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Differentiate Conjugate Gradient Algorithm: Example and Test

Purpose
The conjugate gradient algorithm is sparse linear solver and a good example where checkpointing can be applied (for each iteration). This example is a preliminary version of a new library routine for the conjugate gradient algorithm.

Algorithm
Given a positive definite matrix A ∈ R n × n , a vector b ∈ R n , and tolerance ε , the conjugate gradient algorithm finds an x ∈ R n such that ‖ A x - b ‖ 2 / n ≤ ε 2 (or it terminates at a specified maximum number of iterations).
  1. Input:
    The matrix A ∈ R n × n , the vector b ∈ R n , a tolerance ε ≥ 0 , a maximum number of iterations m , and the initial approximate solution x 0 ∈ R n (can use zero for x 0 ).
  2. Initialize:
    g 0 = A * x 0 - b , d 0 = - g 0 , s 0 = ( g 0 ) T g 0 , k = 0 .
  3. Convergence Check:
    if k = m or s k / n < ε , return k as the number of iterations and x k as the approximate solution.
  4. Next x :
    μ k + 1 = s k / [ ( d k ) T A d k ] , x k + 1 = x k + μ k + 1 d k .
  5. Next g :
    g k + 1 = g k + μ k + 1 A d k , s k + 1 = ( g k + 1 ) T g k + 1 .
  6. Next d :
    d k + 1 = - g k + ( s k + 1 / s k ) d k .
  7. Iterate:
    k = k + 1 , goto Convergence Check. 
 
# include <cppad/cppad.hpp>
# include <cstdlib>
# include <cmath>

namespace { // Begin empty namespace
	using CppAD::AD;

	// A simple matrix multiply c = a * b , where a has n columns 
	// and b has n rows. This should be changed to a function so that
	// it can efficiently handle the case were A is large and sparse.
	template <class Vector> // a simple vector class
	void mat_mul(size_t n, const Vector& a, const Vector& b, Vector& c)
	{	typedef typename Vector::value_type scalar;

		size_t m, p;
		m = a.size() / n;
		p = b.size() / n;

		assert( m * n == a.size() );
		assert( n * p == b.size() );
		assert( m * p == c.size() );

		size_t i, j, k, ij;
		for(i = 0; i < m; i++)
		{	for(j = 0; j < p; j++)
			{	ij    = i * p + j;
				c[ij] = scalar(0);
				for(k = 0; k < n; k++)
					c[ij] = c[ij] + a[i * m + k] * b[k * p + j];
			}
		}
		return;
	}

	// Solve A * x == b to tolerance epsilon or terminate at m interations.
	template <class Vector> // a simple vector class
	size_t conjugate_gradient(
		size_t         m       , // input
		double         epsilon , // input
		const Vector&  A       , // input
		const Vector&  b       , // input
		Vector&        x       ) // input / output
	{	typedef typename Vector::value_type scalar;
		scalar mu, s_previous;
		size_t i, k;

		size_t n = x.size();
		assert( A.size() == n * n );
		assert( b.size() == n );

		Vector g(n), d(n), s(1), Ad(n), dAd(1);

		// g = A * x
		mat_mul(n, A, x, g);
		for(i = 0; i < n; i++)
		{	// g = A * x - b
			g[i] = g[i] - b[i];

			// d = - g
			d[i] = -g[i];
		}
		// s = g^T * g
		mat_mul(n, g, g, s);

		for(k = 0; k < m; k++)
		{	s_previous = s[0];
			if( s_previous < epsilon )
				return k;

			// Ad = A * d
			mat_mul(n, A, d, Ad);

			// dAd = d^T * A * d
			mat_mul(n, d, Ad, dAd);

			// mu = s / d^T * A * d
			mu = s_previous / dAd[0];

			// g = g + mu * A * d 
			for(i = 0; i < n; i++)
			{	x[i] = x[i] + mu * d[i];
				g[i] = g[i] + mu * Ad[i];
			}

			// s = g^T * g
			mat_mul(n, g, g, s);

			// d = - g + (s / s_previous) * d
			for(i = 0; i < n; i++)
				d[i] = - g[i] + ( s[0] / s_previous) * d[i];
		}
		return m;
	}

} // End empty namespace

bool conj_grad(void)
{	bool ok = true;

	// ----------------------------------------------------------------------
	// Setup
	// ----------------------------------------------------------------------
	using CppAD::AD;
	using CppAD::NearEqual;
	using CppAD::vector;
	using std::cout;
	using std::endl;
	size_t i, j;


	// size of the vectors  
	size_t n  = 40;
	vector<double> D(n * n), Dt(n * n), A(n * n), x(n), b(n), c(n);
	vector< AD<double> > a_A(n * n), a_x(n), a_b(n);

	// D = diagonally dominant matrix
	// c = vector of ones
	for(i = 0; i < n; i++)
	{	c[i] = 1.;
		double sum = 0;
		for(j = 0; j < n; j++) if( i != j )
		{	D[ i * n + j ] = std::rand() / double(RAND_MAX);
			Dt[j * n + i ] = D[i * n + j ];
			sum           += D[i * n + j ];
		}
		Dt[ i * n + i ] = D[ i * n + i ] = sum * 1.1;
	}

	// A = D^T * D
	mat_mul(n, Dt, D, A);

	// b = D^T * c
	mat_mul(n, Dt, c, b);

	// copy from double to AD<double>
	for(i = 0; i < n; i++)
	{	a_b[i] = b[i];
		for(j = 0; j < n; j++)
			a_A[ i * n + j ] = A[ i * n + j ];
	}

	// ---------------------------------------------------------------------
	// Record the function f : b -> x
	// ---------------------------------------------------------------------
	// Make b the independent variable vector
	Independent(a_b);

	// Solve A * x = b using conjugate gradient method
	double epsilon = 1e-7;
	for(i = 0; i < n; i++)
			a_x[i] = AD<double>(0);
	size_t m = n + 1;
	size_t k = conjugate_gradient(m, epsilon, a_A, a_b, a_x);

	// create f_cg: b -> x and stop tape recording
	CppAD::ADFun<double> f(a_b, a_x);

	// ---------------------------------------------------------------------
	// Check for correctness
	// ---------------------------------------------------------------------

	// conjugate gradient should converge with in n iterations
	ok &= (k <= n);

	// accuracy to which we expect values to agree
	double delta = 10. * epsilon * std::sqrt( double(n) );

	// copy x from AD<double> to double
	for(i = 0; i < n; i++)
		x[i] = Value( a_x[i] );

	// check c = A * x
	mat_mul(n, A, x, c);
	for(i = 0; i < n; i++)
		ok &= NearEqual(c[i] , b[i],  delta , delta);

	// forward computation of partials w.r.t. b[0]
	vector<double> db(n), dx(n);
	for(j = 0; j < n; j++)
		db[j] = 0.;
	db[0] = 1.;

	// check db = A * dx 
	delta = 5. * delta;
	dx = f.Forward(1, db);
	mat_mul(n, A, dx, c);
	for(i = 0; i < n; i++)
		ok   &= NearEqual(c[i], db[i], delta, delta);

	return ok;
}
Input File: example/conj_grad.cpp