these must be converted to Q)


qrhouse.m
% 
% Compute the QR factorization of a matrix A without column pivoting
% using Householder transformations
% 
% function [V,R] = qrhouse(A)
%
% 
% A = mxn matrix (assumed to have full column rank)
%
% V = [v1 v2 ... vn] = matrix of Householder vectors 
%           (must be converted to Q ) using qrmakeq.
%           
% R = upper triangular matrix


qrmakeq.m
% 
% Convert the V matrix returned by qrhouse into a Q matrix
% 
% function Q = qrmakeq(V)
%
% V = [v1 v2 .... vr], Householder vectors produced by qrhouse
%
% Q = [Q1 Q2 ... Qr], the orthogonal matrix in the QR factorization


qrmakeqgiv.m
% 
% Given thetac and thetas containing rotation parameters from Givens rotations,
% (produced using qrqrgivens), compute Q
% function Q = qrmakeqgiv(thetac,thetas)
% 
% thetac = cosine component of Givens rotation
% thetas = sin component of Givens rotation
%
% Q = orthogonal matrix formed by Givens rotations


qrqtb.m
% 
% Given a matrix V containing Householder vectors as columns
% (produced using qrhouse), compute Q^H B, where Q is formed (implicitly)
% from the Householder vectors in V.
%
% function [B] = qrqtb(B,V)
% 
% B = matrix to be operated on
% V = matrix of Household vectors (as columns)
%
% output: B = Q^H B


qrqtbgiv.m
% 
% Given thetac and thetas containing rotation parameters from Givens rotations,
% (produced using qrqrgivens), compute Q^H B, where Q is formed (implicitly)
% from the rotations in Theta.
%
% function [B] = qrqtbgiv(B,thetac,thetas)
%
% B = matrix to be opererated on
% thetac = cosine component of rotations from Givens rotations
% thetas = sine component of rotations from Givens rotations
%
% Output: B <-- Q^H B


qrtheta.m
% 
% Given x and y, compute cos(theta) and sin(theta) for a Givens rotation
%
% function [c,s] = qrtheta(x,y)
%
% (x, y) = point to determine rotation
%
% c = cos(theta),   s=sin(theta)


reducefree.m
% 
% Perform elimination on the free variables in a linear programming problem
%
% function [A,b,c,value,savefree,nfree] = reducefree(A,b,c,freevars)
% 
% A,b,c = parameters from linear programming problem
% freevars = list of free variables
%
% A,b,c = new parameters for linear programming problem (with free variables
%         eliminated)
% value = value of linear program
% savefree = tableau information for restoring free variables
% nfree = number of free variables found


refltodir.m
% 
% Convert from a set of reflection coefficients kappa(1)...kappa(m)
% to FIR filter coefficients in direct form
%
% function a = refltodir(kappa)
%
% kappa = vector of reflection coefficients
%
% a = output filter coefficients = [1 a(1) a(2) ... a(m)]


restorefree.m
% 
% Restore the free variables by back substitution
%
% function x = restorefree(inx,savefree,freevars)
% 
% inx = linear programming solution (without free variables)
% savefree = information from tableau for substitution
% freevars = list of free variables
% 
% x = linear programming solution (including free variables)


rls.m
%
% Given a scalar input signal x and a desired scalar signal d,
% compute an RLS update of the weight vector h.
%
% This function must be initialized by rlsinit
%
% function [h,eap] = rls(x,d)
%
% x = input signal
% d = desired signal
%
% h = updated filter weight vector
% eap = (optional) a-priori estimation error


rlsinit.m
%
% Initialize the RLS filter
%
% function rlsinit(m,delta)


simplex1.m
% 
% Find the solution of a linear programming problem in standard form
%  minimize c'x
%  subject to Ax=b
%             x >= 0
%
% function [x,value,w] = simplex1(A,b,c,freevars)
% 
% A, b, c: system problem
% freevars = (optional) list of free variables in problem
%
% x = solution
% value = value of solution
% w = (optiona)l solution of the dual problem.
%    If w is used as a return value, then the dual problem is also solved.
%    (In this implementation, the dual problem cannot be solved when free
%    variables are employed.)


sor.m
% 
% Produce an updated solution x to Ax = b successive over-relaxation
% A must be Hermitian positive definite
%
% function x = sor(A,x,b)
%
% A = input matrix
% x = initial solution
% b = right-hand side
% omega = relaxation parameter
% 
% Output x= updated solution


sysidsvd.m
% 
% given a sequence of impulse responses in h
% identify a system (A,B,C)
%
% function [A,B,C] = sysidsvd(h,tol)
% h = impulse responses (a cell array)
% tol = (optional) tolerance used in rank determination
% ord = system order (overrides tolerance)
%
% (A,B,C) = estimated system matrix parameters


testet.m
% testet.m
% Test the emission tomography code
% This script loads an image file, plots it, then calls the 
% code to test the tomographic reconstruction


testirwls.m
% Test the irwls routine for a filter design problem
% After [Burrus 1994, p. 2934]


testnn10.m
% Test the neural network on a pattern recognition problem
%


tls.m
% 
% determine the total least-squares solution of Ax=b
%
% function x = tls(A,b)
%
% A = system matrix
% b = right-hand side
%
% x = TLS solution to Ax=b


tohankel.m
% 
% Determine the matrix nearest to A which is Hankel and has rank r
% using the composite mapping algorithm
%
% function A = tohankel(A,r)
%
% A = input/output matrix
% r = desired rank
%
% Ouptut A = Hankel matrix with desired properties
% d = norm of difference between matrices


tohanktoep.m
% 
% Determine the matrix nearest to A which is the stack
%  [ Hankel
%    Toeplitz ] 
% with rank r using the composite mapping algorithm
%
% function A = tohantoep(A,r)
%
% A = input matrix
% r = desired rank
%
% output: A = matrix with desired properties
% normlist = (optional) vector of errors


tokarmarker.m
% 
% Given a linear programming problem in standard form
%  min       c'x
% subject to Ax = b,  x >= 0
%
% return new values of A and c in "Karmarker standard form"
%  min       c'x
% subject to Ax=0,  sum(x)=n, x >= 0
%
% function [Anew,cnew] = tokarmarker(A,b,c)
%
% (A,b,c) = matrices in standard form
%
% Anew, cnew = matrices in Karmarker standard form


tostoch.m
% 
% Determine the matrix nearest to A which is stochastic using
% the composite mapping algorithm
%
% function A = tostoch(A)
% A = input matrix
%
% Output: A = nearest stochastic A


tridiag.m
% 
% tridiagonalize the real symmetric matrix A
% 
% function [T,Q] = tridiag(A)
%
% A = matrix whose tridiagonal form is sought
%
% T = tridiagonal matrix
% Q = (optional) orthogonal transformation


vitbestcost.m
% 
% Returns the best cost so far in the Viterbi algorithm
%
% function c = vitbestcost


viterbi1.m
% 
% Run the Viterbi algorithm on the input for one branch
% Before calling this function, you must call initvit1
%
% function [p] = viterbi1(r)
%
% r = input value (scalar or vector)
%
% p = 0 if number of branches in window is not enough
% p = statenum on path if enough branches in window
%
% Call vitflush to clear out path when finished


vitflush.m
% 
% Flush out the rest of the viterbi paths
%
% function [plist] = vitflush(termnode)
%
% termnode = 0 or list of allowable terminal nodes
%    If termnode==0, then choose path with lowest cost
%    Otherwise, choose path with best cost among termnode
%
% plist = list of paths from trellis


warp.m
% 
% find the dynamic warping between A and B (which may not be of the
% same length)
%
% function [path] = warp(A,B)
%
% A = cells of the vectors, A{1}, A{2}, ..., A{M}
% B = cells of the vectors, B{1}, B{2}, ..., B{N}
%
% path = Kx2 array of (i,j) correspondence


warshall.m
% 
% Find the transitive closure of the graph represented by the adjacency
% matrix A
%
% function Anew = warshall(A)
%
% A = adjacency matrix of graph
% 
% Anew = adjacency matrix for transitive closure of graph


wavecoeff.m
% Coefficients for Daubechies wavelets
%


wavetest.m
% Test the wavelet transform in matrix notation


wavetesto.m
% Test the wavelet transform in wavelet notation (alternate indexing)


wavetrans.m
%
% Compute the (nonperiodized) discrete wavelet traqnsform 
%
% function [C,ap] = wavetrans(c,coeff,J)
%
% c = data to be transformed
% coeff = wavelet transform coefficients
% J = number of levels of transform
%    If J is specified, then J levels of the transform are computed.  
%    Otherwise, the largest possible number of levels are used.
%
% C = transformed data
%     The output is stacked in C in wavelet-coefficient first order,
%     C = [d1 d2 ... dJ cJ]
% ap indexes the start of the coefficients for the jth level,
%     ap(j+1) indexes the start of the coefficients for the jth level,
%     except that ap(1) indicates the number of original datapoints
%
% This function simply stacks up the data for a call to the function wave


wavetransper.m
%
% Compute the periodized discrete wavelet transform of the data
%
% function [C] = wavetransper(c,coeff,J)
%
% c = data to be transformed
% coeff = transform coefficients
% J indicates how many levels of the transform to compute.
%    If length(c) is not a power of 2, J must be specified.
%
% C = output vector
%    The output is stacked in C in wavelet-coefficient first order,
%    C = [d1 d2 ... dJ cJ]


wftest.m
% Test the Wiener filter equalizer for a first-order signal and first-order
% channel


zerorow.m
% 
% Zero a row by a series of Givens rotations
%
% function [B,U] = zerorow(B,f,U)
%
% B = matrix to have row zeroed
% f = vector of row indices that are zero on the diagonal
% U = (optional) rotation matrix
%
% B = modified matrix
% U = (optional) rotation matrix


***************************************************************
Directory: misc
***************************************************************
H.m
% 
% Compute the binary entropy function
% 
% function h = H(p)
%
% p = crossover probability
%
% h = binary entropy


b2n.m
% convert an m-bit binary sequence b to an integer


bernapprox.m
% Plot Bernstein polynomials


bernpoly.m
% 
% compute the Bernstein polynomial g_{nk}(t)
%
% function g = bernpoly(n,k,t)
%
% n = degree
% k = order
% t = location
% 
% g = value


bsplineval1.m
% 
% When f(x) = sum_i c_i^k B_i^k(x)
% where B_i^k is the kth order spline across t_i,t_{i+1},\ldots,t_{i+k+1}
% This function evaluates f(x) for a given x.
% (See Kincaid-Cheney, p. 396)
%
% function s = bsplineval1(c,t,x,k)
%
% c = set of coefficients
% t = knot set
% x = value.  x must fall in range of knots, t_i <= x < t_{i+1}
% k = order
%
% s = spline value


chebinterp.m
% data for  Chebyshev interpolation example


chi2.m
% 
% Compute the pdf for a chi-squared random variable
%
% function f = chi2(x,n)
%
% x = value
% n = order of chi-squared
%
% f = pdf value


compmap1.m
% Test the composite mapping algorithm on the positive sequence


compmap4.m
% test the positive semi-definite mapping


computeremez.m
% Test the remez algorithm


crtgamma.m
% 
% Compute the gammas for the CRT
%
% function gamma = crtgamma(m)
%
% m = set of modulos
%
% gamm = set of gammas


crtgammapoly.m
% 
% Compute the gammas for the CRT for polynomials
%
% function gamma = crtgammapoly(m)
%
% m = set of modulos (polynomials)
%
% gamma = set of gammas (polynomials)


crypttest.m
% test the cryptographic example


d2b.m
% convert n to an m-bit binary representation


diagstack.m
% Stack matrices diagonally:
% D = [X 0
%      0 Y];
%
% function D = diagstack(X,Y)
% X, Y = input matrices
%
% D = diagonal stack


discapprox.m
% find a discrete approximating polynomial


divdiff.m
% 
% Compute the upper row of a divided difference table
%
% function c = divdiff(ts,fs)
%
% ts = sample locations
% fs = function values
%
% c = set of divided differences


eigfilcon0.m
%
% Find eigenilter constrained so that response is 0 at some frequencies
%
% function h = eigfilcon(wp,ws,N,alpha, Wo)
%
% wp = passband frequency
% ws = stopband frequency
% m = number of coefficients (must be ODD)
% alpha = tradeoff parameter between stopband and passband
% Wo = list of constraint frequencies at which response is 0
%
% h = filter coefficients


eigfilcon0new.m
%
% Find eigenilter constrained so that response is 0 at some frequencies
%
% function h = eigfilcon(wp,ws,N,alpha, Wo)
%
% wp = passband frequency
% ws = stopband frequency
% m = number of coefficients (must be ODD)
% alpha = tradeoff parameter between stopband and passband
% Wo = list of constraint frequencies at which response is 0
%
% h = filter coefficients


elem.m
%
% Return an elementary matrix E_{rs} of size mxn


fact.m
% compute the factorial


findprim5.m
% Find a primitive polynomial in GF(5)


fromcrt.m
% 
% Given a sequence [y1,y2,\ldots,y2] that is a representation 
% of an integer x in CRT form, convert back to x.
%
% function x = fromcrt(y,m)