aggiunto note, piccoli fix a eigenvalues, aggiunto polinomio di chebyshev

functions/eigenvalues_qr.m

@@ -1,10 +1,10 @@
-function [ values, err ] = eigenvalues( A, ite, toll )
+function [ values, err, A ] = eigenvalues_qr( A, ite, toll )
 
 i=0;
 err = inf;
 while (i < ite) && (err > toll)
     [q,r] = qr(A);
-    A1= r * q
+    A1= r * q;
     err = norm(diag(A1) - diag(A),inf) / norm (diag(A1),inf);
     A = A1;
     i=i+1;

functions/external/ChebyshevRoots.m

@@ -0,0 +1,154 @@
+%{
+Copyright (c) 2009, Russell Francis
+All rights reserved.
+
+Redistribution and use in source and binary forms, with or without
+modification, are permitted provided that the following conditions are
+met:
+
+    * Redistributions of source code must retain the above copyright
+      notice, this list of conditions and the following disclaimer.
+    * Redistributions in binary form must reproduce the above copyright
+      notice, this list of conditions and the following disclaimer in
+      the documentation and/or other materials provided with the distribution
+
+THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
+AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
+IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
+ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE
+LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
+CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
+SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
+INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
+CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
+ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
+POSSIBILITY OF SUCH DAMAGE.
+%}
+
+function x = ChebyshevRoots( n, type, range )
+%   USAGE: x = ChebyshevRoots( n [,type [, range]] )
+%
+% This method returns the values of the roots of the Chebyshev polynomial, 
+% either type one or type two of degree n.  In the literature, these are 
+% often referred to as T_n(x) for type 1 or U_n(x) for type 2.  If the
+% optional parameter type is omitted, it is assumed to be type 1.  This
+% function also supports scaling and translating the roots of the
+% polynomial to lie within a range specified.  The polynomials T_n(x) and
+% U_n(x) have n roots which lie within [1,-1].  It is often useful to scale
+% and translate these roots to have the same relative distance but lie over
+% a different range so that they may be used as the nodes for interpolation
+% of a function which does not lie within this [-1, 1] range.
+%
+%   PARAMETERS:
+%       n - The degree of the Chebyshev polynomial whose roots we wish to
+%       find.
+%
+%       type [optional] - Either 'Tn' for type 1 or 'Un' for type 2
+%       depending on whether you wish to generate the roots of the type 1
+%       or type 2 polynomial.  'Tn' is the default if this parameter is 
+%       omitted.
+%
+%       range [optional] - The Chebyshev polynomial is defined over [-1, 1]
+%       this parameter allows the roots of the polynomial to be translated
+%       to be within the range specified.  ie. The relative distance of the
+%       Chebyshev nodes to each other will be the same but their values
+%       will span the provided range instead of the range [-1, 1].
+%
+%   RETURNS:
+%       A vector with the n roots of the Chebyshev polynomial T_n(x) or
+%       U_n(x) of degree n, optionally scaled to lie within the range 
+%       specified.
+%
+%   AUTHOR: 
+%       Russell Francis <rf358197@ohio.edu>
+%
+%   THANKS:
+%       John D'Errico - Provided reference to the Abramowitz and Stegun book 
+%       which provides a rigorous definition of the two types of Chebyshev
+%       polynomials and is suggested reading particularly chapter 22 for
+%       interested parties.
+%
+%   REFERENCES:
+%
+% [1] Abramowitz, M.; Stegun, I.A. Handbook of Mathematical Functions with 
+% Formulas, Graphs, and Mathematical Tables. U.S. Department of Commerce.
+% Online version available at:
+% http://www.knovel.com/knovel2/Toc.jsp?BookID=528&VerticalID=0
+%
+% [2] http://en.wikipedia.org/wiki/Chebyshev_polynomials
+%
+% [3] Burden, Richard L.; Faires, J. Douglas Numerical Analysis 8th ed. 
+% pages 503-512.
+%
+
+%%
+% Verify our parameters.
+%%
+
+% The degree must be specified and must be greater than or equal to 1.
+if( nargin() < 1 )
+    error( 'You must provide the parameter n which is the degree of the polynomial you wish to calculate the roots of.' );
+else
+    if( n < 1 )
+        error( 'The parameter n must be greater than or equal to 1.' );
+    end
+end
+
+% The type of the Chebyshev polynomial to calculate the roots of, optional
+% and defaults to T_n(x)
+if( nargin() < 2)
+    type = 'Tn';
+else
+    if( (strcmp( type, 'Un' ) ~= 1 ) && (strcmp( type, 'Tn' ) ~= 1) )
+        error( 'The type parameter which was specified is not valid!, Please specify either "Tn" or "Un"' );
+    end        
+end
+
+% The range which we wish to scale and translate the result to, optional
+% and the default is to not scale or translate the result.
+if( nargin() < 3 )
+    range = [-1 1];
+else
+    if( length( range ) ~= 2 )
+        error( 'The parameter range must contain two values.' );
+    end
+
+    if( range(1) == range(2) )
+        error( 'The parameter range must contain two distinct values.' );
+    end
+        
+    range = sort(range);    
+end
+
+%%
+% Begin to compute our Chebyshev node values.
+%%
+if( n == 1 )
+    x = [0];
+else
+    if( strcmp( type, 'Tn' ) == 1 )
+        x = [(pi/(2*n)):(pi/n):pi];
+    else
+        x = [(pi/(n+1)):(pi/(n+1)):((n*pi)/(n+1))];
+    end
+    x = sort( cos(x) );
+end
+
+%%
+% x now contains the roots of the nth degree Chebyshev polynomial,
+% we need to scale and translate the result if necessary.
+%%
+if( (range(1) ~= -1) || (range(2) ~= 1) )
+    M = eye(n+1);    
+    % Calculate the scaling factor to apply to the nodes.
+    sf = (range(2) - range(1)) / 2;
+    % Calculate the translation to apply to the nodes.
+    tl = range(1) + sf;
+    % Generate our transformation matrix.
+    M(1:n,1:n) = M(1:n,1:n) * sf;
+    M(n+1,1:n) = tl;
+    % Apply to our earlier result.
+    x = [x 1] * M;
+    x = x(1:n);
+end
+return; % The x values of the Chebyshev nodes.

samples/autovalori,eig ,qr e power_method.m

@@ -0,0 +1,24 @@
+eig(a)   autovalori della matrice a
+
+qr(a) fattorizzazione matric a in q*r con q ortogonale(inversa e trasposta uguale) e r triangolare superiore
+    2/3 n 3 operazioni.
+    
+eningenvalues_qr(a, iterazioni,tolleranza) ccalcola tuttgli gli autovalori usando qr
+    se a ha autovalori reali la matrice converge a triangolare superiore
+        e si richiede che abs(a1) > abs(a2) > abs(a3) ...
+        la velocita con cui gli elementi sotto la diagonale convergono a 0 e' bassa se gli autovalori  sono molto vicini tra loro    
+    se a ha autovalori complessi la matrice converge a quasi triangolare superiore con elmenti non zero sotto la diagonale
+
+    extra:
+            Teorema: A reale simmetrica, esiste Q ortogonale t.c.
+        D = QTA Q diagonale
+        (la trasformazione non avviene in un numero finito di passi)
+
+        Teorema: A reale, esiste Q ortogonale t.c. QTA Q <EFBFBD> di
+        2
+        Hessenberg (tridiagonale se A <EFBFBD> simmetrica)
+        (la trasformazione avviene in un numero finito di passi)
+
+    
+power_method funziona solo se il massimo in valore assoluto degli autovalori di una matrice e' unico
+ e' lento se gli autovalori  sono molto vicini tra loro

samples/chebyshev.m

@@ -0,0 +1,2 @@
+ChebyshevRoots(6,'Tn',[0.1 0.6])
+input: grado, tipo, range

samples/plot,functions,polyval.m

@@ -1,10 +1,15 @@
-fplot( @(x) x^2 * cos(x), [-4 4],'r*' )
+f=@(x) x^2 * cos(x)   usa ./ .* .^ per divisioni moltiplicazioni elevazioni elemento per elemento
+fplot(f, [-4 4],'r*' )
 hold on
 
-p= [1,2,3,4,5] //coefficinti del polinomio
 x=1:0.1:10  oppure x=linspace(-5,5,100) 100 punti tra -5,5
-y=polyval(p,x)
+y=f(x)
-plot(x,y)
-
 
 p= polyfit(x,y,n) // n <EFBFBD> il grado, 1 in meno dei punti
+p= [1,2,3,4,5] //coefficinti del polinomio
+
+xx=1:0.01:10 
+yy=polyval(p,xx)
+plot(xx,yy)
+
+

samples/quadratura e integrali.m

@@ -0,0 +1,4 @@
+f = @(x) x.^2 + 2;
+integral(f,-6,6)   integrale matlab
+quad(f,-6,6) o quad(f,-6,6,1e-5)  quadratura integrale matlab
+

samples/spline cubic.m

@@ -0,0 +1,5 @@
+x=[-2 0 2 3 4 5];
+y=[4 0 -4 -30 -40 -30];
+xx=-2:0.1:5;
+yy=spline(x,y,xx);
+plot(xx,yy,'g')

samples/spline.m

@@ -1,15 +0,0 @@
-x=[-2 0 2 3 4 5];
-y=[4 0 -4 -30 -40 -30];
-plot(x,y,'sr')
-hold on
-p=polyfit(x,y,5)
-
-p =
-
-   -0.2286    2.5333   -6.2286  -10.1333   26.5714    0.0000
-
-xx=-2:0.1:5;
-yy=polyval(p,xx);
-plot(xx,yy,'b')
-ys=spline(x,y,xx);
-plot(xx,ys,'g')