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Up-> CppAD AD VecAD vec_ad.cpp

vec_ad.cpp

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# 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;
}