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@(@\newcommand{\W}[1]{ \; #1 \; }
\newcommand{\R}[1]{ {\rm #1} }
\newcommand{\B}[1]{ {\bf #1} }
\newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} }
\newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} }
\newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} }
\newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }@)@Atomic Eigen Cholesky Factorization: Example and Test
Description
The ADFun
function object
f
for this example is
@[@
f(x)
=
\R{chol} \left( \begin{array}{cc}
x_0 & x_1 \\
x_1 & x_2
\end{array} \right)
=
\frac{1}{ \sqrt{x_0} }
\left( \begin{array}{cc}
x_0 & 0 \\
x_1 & \sqrt{ x_0 x_2 - x_1 x_1 }
\end{array} \right)
@]@
where the matrix is positive definite; i.e.,
@(@
x_0 > 0
@)@, @(@
x_2 > 0
@)@ and
@(@
x_0 x_2 - x_1 x_1 > 0
@)@.
Contents
Use Atomic Function
# include <cppad/cppad.hpp>
# include <cppad/example/eigen_cholesky.hpp>
bool eigen_cholesky(void)
{
typedef double scalar;
typedef typename atomic_eigen_cholesky<scalar>::ad_scalar ad_scalar;
typedef typename atomic_eigen_cholesky<scalar>::ad_matrix ad_matrix;
//
bool ok = true;
scalar eps = 10. * std::numeric_limits<scalar>::epsilon();
using CppAD::NearEqual;
//
Constructor
// -------------------------------------------------------------------
// object that computes cholesky factor of a matrix
atomic_eigen_cholesky<scalar> cholesky;
// -------------------------------------------------------------------
// declare independent variable vector x
size_t n = 3;
CPPAD_TESTVECTOR(ad_scalar) ad_x(n);
ad_x[0] = 2.0;
ad_x[1] = 0.5;
ad_x[2] = 3.0;
CppAD::Independent(ad_x);
// -------------------------------------------------------------------
// A = [ x[0] x[1] ]
// [ x[1] x[2] ]
size_t nr = 2;
ad_matrix ad_A(nr, nr);
ad_A(0, 0) = ad_x[0];
ad_A(1, 0) = ad_x[1];
ad_A(0, 1) = ad_x[1];
ad_A(1, 1) = ad_x[2];
// -------------------------------------------------------------------
// use atomic operation to L such that A = L * L^T
ad_matrix ad_L = cholesky.op(ad_A);
// -------------------------------------------------------------------
// declare the dependent variable vector y
size_t m = 3;
CPPAD_TESTVECTOR(ad_scalar) ad_y(m);
ad_y[0] = ad_L(0, 0);
ad_y[1] = ad_L(1, 0);
ad_y[2] = ad_L(1, 1);
CppAD::ADFun<scalar> f(ad_x, ad_y);
// -------------------------------------------------------------------
// check zero order forward mode
CPPAD_TESTVECTOR(scalar) x(n), y(m);
x[0] = 2.0;
x[1] = 0.5;
x[2] = 5.0;
y = f.Forward(0, x);
scalar check;
check = std::sqrt( x[0] );
ok &= NearEqual(y[0], check, eps, eps);
check = x[1] / std::sqrt( x[0] );
ok &= NearEqual(y[1], check, eps, eps);
check = std::sqrt( x[2] - x[1] * x[1] / x[0] );
ok &= NearEqual(y[2], check, eps, eps);
// -------------------------------------------------------------------
// check first order forward mode
CPPAD_TESTVECTOR(scalar) x1(n), y1(m);
//
// partial w.r.t. x[0]
x1[0] = 1.0;
x1[1] = 0.0;
x1[2] = 0.0;
//
y1 = f.Forward(1, x1);
check = 1.0 / (2.0 * std::sqrt( x[0] ) );
ok &= NearEqual(y1[0], check, eps, eps);
//
check = - x[1] / (2.0 * x[0] * std::sqrt( x[0] ) );
ok &= NearEqual(y1[1], check, eps, eps);
//
check = std::sqrt( x[2] - x[1] * x[1] / x[0] );
check = x[1] * x[1] / (x[0] * x[0] * 2.0 * check);
ok &= NearEqual(y1[2], check, eps, eps);
//
// partial w.r.t. x[1]
x1[0] = 0.0;
x1[1] = 1.0;
x1[2] = 0.0;
//
y1 = f.Forward(1, x1);
ok &= NearEqual(y1[0], 0.0, eps, eps);
//
check = 1.0 / std::sqrt( x[0] );
ok &= NearEqual(y1[1], check, eps, eps);
//
check = std::sqrt( x[2] - x[1] * x[1] / x[0] );
check = - 2.0 * x[1] / (2.0 * check * x[0] );
ok &= NearEqual(y1[2], check, eps, eps);
//
// partial w.r.t. x[2]
x1[0] = 0.0;
x1[1] = 0.0;
x1[2] = 1.0;
//
y1 = f.Forward(1, x1);
ok &= NearEqual(y1[0], 0.0, eps, eps);
ok &= NearEqual(y1[1], 0.0, eps, eps);
//
check = std::sqrt( x[2] - x[1] * x[1] / x[0] );
check = 1.0 / (2.0 * check);
ok &= NearEqual(y1[2], check, eps, eps);
// -------------------------------------------------------------------
// check second order forward mode
CPPAD_TESTVECTOR(scalar) x2(n), y2(m);
//
// second partial w.r.t x[2]
x2[0] = 0.0;
x2[1] = 0.0;
x2[2] = 0.0;
y2 = f.Forward(2, x2);
ok &= NearEqual(y2[0], 0.0, eps, eps);
ok &= NearEqual(y2[1], 0.0, eps, eps);
//
check = std::sqrt( x[2] - x[1] * x[1] / x[0] ); // funciton value
check = - 1.0 / ( 4.0 * check * check * check ); // second derivative
check = 0.5 * check; // taylor coefficient
ok &= NearEqual(y2[2], check, eps, eps);
// -------------------------------------------------------------------
// check first order reverse mode
CPPAD_TESTVECTOR(scalar) w(m), d1w(n);
w[0] = 0.0;
w[1] = 0.0;
w[2] = 1.0;
d1w = f.Reverse(1, w);
//
// partial of f[2] w.r.t x[0]
scalar f2 = std::sqrt( x[2] - x[1] * x[1] / x[0] );
scalar f2_x0 = x[1] * x[1] / (2.0 * f2 * x[0] * x[0] );
ok &= NearEqual(d1w[0], f2_x0, eps, eps);
//
// partial of f[2] w.r.t x[1]
scalar f2_x1 = - x[1] / (f2 * x[0] );
ok &= NearEqual(d1w[1], f2_x1, eps, eps);
//
// partial of f[2] w.r.t x[2]
scalar f2_x2 = 1.0 / (2.0 * f2 );
ok &= NearEqual(d1w[2], f2_x2, eps, eps);
// -------------------------------------------------------------------
// check second order reverse mode
CPPAD_TESTVECTOR(scalar) d2w(2 * n);
d2w = f.Reverse(2, w);
//
// check first order results
ok &= NearEqual(d2w[0 * 2 + 0], f2_x0, eps, eps);
ok &= NearEqual(d2w[1 * 2 + 0], f2_x1, eps, eps);
ok &= NearEqual(d2w[2 * 2 + 0], f2_x2, eps, eps);
//
// check second order results
scalar f2_x2_x0 = - 0.5 * f2_x0 / (f2 * f2 );
ok &= NearEqual(d2w[0 * 2 + 1], f2_x2_x0, eps, eps);
scalar f2_x2_x1 = - 0.5 * f2_x1 / (f2 * f2 );
ok &= NearEqual(d2w[1 * 2 + 1], f2_x2_x1, eps, eps);
scalar f2_x2_x2 = - 0.5 * f2_x2 / (f2 * f2 );
ok &= NearEqual(d2w[2 * 2 + 1], f2_x2_x2, eps, eps);
// -------------------------------------------------------------------
// check third order reverse mode
CPPAD_TESTVECTOR(scalar) d3w(3 * n);
d3w = f.Reverse(3, w);
//
// check first order results
ok &= NearEqual(d3w[0 * 3 + 0], f2_x0, eps, eps);
ok &= NearEqual(d3w[1 * 3 + 0], f2_x1, eps, eps);
ok &= NearEqual(d3w[2 * 3 + 0], f2_x2, eps, eps);
//
// check second order results
ok &= NearEqual(d3w[0 * 3 + 1], f2_x2_x0, eps, eps);
ok &= NearEqual(d3w[1 * 3 + 1], f2_x2_x1, eps, eps);
ok &= NearEqual(d3w[2 * 3 + 1], f2_x2_x2, eps, eps);
// -------------------------------------------------------------------
scalar f2_x2_x2_x0 = - 0.5 * f2_x2_x0 / (f2 * f2);
f2_x2_x2_x0 += f2_x2 * f2_x0 / (f2 * f2 * f2);
ok &= NearEqual(d3w[0 * 3 + 2], 0.5 * f2_x2_x2_x0, eps, eps);
scalar f2_x2_x2_x1 = - 0.5 * f2_x2_x1 / (f2 * f2);
f2_x2_x2_x1 += f2_x2 * f2_x1 / (f2 * f2 * f2);
ok &= NearEqual(d3w[1 * 3 + 2], 0.5 * f2_x2_x2_x1, eps, eps);
scalar f2_x2_x2_x2 = - 0.5 * f2_x2_x2 / (f2 * f2);
f2_x2_x2_x2 += f2_x2 * f2_x2 / (f2 * f2 * f2);
ok &= NearEqual(d3w[2 * 3 + 2], 0.5 * f2_x2_x2_x2, eps, eps);
return ok;
}
Input File: example/atomic/eigen_cholesky.cpp