X-Git-Url: https://git.creatis.insa-lyon.fr/pubgit/?p=CreaPhase.git;a=blobdiff_plain;f=octave_packages%2Fspecfun-1.1.0%2Fdoc-cache;fp=octave_packages%2Fspecfun-1.1.0%2Fdoc-cache;h=4bf750a9a88542fabed777bc044467f5f6e14dcf;hp=0000000000000000000000000000000000000000;hb=f5f7a74bd8a4900f0b797da6783be80e11a68d86;hpb=1705066eceaaea976f010f669ce8e972f3734b05 diff --git a/octave_packages/specfun-1.1.0/doc-cache b/octave_packages/specfun-1.1.0/doc-cache new file mode 100644 index 0000000..4bf750a --- /dev/null +++ b/octave_packages/specfun-1.1.0/doc-cache @@ -0,0 +1,742 @@ +# Created by Octave 3.6.1, Sun Mar 11 22:05:14 2012 UTC +# name: cache +# type: cell +# rows: 3 +# columns: 19 +# name: +# type: sq_string +# elements: 1 +# length: 2 +Ci + + +# name: +# type: sq_string +# elements: 1 +# length: 275 + -- Function File: Y = Ci (Z) + Compute the cosine integral function defined by: Inf + / + Ci(x) = | cos(t)/t dt + / + x + + See also: cosint, Si, sinint, expint, expint_Ei + + + + + +# name: +# type: sq_string +# elements: 1 +# length: 80 +Compute the cosine integral function defined by: + Inf + + + + +# name: +# type: sq_string +# elements: 1 +# length: 2 +Si + + +# name: +# type: sq_string +# elements: 1 +# length: 207 + -- Function File: Y = Si (X) + Compute the sine integral defined by: x + / + Si(x) = | sin(t)/t dt + / + 0 + + + + +# name: +# type: sq_string +# elements: 1 +# length: 80 +Compute the sine integral defined by: + x + / + + + + +# name: +# type: sq_string +# elements: 1 +# length: 6 +cosint + + +# name: +# type: sq_string +# elements: 1 +# length: 275 + -- Function File: Y = cosint (Z) + Compute the cosine integral function defined by: Inf + / + cosint(x) = | cos(t)/t dt + / + x + + See also: Ci, Si, sinint, expint, expint_Ei + + + + + +# name: +# type: sq_string +# elements: 1 +# length: 80 +Compute the cosine integral function defined by: + Inf + + + + +# name: +# type: sq_string +# elements: 1 +# length: 5 +dirac + + +# name: +# type: sq_string +# elements: 1 +# length: 99 + -- Function File: Y = dirac(X) + Compute the dirac delta function. + + See also: heaviside + + + + + +# name: +# type: sq_string +# elements: 1 +# length: 33 +Compute the dirac delta function. + + + +# name: +# type: sq_string +# elements: 1 +# length: 7 +ellipke + + +# name: +# type: sq_string +# elements: 1 +# length: 410 + -- Function File: [K, E] = ellipke (M[,TOL]) + Compute complete elliptic integral of first K(M) and second E(M). + + M is either real array or scalar with 0 <= m <= 1 + + TOL will be ignored (MATLAB uses this to allow faster, less + accurate approximation) + + Ref: Abramowitz, Milton and Stegun, Irene A. Handbook of + Mathematical Functions, Dover, 1965, Chapter 17. + + See also: ellipj + + + + + +# name: +# type: sq_string +# elements: 1 +# length: 65 +Compute complete elliptic integral of first K(M) and second E(M). + + + +# name: +# type: sq_string +# elements: 1 +# length: 7 +erfcinv + + +# name: +# type: sq_string +# elements: 1 +# length: 122 + -- Function File: erfcinv (X) + Compute the inverse complementary error function. + + See also: erfc, erf, erfinv + + + + + +# name: +# type: sq_string +# elements: 1 +# length: 49 +Compute the inverse complementary error function. + + + +# name: +# type: sq_string +# elements: 1 +# length: 6 +expint + + +# name: +# type: sq_string +# elements: 1 +# length: 251 + -- Function File: Y = expint (X) + Compute the exponential integral, infinity + / + expint(x) = | exp(t)/t dt + / + x + + See also: expint_E1, expint_Ei + + + + + +# name: +# type: sq_string +# elements: 1 +# length: 80 +Compute the exponential integral, + infinity + + + + +# name: +# type: sq_string +# elements: 1 +# length: 9 +expint_E1 + + +# name: +# type: sq_string +# elements: 1 +# length: 251 + -- Function File: Y = expint_E1 (X) + Compute the exponential integral, infinity + / + expint(x) = | exp(t)/t dt + / + x + + See also: expint, expint_Ei + + + + + +# name: +# type: sq_string +# elements: 1 +# length: 80 +Compute the exponential integral, + infinity + + + + +# name: +# type: sq_string +# elements: 1 +# length: 9 +expint_Ei + + +# name: +# type: sq_string +# elements: 1 +# length: 263 + -- Function File: Y = expint_Ei (X) + Compute the exponential integral, infinity + / + expint_Ei(x) = - | exp(t)/t dt + / + -x + + See also: expint, expint_E1 + + + + + +# name: +# type: sq_string +# elements: 1 +# length: 80 +Compute the exponential integral, + infinity + + + + +# name: +# type: sq_string +# elements: 1 +# length: 9 +heaviside + + +# name: +# type: sq_string +# elements: 1 +# length: 400 + -- Function File: heaviside(X) + -- Function File: heaviside(X, ZERO_VALUE) + Compute the Heaviside step function. + + The Heaviside function is defined as + + Heaviside (X) = 1, X > 0 + Heaviside (X) = 0, X < 0 + + The value of the Heaviside function at X = 0 is by default 0.5, + but can be changed via the optional second input argument. + + See also: dirac + + + + + +# name: +# type: sq_string +# elements: 1 +# length: 36 +Compute the Heaviside step function. + + + +# name: +# type: sq_string +# elements: 1 +# length: 8 +laguerre + + +# name: +# type: sq_string +# elements: 1 +# length: 171 + -- Function File: Y = laguerre (X,N) + -- Function File: [Y P]= laguerre (X,N) + Compute the value of the Laguerre polynomial of order N for each + element of X + + + + + +# name: +# type: sq_string +# elements: 1 +# length: 78 +Compute the value of the Laguerre polynomial of order N for each +element of X + + + + +# name: +# type: sq_string +# elements: 1 +# length: 8 +lambertw + + +# name: +# type: sq_string +# elements: 1 +# length: 1029 + -- Function File: X = lambertw (Z) + -- Function File: X = lambertw (Z, N) + Compute the Lambert W function of Z. + + This function satisfies W(z).*exp(W(z)) = z, and can thus be used + to express solutions of transcendental equations involving + exponentials or logarithms. + + N must be integer, and specifies the branch of W to be computed; + W(z) is a shorthand for W(0,z), the principal branch. Branches 0 + and -1 are the only ones that can take on non-complex values. + + If either N or Z are non-scalar, the function is mapped to each + element; both may be non-scalar provided their dimensions agree. + + This implementation should return values within 2.5*eps of its + counterpart in Maple V, release 3 or later. Please report any + discrepancies to the author, Nici Schraudolph + . + + For further details, see: + + Corless, Gonnet, Hare, Jeffrey, and Knuth (1996), `On the Lambert + W Function', Advances in Computational Mathematics 5(4):329-359. + + + + +# name: +# type: sq_string +# elements: 1 +# length: 36 +Compute the Lambert W function of Z. + + + +# name: +# type: sq_string +# elements: 1 +# length: 9 +laplacian + + +# name: +# type: sq_string +# elements: 1 +# length: 3694 + LAPLACIAN Sparse Negative Laplacian in 1D, 2D, or 3D + + [~,~,A]=LAPLACIAN(N) generates a sparse negative 3D Laplacian matrix + with Dirichlet boundary conditions, from a rectangular cuboid regular + grid with j x k x l interior grid points if N = [j k l], using the + standard 7-point finite-difference scheme, The grid size is always + one in all directions. + + [~,~,A]=LAPLACIAN(N,B) specifies boundary conditions with a cell array + B. For example, B = {'DD' 'DN' 'P'} will Dirichlet boundary conditions + ('DD') in the x-direction, Dirichlet-Neumann conditions ('DN') in the + y-direction and period conditions ('P') in the z-direction. Possible + values for the elements of B are 'DD', 'DN', 'ND', 'NN' and 'P'. + + LAMBDA = LAPLACIAN(N,B,M) or LAPLACIAN(N,M) outputs the m smallest + eigenvalues of the matrix, computed by an exact known formula, see + http://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors_of_the_second_derivative + It will produce a warning if the mth eigenvalue is equal to the + (m+1)th eigenvalue. If m is absebt or zero, lambda will be empty. + + [LAMBDA,V] = LAPLACIAN(N,B,M) also outputs orthonormal eigenvectors + associated with the corresponding m smallest eigenvalues. + + [LAMBDA,V,A] = LAPLACIAN(N,B,M) produces a 2D or 1D negative + Laplacian matrix if the length of N and B are 2 or 1 respectively. + It uses the standard 5-point scheme for 2D, and 3-point scheme for 1D. + + % Examples: + [lambda,V,A] = laplacian([100,45,55],{'DD' 'NN' 'P'}, 20); + % Everything for 3D negative Laplacian with mixed boundary conditions. + laplacian([100,45,55],{'DD' 'NN' 'P'}, 20); + % or + lambda = laplacian([100,45,55],{'DD' 'NN' 'P'}, 20); + % computes the eigenvalues only + + [~,V,~] = laplacian([200 200],{'DD' 'DN'},30); + % Eigenvectors of 2D negative Laplacian with mixed boundary conditions. + + [~,~,A] = laplacian(200,{'DN'},30); + % 1D negative Laplacian matrix A with mixed boundary conditions. + + % Example to test if outputs correct eigenvalues and vectors: + [lambda,V,A] = laplacian([13,10,6],{'DD' 'DN' 'P'},30); + [Veig D] = eig(full(A)); lambdaeig = diag(D(1:30,1:30)); + max(abs(lambda-lambdaeig)) %checking eigenvalues + subspace(V,Veig(:,1:30)) %checking the invariant subspace + subspace(V(:,1),Veig(:,1)) %checking selected eigenvectors + subspace(V(:,29:30),Veig(:,29:30)) %a multiple eigenvalue + + % Example showing equivalence between laplacian.m and built-in MATLAB + % DELSQ for the 2D case. The output of the last command shall be 0. + A1 = delsq(numgrid('S',32)); % input 'S' specifies square grid. + [~,~,A2] = laplacian([30,30]); + norm(A1-A2,inf) + + Class support for inputs: + N - row vector float double + B - cell array + M - scalar float double + + Class support for outputs: + lambda and V - full float double, A - sparse float double. + + Note: the actual numerical entries of A fit int8 format, but only + double data class is currently (2010) supported for sparse matrices. + + This program is designed to efficiently compute eigenvalues, + eigenvectors, and the sparse matrix of the (1-3)D negative Laplacian + on a rectangular grid for Dirichlet, Neumann, and Periodic boundary + conditions using tensor sums of 1D Laplacians. For more information on + tensor products, see + http://en.wikipedia.org/wiki/Kronecker_sum_of_discrete_Laplacians + For 2D case in MATLAB, see + http://www.mathworks.com/access/helpdesk/help/techdoc/ref/kron.html. + + This code is also part of the BLOPEX package: + http://en.wikipedia.org/wiki/BLOPEX or directly + http://code.google.com/p/blopex/ + + + +# name: +# type: sq_string +# elements: 1 +# length: 56 + LAPLACIAN Sparse Negative Laplacian in 1D, 2D, or 3D + + + + +# name: +# type: sq_string +# elements: 1 +# length: 8 +multinom + + +# name: +# type: sq_string +# elements: 1 +# length: 588 + -- Function File: [Y ALPHA] = multinom (X, N) + -- Function File: [Y ALPHA] = multinom (X, N,SORT) + Returns the terms (monomials) of the multinomial expansion of + degree n. + + (x1 + x2 + ... + xm)^N + + X is a nT-by-m matrix where each column represents a different + variable, the output Y has the same format. The order of the + terms is inherited from multinom_exp and can be controlled through + the optional argument SORT and is passed to the function `sort'. + The exponents are returned in ALPHA. + + See also: multinom_exp, multinom_coeff, sort + + + + + +# name: +# type: sq_string +# elements: 1 +# length: 71 +Returns the terms (monomials) of the multinomial expansion of degree n. + + + +# name: +# type: sq_string +# elements: 1 +# length: 14 +multinom_coeff + + +# name: +# type: sq_string +# elements: 1 +# length: 937 + -- Function File: [C ALPHA] = multinom_coeff (M, N) + -- Function File: [C ALPHA] = multinom_coeff (M, N,ORDER) + Produces the coefficients of the multinomial expansion + + (x1 + x2 + ... + xm).^n + + For example, for m=3, n=3 the expansion is + + (x1+x2+x3)^3 = + = x1^3 + x2^3 + x3^3 + + + 3 x1^2 x2 + 3 x1^2 x3 + 3 x2^2 x1 + 3 x2^2 x3 + + + 3 x3^2 x1 + 3 x3^2 x2 + 6 x1 x2 x3 + + and the coefficients are [6 3 3 3 3 3 3 1 1 1]. + + The order of the coefficients is defined by the optinal argument + ORDER. It is passed to the function `multion_exp'. See the help + of that function for explanation. The multinomial coefficients + are generated using + + / \ + | n | n! + | | = ------------------------ + | k | k(1)!k(2)! ... k(end)! + \ / + + See also: multinom, multinom_exp + + + + + +# name: +# type: sq_string +# elements: 1 +# length: 55 +Produces the coefficients of the multinomial expansion + + + + +# name: +# type: sq_string +# elements: 1 +# length: 12 +multinom_exp + + +# name: +# type: sq_string +# elements: 1 +# length: 700 + -- Function File: ALPHA = multinom_exp (M, N) + -- Function File: ALPHA = multinom_exp (M, N,SORT) + Returns the exponents of the terms in the multinomial expansion + + (x1 + x2 + ... + xm).^N + + For example, for m=2, n=3 the expansion has the terms + + x1^3, x2^3, x1^2*x2, x1*x2^2 + + then `alpha = [3 0; 2 1; 1 2; 0 3]'; + + The optional argument SORT is passed to function `sort' to sort + the exponents by the maximum degree. The example above calling ` + multinom(m,n,"ascend")' produces + + `alpha = [2 1; 1 2; 3 0; 0 3]'; + + calling ` multinom(m,n,"descend")' produces + + `alpha = [3 0; 0 3; 2 1; 1 2]'; + + See also: multinom, multinom_coeff, sort + + + + + +# name: +# type: sq_string +# elements: 1 +# length: 64 +Returns the exponents of the terms in the multinomial expansion + + + + +# name: +# type: sq_string +# elements: 1 +# length: 3 +psi + + +# name: +# type: sq_string +# elements: 1 +# length: 201 + -- Function File: Y = psi (X) + Compute the psi function, for each value of X. + + d + psi(x) = __ log(gamma(x)) + dx + + See also: gamma, gammainc, gammaln + + + + + +# name: +# type: sq_string +# elements: 1 +# length: 46 +Compute the psi function, for each value of X. + + + +# name: +# type: sq_string +# elements: 1 +# length: 6 +sinint + + +# name: +# type: sq_string +# elements: 1 +# length: 96 + -- Function File: Y = sinint (X) + Compute the sine integral function. + + See also: Si + + + + + +# name: +# type: sq_string +# elements: 1 +# length: 35 +Compute the sine integral function. + + + +# name: +# type: sq_string +# elements: 1 +# length: 4 +zeta + + +# name: +# type: sq_string +# elements: 1 +# length: 95 + -- Function File: Z = zeta (T) + Compute the Riemann's Zeta function. + + See also: Si + + + + + +# name: +# type: sq_string +# elements: 1 +# length: 36 +Compute the Riemann's Zeta function. + + + + +