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// SPDX-License-Identifier: EPL-2.0 OR GPL-2.0-or-later
// SPDX-FileCopyrightText: Bradley M. Bell <bradbell@seanet.com>
// SPDX-FileContributor: 2003-22 Bradley M. Bell
// ----------------------------------------------------------------------------
/*
{xrst_begin vec_ad.cpp}
AD Vectors that Record Index Operations: Example and Test
#########################################################
{xrst_literal
// BEGIN C++
// END C++
}
{xrst_end vec_ad.cpp}
*/
// BEGIN C++
# include <cppad/cppad.hpp>
# include <cassert>
namespace {
// return the vector x that solves the following linear system
// a[0] * x[0] + a[1] * x[1] = b[0]
// a[2] * x[0] + a[3] * x[1] = b[1]
// in a way that will record pivot operations on the AD<double> tape
typedef CPPAD_TESTVECTOR(CppAD::AD<double>) Vector;
Vector Solve(const Vector &a , const Vector &b)
{ using namespace CppAD;
assert(a.size() == 4 && b.size() == 2);
// copy the vector b into the VecAD object B
VecAD<double> B(2);
AD<double> u;
for(u = 0; u < 2; u += 1.)
B[u] = b[ size_t( Integer(u) ) ];
// copy the matrix a into the VecAD object A
VecAD<double> A(4);
for(u = 0; u < 4; u += 1.)
A[u] = a [ size_t( Integer(u) ) ];
// tape AD operation sequence that determines the row of A
// with maximum absolute element in column zero
AD<double> zero(0), one(1);
AD<double> rmax = CondExpGt(fabs(a[0]), fabs(a[2]), zero, one);
// divide row rmax by A(rmax, 0)
A[rmax * 2 + 1] = A[rmax * 2 + 1] / A[rmax * 2 + 0];
B[rmax] = B[rmax] / A[rmax * 2 + 0];
A[rmax * 2 + 0] = one;
// subtract A(other,0) times row A(rmax, *) from row A(other,*)
AD<double> other = one - rmax;
A[other * 2 + 1] = A[other * 2 + 1]
- A[other * 2 + 0] * A[rmax * 2 + 1];
B[other] = B[other]
- A[other * 2 + 0] * B[rmax];
A[other * 2 + 0] = zero;
// back substitute to compute the solution vector x.
// Note that the columns of A correspond to rows of x.
// Also note that A[rmax * 2 + 0] is equal to one.
CPPAD_TESTVECTOR(AD<double>) x(2);
x[1] = B[other] / A[other * 2 + 1];
x[0] = B[rmax] - A[rmax * 2 + 1] * x[1];
return x;
}
}
bool vec_ad(void)
{ bool ok = true;
using CppAD::AD;
using CppAD::NearEqual;
double eps99 = 99.0 * std::numeric_limits<double>::epsilon();
// domain space vector
size_t n = 4;
CPPAD_TESTVECTOR(double) x(n);
CPPAD_TESTVECTOR(AD<double>) X(n);
// 2 * identity matrix (rmax in Solve will be 0)
X[0] = x[0] = 2.; X[1] = x[1] = 0.;
X[2] = x[2] = 0.; X[3] = x[3] = 2.;
// declare independent variables and start tape recording
CppAD::Independent(X);
// define the vector b
CPPAD_TESTVECTOR(double) b(2);
CPPAD_TESTVECTOR(AD<double>) B(2);
B[0] = b[0] = 0.;
B[1] = b[1] = 1.;
// range space vector solves X * Y = b
size_t m = 2;
CPPAD_TESTVECTOR(AD<double>) Y(m);
Y = Solve(X, B);
// create f: X -> Y and stop tape recording
CppAD::ADFun<double> f(X, Y);
// By Cramer's rule:
// y[0] = [ b[0] * x[3] - x[1] * b[1] ] / [ x[0] * x[3] - x[1] * x[2] ]
// y[1] = [ x[0] * b[1] - b[0] * x[2] ] / [ x[0] * x[3] - x[1] * x[2] ]
double den = x[0] * x[3] - x[1] * x[2];
double dsq = den * den;
double num0 = b[0] * x[3] - x[1] * b[1];
double num1 = x[0] * b[1] - b[0] * x[2];
// check value
ok &= NearEqual(Y[0] , num0 / den, eps99, eps99);
ok &= NearEqual(Y[1] , num1 / den, eps99, eps99);
// forward computation of partials w.r.t. x[0]
CPPAD_TESTVECTOR(double) dx(n);
CPPAD_TESTVECTOR(double) dy(m);
dx[0] = 1.; dx[1] = 0.;
dx[2] = 0.; dx[3] = 0.;
dy = f.Forward(1, dx);
ok &= NearEqual(dy[0], 0. - num0 * x[3] / dsq, eps99, eps99);
ok &= NearEqual(dy[1], b[1] / den - num1 * x[3] / dsq, eps99, eps99);
// compute the solution for a new x matrix such that pivioting
// on the original rmax row would divide by zero
CPPAD_TESTVECTOR(double) y(m);
x[0] = 0.; x[1] = 2.;
x[2] = 2.; x[3] = 0.;
// new values for Cramer's rule
den = x[0] * x[3] - x[1] * x[2];
dsq = den * den;
num0 = b[0] * x[3] - x[1] * b[1];
num1 = x[0] * b[1] - b[0] * x[2];
// check values
y = f.Forward(0, x);
ok &= NearEqual(y[0] , num0 / den, eps99, eps99);
ok &= NearEqual(y[1] , num1 / den, eps99, eps99);
// forward computation of partials w.r.t. x[1]
dx[0] = 0.; dx[1] = 1.;
dx[2] = 0.; dx[3] = 0.;
dy = f.Forward(1, dx);
ok &= NearEqual(dy[0],-b[1] / den + num0 * x[2] / dsq, eps99, eps99);
ok &= NearEqual(dy[1], 0. + num1 * x[2] / dsq, eps99, eps99);
// reverse computation of derivative of y[0] w.r.t x
CPPAD_TESTVECTOR(double) w(m);
CPPAD_TESTVECTOR(double) dw(n);
w[0] = 1.; w[1] = 0.;
dw = f.Reverse(1, w);
ok &= NearEqual(dw[0], 0. - num0 * x[3] / dsq, eps99, eps99);
ok &= NearEqual(dw[1],-b[1] / den + num0 * x[2] / dsq, eps99, eps99);
ok &= NearEqual(dw[2], 0. + num0 * x[1] / dsq, eps99, eps99);
ok &= NearEqual(dw[3], b[0] / den - num0 * x[0] / dsq, eps99, eps99);
return ok;
}
// END C++
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