\(\newcommand{\W}[1]{ \; #1 \; }\) \(\newcommand{\R}[1]{ {\rm #1} }\) \(\newcommand{\B}[1]{ {\bf #1} }\) \(\newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} }\) \(\newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} }\) \(\newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} }\) \(\newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }\)
conj_grad.cpp¶
View page sourceDifferentiate 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 \(A \in \B{R}^{n \times n}\), a vector \(b \in \B{R}^n\), and tolerance \(\varepsilon\), the conjugate gradient algorithm finds an \(x \in \B{R}^n\) such that \(\| A x - b \|^2 / n \leq \varepsilon^2\) (or it terminates at a specified maximum number of iterations).
Input:
The matrix \(A \in \B{R}^{n \times n}\), the vector \(b \in \B{R}^n\), a tolerance \(\varepsilon \geq 0\), a maximum number of iterations \(m\), and the initial approximate solution \(x^0 \in \B{R}^n\) (can use zero for \(x^0\)).
Initialize:
\(g^0 = A * x^0 - b\), \(d^0 = - g^0\), \(s_0 = ( g^0 )^\R{T} g^0\), \(k = 0\).
Convergence Check:
if \(k = m\) or \(\sqrt{ s_k / n } < \varepsilon\), return \(k\) as the number of iterations and \(x^k\) as the approximate solution.
Next \(x\):
\(\mu_{k+1} = s_k / [ ( d^k )^\R{T} A d^k ]\), \(x^{k+1} = x^k + \mu_{k+1} d^k\).
Next \(g\):
\(g^{k+1} = g^k + \mu_{k+1} A d^k\), \(s_{k+1} = ( g^{k+1} )^\R{T} g^{k+1}\).
Next \(d\):
\(d^{k+1} = - g^k + ( s_{k+1} / s_k ) d^k\).
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 iterations.
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;
}