aggiunto note, piccoli fix a eigenvalues, aggiunto polinomio di chebyshev
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function [ values, err ] = eigenvalues( A, ite, toll )
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function [ values, err, A ] = eigenvalues_qr( A, ite, toll )
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i=0;
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err = inf;
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while (i < ite) && (err > toll)
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[q,r] = qr(A);
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A1= r * q
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A1= r * q;
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err = norm(diag(A1) - diag(A),inf) / norm (diag(A1),inf);
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A = A1;
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i=i+1;
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154
functions/external/ChebyshevRoots.m
vendored
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154
functions/external/ChebyshevRoots.m
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%{
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Copyright (c) 2009, Russell Francis
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All rights reserved.
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Redistribution and use in source and binary forms, with or without
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modification, are permitted provided that the following conditions are
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met:
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* Redistributions of source code must retain the above copyright
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notice, this list of conditions and the following disclaimer.
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* Redistributions in binary form must reproduce the above copyright
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notice, this list of conditions and the following disclaimer in
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the documentation and/or other materials provided with the distribution
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THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
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AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
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IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
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ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE
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LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
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CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
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SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
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INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
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CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
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ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
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POSSIBILITY OF SUCH DAMAGE.
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%}
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function x = ChebyshevRoots( n, type, range )
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% USAGE: x = ChebyshevRoots( n [,type [, range]] )
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%
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% This method returns the values of the roots of the Chebyshev polynomial,
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% either type one or type two of degree n. In the literature, these are
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% often referred to as T_n(x) for type 1 or U_n(x) for type 2. If the
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% optional parameter type is omitted, it is assumed to be type 1. This
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% function also supports scaling and translating the roots of the
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% polynomial to lie within a range specified. The polynomials T_n(x) and
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% U_n(x) have n roots which lie within [1,-1]. It is often useful to scale
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% and translate these roots to have the same relative distance but lie over
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% a different range so that they may be used as the nodes for interpolation
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% of a function which does not lie within this [-1, 1] range.
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%
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% PARAMETERS:
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% n - The degree of the Chebyshev polynomial whose roots we wish to
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% find.
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%
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% type [optional] - Either 'Tn' for type 1 or 'Un' for type 2
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% depending on whether you wish to generate the roots of the type 1
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% or type 2 polynomial. 'Tn' is the default if this parameter is
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% omitted.
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%
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% range [optional] - The Chebyshev polynomial is defined over [-1, 1]
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% this parameter allows the roots of the polynomial to be translated
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% to be within the range specified. ie. The relative distance of the
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% Chebyshev nodes to each other will be the same but their values
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% will span the provided range instead of the range [-1, 1].
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%
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% RETURNS:
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% A vector with the n roots of the Chebyshev polynomial T_n(x) or
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% U_n(x) of degree n, optionally scaled to lie within the range
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% specified.
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%
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% AUTHOR:
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% Russell Francis <rf358197@ohio.edu>
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%
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% THANKS:
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% John D'Errico - Provided reference to the Abramowitz and Stegun book
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% which provides a rigorous definition of the two types of Chebyshev
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% polynomials and is suggested reading particularly chapter 22 for
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% interested parties.
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%
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% REFERENCES:
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%
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% [1] Abramowitz, M.; Stegun, I.A. Handbook of Mathematical Functions with
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% Formulas, Graphs, and Mathematical Tables. U.S. Department of Commerce.
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% Online version available at:
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% http://www.knovel.com/knovel2/Toc.jsp?BookID=528&VerticalID=0
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%
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% [2] http://en.wikipedia.org/wiki/Chebyshev_polynomials
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%
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% [3] Burden, Richard L.; Faires, J. Douglas Numerical Analysis 8th ed.
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% pages 503-512.
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%
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%%
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% Verify our parameters.
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%%
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% The degree must be specified and must be greater than or equal to 1.
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if( nargin() < 1 )
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error( 'You must provide the parameter n which is the degree of the polynomial you wish to calculate the roots of.' );
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else
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if( n < 1 )
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error( 'The parameter n must be greater than or equal to 1.' );
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end
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end
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% The type of the Chebyshev polynomial to calculate the roots of, optional
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% and defaults to T_n(x)
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if( nargin() < 2)
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type = 'Tn';
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else
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if( (strcmp( type, 'Un' ) ~= 1 ) && (strcmp( type, 'Tn' ) ~= 1) )
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error( 'The type parameter which was specified is not valid!, Please specify either "Tn" or "Un"' );
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end
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end
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% The range which we wish to scale and translate the result to, optional
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% and the default is to not scale or translate the result.
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if( nargin() < 3 )
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range = [-1 1];
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else
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if( length( range ) ~= 2 )
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error( 'The parameter range must contain two values.' );
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end
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if( range(1) == range(2) )
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error( 'The parameter range must contain two distinct values.' );
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end
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range = sort(range);
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end
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%%
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% Begin to compute our Chebyshev node values.
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%%
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if( n == 1 )
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x = [0];
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else
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if( strcmp( type, 'Tn' ) == 1 )
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x = [(pi/(2*n)):(pi/n):pi];
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else
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x = [(pi/(n+1)):(pi/(n+1)):((n*pi)/(n+1))];
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end
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x = sort( cos(x) );
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end
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%%
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% x now contains the roots of the nth degree Chebyshev polynomial,
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% we need to scale and translate the result if necessary.
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%%
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if( (range(1) ~= -1) || (range(2) ~= 1) )
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M = eye(n+1);
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% Calculate the scaling factor to apply to the nodes.
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sf = (range(2) - range(1)) / 2;
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% Calculate the translation to apply to the nodes.
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tl = range(1) + sf;
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% Generate our transformation matrix.
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M(1:n,1:n) = M(1:n,1:n) * sf;
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M(n+1,1:n) = tl;
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% Apply to our earlier result.
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x = [x 1] * M;
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x = x(1:n);
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end
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return; % The x values of the Chebyshev nodes.
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24
samples/autovalori,eig ,qr e power_method.m
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24
samples/autovalori,eig ,qr e power_method.m
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eig(a) autovalori della matrice a
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qr(a) fattorizzazione matric a in q*r con q ortogonale(inversa e trasposta uguale) e r triangolare superiore
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2/3 n 3 operazioni.
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eningenvalues_qr(a, iterazioni,tolleranza) ccalcola tuttgli gli autovalori usando qr
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se a ha autovalori reali la matrice converge a triangolare superiore
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e si richiede che abs(a1) > abs(a2) > abs(a3) ...
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la velocita con cui gli elementi sotto la diagonale convergono a 0 e' bassa se gli autovalori sono molto vicini tra loro
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se a ha autovalori complessi la matrice converge a quasi triangolare superiore con elmenti non zero sotto la diagonale
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extra:
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Teorema: A reale simmetrica, esiste Q ortogonale t.c.
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D = QTA Q diagonale
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(la trasformazione non avviene in un numero finito di passi)
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Teorema: A reale, esiste Q ortogonale t.c. QTA Q <EFBFBD> di
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2
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Hessenberg (tridiagonale se A <EFBFBD> simmetrica)
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(la trasformazione avviene in un numero finito di passi)
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power_method funziona solo se il massimo in valore assoluto degli autovalori di una matrice e' unico
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e' lento se gli autovalori sono molto vicini tra loro
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2
samples/chebyshev.m
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2
samples/chebyshev.m
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ChebyshevRoots(6,'Tn',[0.1 0.6])
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input: grado, tipo, range
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@@ -1,10 +1,15 @@
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fplot( @(x) x^2 * cos(x), [-4 4],'r*' )
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f=@(x) x^2 * cos(x) usa ./ .* .^ per divisioni moltiplicazioni elevazioni elemento per elemento
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fplot(f, [-4 4],'r*' )
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hold on
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p= [1,2,3,4,5] //coefficinti del polinomio
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x=1:0.1:10 oppure x=linspace(-5,5,100) 100 punti tra -5,5
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y=polyval(p,x)
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plot(x,y)
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y=f(x)
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p= polyfit(x,y,n) // n <EFBFBD> il grado, 1 in meno dei punti
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p= [1,2,3,4,5] //coefficinti del polinomio
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xx=1:0.01:10
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yy=polyval(p,xx)
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plot(xx,yy)
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p= polyfit(x,y,n) // n <EFBFBD> il grado, 1 in meno dei punti
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4
samples/quadratura e integrali.m
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4
samples/quadratura e integrali.m
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f = @(x) x.^2 + 2;
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integral(f,-6,6) integrale matlab
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quad(f,-6,6) o quad(f,-6,6,1e-5) quadratura integrale matlab
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5
samples/spline cubic.m
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5
samples/spline cubic.m
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x=[-2 0 2 3 4 5];
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y=[4 0 -4 -30 -40 -30];
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xx=-2:0.1:5;
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yy=spline(x,y,xx);
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plot(xx,yy,'g')
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@@ -1,15 +0,0 @@
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x=[-2 0 2 3 4 5];
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y=[4 0 -4 -30 -40 -30];
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plot(x,y,'sr')
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hold on
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p=polyfit(x,y,5)
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p =
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-0.2286 2.5333 -6.2286 -10.1333 26.5714 0.0000
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xx=-2:0.1:5;
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yy=polyval(p,xx);
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plot(xx,yy,'b')
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ys=spline(x,y,xx);
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plot(xx,ys,'g')
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