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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/sparse/conj_grad.cpp