--- /dev/null
+# Created by Octave 3.6.1, Sun Mar 11 22:05:14 2012 UTC <root@t61>
+# name: cache
+# type: cell
+# rows: 3
+# columns: 19
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 2
+Ci
+
+
+# name: <cell-element>
+# 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: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 80
+Compute the cosine integral function defined by:
+ Inf
+
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 2
+Si
+
+
+# name: <cell-element>
+# 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: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 80
+Compute the sine integral defined by:
+ x
+ /
+
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 6
+cosint
+
+
+# name: <cell-element>
+# 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: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 80
+Compute the cosine integral function defined by:
+ Inf
+
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 5
+dirac
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 99
+ -- Function File: Y = dirac(X)
+ Compute the dirac delta function.
+
+ See also: heaviside
+
+
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 33
+Compute the dirac delta function.
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 7
+ellipke
+
+
+# name: <cell-element>
+# 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: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 65
+Compute complete elliptic integral of first K(M) and second E(M).
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 7
+erfcinv
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 122
+ -- Function File: erfcinv (X)
+ Compute the inverse complementary error function.
+
+ See also: erfc, erf, erfinv
+
+
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 49
+Compute the inverse complementary error function.
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 6
+expint
+
+
+# name: <cell-element>
+# 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: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 80
+Compute the exponential integral,
+ infinity
+
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 9
+expint_E1
+
+
+# name: <cell-element>
+# 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: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 80
+Compute the exponential integral,
+ infinity
+
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 9
+expint_Ei
+
+
+# name: <cell-element>
+# 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: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 80
+Compute the exponential integral,
+ infinity
+
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 9
+heaviside
+
+
+# name: <cell-element>
+# 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: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 36
+Compute the Heaviside step function.
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 8
+laguerre
+
+
+# name: <cell-element>
+# 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: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 78
+Compute the value of the Laguerre polynomial of order N for each
+element of X
+
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 8
+lambertw
+
+
+# name: <cell-element>
+# 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
+ <schraudo@inf.ethz.ch>.
+
+ 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: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 36
+Compute the Lambert W function of Z.
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 9
+laplacian
+
+
+# name: <cell-element>
+# 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: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 56
+ LAPLACIAN Sparse Negative Laplacian in 1D, 2D, or 3D
+
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 8
+multinom
+
+
+# name: <cell-element>
+# 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: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 71
+Returns the terms (monomials) of the multinomial expansion of degree n.
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 14
+multinom_coeff
+
+
+# name: <cell-element>
+# 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: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 55
+Produces the coefficients of the multinomial expansion
+
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 12
+multinom_exp
+
+
+# name: <cell-element>
+# 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: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 64
+Returns the exponents of the terms in the multinomial expansion
+
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 3
+psi
+
+
+# name: <cell-element>
+# 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: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 46
+Compute the psi function, for each value of X.
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 6
+sinint
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 96
+ -- Function File: Y = sinint (X)
+ Compute the sine integral function.
+
+ See also: Si
+
+
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 35
+Compute the sine integral function.
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 4
+zeta
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 95
+ -- Function File: Z = zeta (T)
+ Compute the Riemann's Zeta function.
+
+ See also: Si
+
+
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 36
+Compute the Riemann's Zeta function.
+
+
+
+
+
git://git.creatis.insa-lyon.fr / CreaPhase.git / blobdiff