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// Scilab ( http://www.scilab.org/ ) - This file is part of Scilab
// Copyright (C) ? - 2008 - Rainer von Seggern
// Copyright (C) ? - 2008 - Bruno Pincon
// Copyright (C) 2009 - INRIA - Michael Baudin
// Copyright (C) 2010-2011 - DIGITEO - Michael Baudin
//
// This file must be used under the terms of the CeCILL.
// This source file is licensed as described in the file COPYING, which
// you should have received as part of this distribution. The terms
// are also available at
// http://www.cecill.info/licences/Licence_CeCILL_V2.1-en.txt
function [J, H] = numderivative(varargin)
//
// Check input arguments
[lhs, rhs] = argn();
if (rhs < 2 | rhs > 6) then
error(msprintf(gettext("%s: Wrong number of input arguments: %d to %d expected.\n"), "numderivative", 2, 6));
end
if (lhs < 1 | lhs > 2) then
error(msprintf(gettext("%s: Wrong number of output arguments: %d to %d expected.\n"), "numderivative", 1, 2));
end
//
// Get input arguments
__numderivative_f__ = varargin(1)
if and(type(__numderivative_f__) <> [11 13 15 130]) then
// Must be a function (uncompiled or compiled) or a list
error(msprintf(gettext("%s: Wrong type for argument #%d: Function or list expected.\n"), "numderivative", 1));
end
if type(__numderivative_f__) == 15 then
// List case
// Check that the first element in the list is a function
if and(type(__numderivative_f__(1)) <> [11 13]) then
error(msprintf(gettext("%s: Wrong type for argument #%d: Function expected in first element of list.\n"), "numderivative", 1));
end
if length(__numderivative_f__) < 2 then
error(msprintf(gettext("%s: Wrong number of elements in input argument #%d: At least %d elements expected, but current number is %d.\n"), "numderivative", 1, 2, length(__numderivative_f__)));
end
end
//
// Manage x, to get the size n.
x = varargin(2);
if type(x) ~= 1 then
error(msprintf(gettext("%s: Wrong type for argument #%d: Matrix expected.\n"), "numderivative", 2));
end
[n, p] = size(x);
if (n <> 1 & p <> 1) then
error(msprintf(gettext("%s: Wrong size for input argument #%d: Vector expected.\n"), "numderivative", 2));
end
// Make x a column vector, if required
if p <> 1 then
x = x(:);
[n, p] = size(x);
end
//
// Manage h: make it a column vector, if required.
h = [];
if rhs >= 3 then
h = varargin(3);
if type(h) ~= 1 then
error(msprintf(gettext("%s: Wrong type for argument #%d: Matrix expected.\n"), "numderivative", 3));
end
if h <> [] then
if size(h, "*") <> 1 then
[nrows, ncols] = size(h);
if (nrows <> 1 & ncols <> 1) then
error(msprintf(gettext("%s: Wrong size for input argument #%d: Vector expected.\n"), "numderivative", 3));
end
if ncols <> 1 then
h = h(:);
end
if or(size(h) <> [n 1]) then
error(msprintf(gettext("%s: Incompatible input arguments #%d and #%d: Same sizes expected.\n"), "numderivative", 3, 1));
end
end
if or(h < 0) then
error(msprintf(gettext("%s: Wrong value for input argument #%d: Must be > %d.\n"), "numderivative", 3, 0));
end
end
end
order = 2;
if (rhs >= 4 & varargin(4) <> []) then
order = varargin(4);
if type(order) ~= 1 then
error(msprintf(gettext("%s: Wrong type for argument #%d: Matrix expected.\n"), "numderivative", 4));
end
if or(size(order) <> [1 1]) then
error(msprintf(gettext("%s: Wrong size for input argument #%d: %d-by-%d matrix expected.\n"), "numderivative", 4, 1, 1));
end
if and(order <> [1 2 4]) then
error(msprintf(gettext("%s: Wrong value for input argument #%d: Must be in the set {%s}.\n"), "numderivative", 4, "1, 2, 4"));
end
end
H_form = "default";
if (rhs >= 5 & varargin(5) <> []) then
H_form = varargin(5);
if type(H_form) ~= 10 then
error(msprintf(gettext("%s: Wrong type for input argument #%d: String array expected.\n"), "numderivative", 5));
end
if or(size(H_form) <> [1 1]) then
error(msprintf(gettext("%s: Wrong size for input argument #%d: %d-by-%d matrix expected.\n"), "numderivative", 5, 1, 1));
end
if and(H_form <> ["default" "blockmat" "hypermat"]) then
error(msprintf(gettext("%s: Wrong value for input argument #%d: Must be in the set {%s}.\n"), "numderivative", 5, "default, blockmat, hypermat"));
end
end
Q = eye(n, n);
Q_not_given = %t;
if (rhs >= 6 & varargin(6) <> []) then
Q = varargin(6);
Q_not_given = %f;
if type(Q) ~= 1 then
error(msprintf(gettext("%s: Wrong type for argument #%d: Matrix expected.\n"), "numderivative", 6));
end
if or(size(Q) <> [n n]) then
error(msprintf(gettext("%s: Wrong size for input argument #%d: %d-by-%d matrix expected.\n"), "numderivative", 6, n ,n));
end
if norm(clean(Q*Q'-eye(n, n))) > 0 then
error(msprintf(gettext("%s: Q must be orthogonal.\n"), "numderivative"));
end
end
//
// Proceed...
if h == [] then
h_not_given = %t;
else
h_not_given = %f;
// If h is scalar, expand to the same size as x.
if size(h) == [1 1] then
h = h * ones(x);
end
end
//
// Compute Jacobian
if ( h_not_given ) then
h = numderivative_step(x, order, 1);
end
J = numderivative_deriv1(__numderivative_f__, x, h, order, Q);
//
// Quick return if possible
if lhs == 1 then
return
end
m = size(J, 1);
//
// Compute Hessian matrix
if ( h_not_given ) then
h = numderivative_step(x, order, 2);
end
funForHList = list(numderivative_funForH, __numderivative_f__, h, order, Q);
if ~Q_not_given then
H = numderivative_deriv1(funForHList, x, h, order, Q);
else
H = numderivative_deriv2(funForHList, x, h, order, Q);
end
//
// At this point, H is a m*n-by-n block matrix.
// Update the format of the Hessian
if H_form == "default" then
// H has the old scilab form
H = matrix(H', n*n, m)'
end
if H_form == "hypermat" then
if m > 1 then
// H is a hypermatrix if m > 1
H = H';
H = hypermat([n n m], H(:));
end
end
endfunction
//
// numderivative_step --
// Returns the step for given x, given order and given derivative:
// d = 1 is for Jacobian
// d = 2 is for Hessian
// Uses the optimal step.
// Then scale the step depending on abs(x).
function h = numderivative_step(x, order, d)
n = size(x, "*");
select d
case 1
// For Jacobian
select order
case 1
hdefault = sqrt(%eps);
case 2
hdefault = %eps^(1/3);
case 4
hdefault = %eps^(1/5);
else
lclmsg = gettext("%s: Unknown value %s for option %s.\n");
error(msprintf(lclmsg,"numderivative_step", string(d), "d"));
end
case 2
// For Hessian
select order
case 1
hdefault = %eps^(1/3);
case 2
hdefault = %eps^(1/4);
case 4
hdefault = %eps^(1/6);
else
lclmsg = gettext("%s: Unknown value %s for option %s.\n");
error(msprintf(lclmsg, "numderivative_step", string(d), "d"));
end
else
lclmsg = gettext("%s: Unknown value %s for option %s.\n");
error(msprintf(lclmsg, "numderivative_step", string(order), "order"));
end
// Convert this scalar into a vector, with same size as x
// For zero entries in x, use the default.
// For nonzero entries in x, scales by abs(x).
h = hdefault * abs(x);
h(x==0) = hdefault;
endfunction
//
// numderivative_funForH --
// Returns the numerical derivative of __numderivative_f__.
// This function is called to compute the numerical Hessian.
function J = numderivative_funForH(x, __numderivative_f__, h, order, Q)
// Transpose !
J = numderivative_deriv1(__numderivative_f__, x, h, order, Q)';
J = J(:);
endfunction
// numderivative_deriv1 --
// Computes the numerical gradient of __numderivative_f__, using the given step h.
// This function is used for the computation of the jacobian matrix.
function g = numderivative_deriv1(__numderivative_f__, x, h, order, Q)
n = size(x, "*");
%Dy = []; // At this point, we do not know 'm' yet, so we cannot allocate Dy.
select order
case 1
D = Q * diag(h);
y = numderivative_evalf(__numderivative_f__, x);
for i=1:n
d = D(:, i);
yplus = numderivative_evalf(__numderivative_f__, x+d);
Dyi = (yplus-y)/h(i);
%Dy = [%Dy Dyi];
end
g = %Dy*Q';
case 2
D = Q * diag(h);
for i=1:n
d = D(:, i);
yplus = numderivative_evalf(__numderivative_f__, x+d);
yminus = numderivative_evalf(__numderivative_f__, x-d);
Dyi = (yplus-yminus)/(2*h(i));
%Dy = [%Dy Dyi];
end
g = %Dy*Q';
case 4
D = Q * diag(h);
for i=1:n
d = D(:, i);
yplus = numderivative_evalf(__numderivative_f__, x+d);
yminus = numderivative_evalf(__numderivative_f__, x-d);
yplus2 = numderivative_evalf(__numderivative_f__, x+2*d);
yminus2 = numderivative_evalf(__numderivative_f__, x-2*d);
dFh = (yplus-yminus)/(2*h(i));
dF2h = (yplus2-yminus2)/(4*h(i));
Dyi = (4*dFh - dF2h)/3;
%Dy = [%Dy Dyi];
end
g = %Dy*Q';
end
endfunction
// numderivative_deriv2 --
// Computes the numerical gradient of the argument __numderivative_f__, using the given step h.
// This function is used for the computation of the hessian matrix, to take advantage of its symmetry
function g = numderivative_deriv2(__numderivative_f__, x, h, order, Q)
n = size(x, "*");
%Dy = zeros(m*n, n); // 'm' is known at this point, so we can allocate Dy to reduce memory operations
select order
case 1
D = Q * diag(h);
y = numderivative_evalf(__numderivative_f__, x);
for i=1:n
d = D(:, i);
yplus = numderivative_evalf(__numderivative_f__, x+d);
for j=0:m-1
Dyi(1+j*n:i-1+j*n) = %Dy(i+j*n, 1:i-1)'; // Retrieving symmetric elements (will not be done for the first vector)
Dyi(i+j*n:(j+1)*n) = (yplus(i+j*n:(j+1)*n)-y(i+j*n:(j+1)*n))/h(i); // Computing the new ones
end
%Dy(:, i) = Dyi;
end
g = %Dy*Q';
case 2
D = Q * diag(h);
for i=1:n
d = D(:, i);
yplus = numderivative_evalf(__numderivative_f__, x+d);
yminus = numderivative_evalf(__numderivative_f__, x-d);
for j=0:m-1
Dyi(1+j*n:i-1+j*n) = %Dy(i+j*n, 1:i-1)'; // Retrieving symmetric elements (will not be done for the first vector)
Dyi(i+j*n:(j+1)*n) = (yplus(i+j*n:(j+1)*n)-yminus(i+j*n:(j+1)*n))/(2*h(i)); // Computing the new ones
end
%Dy(:, i) = Dyi;
end
g = %Dy*Q';
case 4
D = Q * diag(h);
for i=1:n
d = D(:, i);
yplus = numderivative_evalf(__numderivative_f__, x+d);
yminus = numderivative_evalf(__numderivative_f__, x-d);
yplus2 = numderivative_evalf(__numderivative_f__, x+2*d);
yminus2 = numderivative_evalf(__numderivative_f__, x-2*d);
for j=0:m-1
dFh(1+j*n:i-1+j*n) = %Dy(i+j*n, 1:i-1)'; // Retrieving symmetric elements (will not be done for the first vector)
dFh(i+j*n:(j+1)*n) = (yplus(i+j*n:(j+1)*n)-yminus(i+j*n:(j+1)*n))/(2*h(i)); // Computing the new ones
dF2h(1+j*n:i-1+j*n) = %Dy(i+j*n, 1:i-1)'; // Retrieving symmetric elements (will not be done for the first vector)
dF2h(i+j*n:(j+1)*n) = (yplus2(i+j*n:(j+1)*n)-yminus2(i+j*n:(j+1)*n))/(4*h(i)); // Computing the new ones
end
Dyi = (4*dFh - dF2h)/3;
%Dy(:, i) = Dyi;
end
g = %Dy*Q';
end
endfunction
// numderivative_evalf --
// Computes the value of __numderivative_f__ at the point x.
// The argument __numderivative_f__ can be a function (macro or linked code) or a list.
function y = numderivative_evalf(__numderivative_f__, x)
if type(__numderivative_f__) == 15 then
// List case
__numderivative_fun__ = __numderivative_f__(1);
instr = "y = __numderivative_fun__(x, __numderivative_f__(2:$))";
elseif or(type(__numderivative_f__) == [11 13 130]) then
// Function case
instr = "y = __numderivative_f__(x)";
else
error(msprintf(gettext("%s: Wrong type for input argument #%d: A function expected.\n"), "numderivative", 1));
end
ierr = execstr(instr, "errcatch")
if ierr <> 0 then
lamsg = lasterror();
lclmsg = "%s: Error while evaluating the function: ""%s""\n";
error(msprintf(gettext(lclmsg), "numderivative", lamsg));
end
endfunction
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