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annotate toolbox/pnorm.m @ 2:c124219d7bfa draft

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Re-add the 1995 toolbox after noticing the statement in the ~higham/mctoolbox/ webpage.
author Antonio Pino Robles <data.script93@gmail.com>
date Thu, 07 May 2015 18:36:24 +0200
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1 function [est, x, k] = pnorm(A, p, tol, noprint)
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2 %PNORM Estimate of matrix p-norm (1 <= p <= inf).
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3 % [EST, x, k] = PNORM(A, p, TOL) estimates the Holder p-norm of a
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4 % matrix A, using the p-norm power method with a specially
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5 % chosen starting vector.
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6 % TOL is a relative convergence tolerance (default 1E-4).
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7 % Returned are the norm estimate EST (which is a lower bound for the
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8 % exact p-norm), the corresponding approximate maximizing vector x,
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9 % and the number of power method iterations k.
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10 % A nonzero fourth argument causes trace output to the screen.
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11 % If A is a vector, this routine simply returns NORM(A, p).
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12 %
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13 % See also NORM, NORMEST.
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14
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15 % Note: The estimate is exact for p = 1, but is not always exact for
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16 % p = 2 or p = inf. Code could be added to treat p = 2 and p = inf
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17 % separately.
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18 %
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19 % Calls DUAL and SEQA.
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20 %
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21 % Reference:
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22 % N.J. Higham, Estimating the matrix p-norm,
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23 % Numer. Math., 62 (1992), pp. 539-555.
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24
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25 if nargin < 2, error('Must specify norm via second parameter.'), end
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26 [m,n] = size(A);
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27 if min(m,n) == 1, est = norm(A,p); return, end
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28
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29 if nargin < 4, noprint = 0; end
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30 if nargin < 3, tol = 1e-4; end
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31
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32 % Stage I. Use Algorithm OSE to get starting vector x for power method.
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33 % Form y = B*x, at each stage choosing x(k) = c and scaling previous
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34 % x(k+1:n) by s, where norm([c s],p)=1.
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35
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36 sm = 9; % Number of samples.
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37 y = zeros(m,1); x = zeros(n,1);
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38
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39 for k=1:n
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40
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41 if k == 1
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42 c = 1; s = 0;
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43 else
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44 W = [A(:,k) y];
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45
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46 if p == 2 % Special case. Solve exactly for 2-norm.
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47 [U,S,V] = svd(full(W));
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48 c = V(1,1); s = V(2,1);
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49
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50 else
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51
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52 fopt = 0;
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53 for th=seqa(0,pi,sm)
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54 c1 = cos(th); s1 = sin(th);
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55 nrm = norm([c1 s1],p);
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56 c1 = c1/nrm; s1 = s1/nrm; % [c1 s1] has unit p-norm.
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57 f = norm( W*[c1 s1]', p );
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58 if f > fopt
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59 fopt = f;
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60 c = c1; s = s1;
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61 end
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62 end
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63
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64 end
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65 end
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66
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67 x(k) = c;
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68 y = x(k)*A(:,k) + s*y;
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69 if k > 1, x(1:k-1) = s*x(1:k-1); end
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70
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71 end
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72
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73 est = norm(y,p);
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74 if noprint, fprintf('Alg OSE: %9.4e\n', est), end
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75
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76 % Stage II. Apply Algorithm PM (the power method).
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77
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78 q = dual(p);
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79 k = 1;
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80
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81 while 1
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82
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83 y = A*x;
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84 est_old = est;
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85 est = norm(y,p);
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86
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87 z = A' * dual(y,p);
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88
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89 if noprint
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90 fprintf('%2.0f: norm(y) = %9.4e, norm(z) = %9.4e', ...
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91 k, norm(y,p), norm(z,q))
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92 fprintf(' rel_incr(est) = %9.4e\n', (est-est_old)/est)
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93 end
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94
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95 if ( norm(z,q) <= z'*x | abs(est-est_old)/est <= tol ) & k > 1
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96 return
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97 end
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98
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99 x = dual(z,q);
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100 k = k + 1;
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101
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102 end