Mercurial > matrix-functions
annotate matrixcomp/pnorm.m @ 0:8f23314345f4 draft
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author | Antonio Pino Robles <data.script93@gmail.com> |
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date | Wed, 06 May 2015 14:56:53 +0200 |
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1 function [est, x, k] = pnorm(A, p, tol, prnt) |
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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 input 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, NORMEST1. |
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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. |
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20 % |
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21 % Reference: |
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22 % N. J. Higham, Estimating the matrix p-norm, Numer. Math., |
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23 % 62 (1992), pp. 539-555. |
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24 % N. J. Higham, Accuracy and Stability of Numerical Algorithms, |
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25 % Second edition, Society for Industrial and Applied Mathematics, |
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26 % Philadelphia, PA, 2002; sec. 15.2. |
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27 |
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28 if nargin < 2, error('Must specify norm via second parameter.'), end |
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29 [m,n] = size(A); |
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30 if min(m,n) == 1, est = norm(A,p); return, end |
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31 |
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32 if nargin < 4, prnt = 0; end |
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33 if nargin < 3 | isempty(tol), tol = 1e-4; end |
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34 |
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35 % Stage I. Use Algorithm OSE to get starting vector x for power method. |
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36 % Form y = B*x, at each stage choosing x(k) = c and scaling previous |
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37 % x(k+1:n) by s, where norm([c s],p)=1. |
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38 |
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39 sm = 9; % Number of samples. |
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40 y = zeros(m,1); x = zeros(n,1); |
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41 |
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42 for k=1:n |
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43 |
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44 if k == 1 |
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45 c = 1; s = 0; |
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46 else |
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47 W = [A(:,k) y]; |
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48 |
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49 if p == 2 % Special case. Solve exactly for 2-norm. |
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50 [U,S,V] = svd(full(W)); |
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51 c = V(1,1); s = V(2,1); |
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52 |
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53 else |
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54 |
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55 fopt = 0; |
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56 for th=linspace(0,pi,sm) |
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57 c1 = cos(th); s1 = sin(th); |
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58 nrm = norm([c1 s1],p); |
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59 c1 = c1/nrm; s1 = s1/nrm; % [c1 s1] has unit p-norm. |
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60 f = norm( W*[c1 s1]', p ); |
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61 if f > fopt |
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62 fopt = f; |
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63 c = c1; s = s1; |
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64 end |
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65 end |
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66 |
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67 end |
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68 end |
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69 |
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70 x(k) = c; |
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71 y = x(k)*A(:,k) + s*y; |
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72 if k > 1, x(1:k-1) = s*x(1:k-1); end |
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73 |
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74 end |
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75 |
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76 est = norm(y,p); |
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77 if prnt, fprintf('Alg OSE: %9.4e\n', est), end |
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78 |
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79 % Stage II. Apply Algorithm PM (the power method). |
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80 |
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81 q = dual(p); |
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82 k = 1; |
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83 |
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84 while 1 |
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85 |
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86 y = A*x; |
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87 est_old = est; |
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88 est = norm(y,p); |
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89 |
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90 z = A' * dual(y,p); |
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91 |
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92 if prnt |
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93 fprintf('%2.0f: norm(y) = %9.4e, norm(z) = %9.4e', ... |
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94 k, norm(y,p), norm(z,q)) |
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95 fprintf(' rel_incr(est) = %9.4e\n', (est-est_old)/est) |
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96 end |
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97 |
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98 if ( norm(z,q) <= z'*x | abs(est-est_old)/est <= tol ) & k > 1 |
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99 return |
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100 end |
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101 |
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102 x = dual(z,q); |
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103 k = k + 1; |
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104 |
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105 end |