bool link_sparse_hessian(
size_t size ,
size_t repeat ,
CppAD::vector<double>& x ,
const CppAD::vector<size_t>& row ,
const CppAD::vector<size_t>& col ,
CppAD::vector<double>& hessian )
{
// -----------------------------------------------------
// setup
typedef vector<double> DblVector;
typedef vector< std::set<size_t> > SetVector;
typedef CppAD::AD<double> ADScalar;
typedef vector<ADScalar> ADVector;
size_t i, j, k;
size_t order = 0; // derivative order corresponding to function
size_t m = 1; // number of dependent variables
size_t n = size; // number of independent variables
size_t K = row.size(); // number of non-zeros in lower triangle
ADVector a_x(n); // AD domain space vector
ADVector a_y(m); // AD range space vector
DblVector w(m); // double range space vector
DblVector hes(K); // non-zeros in lower triangle
CppAD::ADFun<double> f; // AD function object
// weights for hessian calculation (only one component of f)
w[0] = 1.;
// use the unspecified fact that size is non-decreasing between calls
static size_t previous_size = 0;
bool print = (repeat > 1) & (previous_size != size);
previous_size = size;
// declare sparsity pattern
# if USE_SET_SPARSITY
SetVector sparsity(n);
# else
typedef vector<bool> BoolVector;
BoolVector sparsity(n * n);
# endif
// initialize all entries as zero
for(i = 0; i < n; i++)
{ for(j = 0; j < n; j++)
hessian[ i * n + j] = 0.;
}
// ------------------------------------------------------
extern bool global_retape;
if( global_retape) while(repeat--)
{ // choose a value for x
CppAD::uniform_01(n, x);
for(j = 0; j < n; j++)
a_x[j] = x[j];
// declare independent variables
Independent(a_x);
// AD computation of f(x)
CppAD::sparse_hes_fun<ADScalar>(n, a_x, row, col, order, a_y);
// create function object f : X -> Y
f.Dependent(a_x, a_y);
extern bool global_optimize;
if( global_optimize )
{ print_optimize(f, print, "cppad_sparse_hessian_optimize", size);
print = false;
}
// calculate the Hessian sparsity pattern for this function
calc_sparsity(sparsity, f);
// structure that holds some of work done by SparseHessian
CppAD::sparse_hessian_work work;
// calculate this Hessian at this x
f.SparseHessian(x, w, sparsity, row, col, hes, work);
for(k = 0; k < K; k++)
{ hessian[ row[k] * n + col[k] ] = hes[k];
hessian[ col[k] * n + row[k] ] = hes[k];
}
}
else
{ // choose a value for x
CppAD::uniform_01(n, x);
for(j = 0; j < n; j++)
a_x[j] = x[j];
// declare independent variables
Independent(a_x);
// AD computation of f(x)
CppAD::sparse_hes_fun<ADScalar>(n, a_x, row, col, order, a_y);
// create function object f : X -> Y
f.Dependent(a_x, a_y);
extern bool global_optimize;
if( global_optimize )
{ print_optimize(f, print, "cppad_sparse_hessian_optimize", size);
print = false;
}
// calculate the Hessian sparsity pattern for this function
calc_sparsity(sparsity, f);
// declare structure that holds some of work done by SparseHessian
CppAD::sparse_hessian_work work;
while(repeat--)
{ // choose a value for x
CppAD::uniform_01(n, x);
// calculate sparsity at this x
f.SparseHessian(x, w, sparsity, row, col, hes, work);
for(k = 0; k < K; k++)
{ hessian[ row[k] * n + col[k] ] = hes[k];
hessian[ col[k] * n + row[k] ] = hes[k];
}
}
}
return true;
}
/* $$
$head Use Atomic Function$$
$codep */
bool get_started(void)
{ bool ok = true;
using CppAD::AD;
using CppAD::NearEqual;
double eps = 10. * CppAD::numeric_limits<double>::epsilon();
/* $$
$subhead Constructor$$
$codep */
// Create the atomic get_started object
atomic_get_started afun("atomic_get_started");
/* $$
$subhead Recording$$
$codep */
// Create the function f(x)
//
// domain space vector
size_t n = 1;
double x0 = 0.5;
vector< AD<double> > ax(n);
ax[0] = x0;
// declare independent variables and start tape recording
CppAD::Independent(ax);
// range space vector
size_t m = 1;
vector< AD<double> > ay(m);
// call user function and store get_started(x) in au[0]
vector< AD<double> > au(m);
afun(ax, au); // u = 1 / x
// now use AD division to invert to invert the operation
ay[0] = 1.0 / au[0]; // y = 1 / u = x
// create f: x -> y and stop tape recording
CppAD::ADFun<double> f;
f.Dependent (ax, ay); // f(x) = x
/* $$
$subhead forward$$
$codep */
// check function value
double check = x0;
ok &= NearEqual( Value(ay[0]) , check, eps, eps);
// check zero order forward mode
size_t p;
vector<double> x_p(n), y_p(m);
p = 0;
x_p[0] = x0;
y_p = f.Forward(p, x_p);
ok &= NearEqual(y_p[0] , check, eps, eps);
return ok;
}
bool sparse_hessian(void)
{ bool ok = true;
using CppAD::AD;
using CppAD::NearEqual;
size_t i, j, k, ell;
typedef CPPAD_TESTVECTOR(AD<double>) a_vector;
typedef CPPAD_TESTVECTOR(double) d_vector;
typedef CPPAD_TESTVECTOR(size_t) i_vector;
typedef CPPAD_TESTVECTOR(bool) b_vector;
typedef CPPAD_TESTVECTOR(std::set<size_t>) s_vector;
double eps = 10. * CppAD::numeric_limits<double>::epsilon();
// domain space vector
size_t n = 12; // must be greater than or equal 3; see n_sweep below
a_vector a_x(n);
for(j = 0; j < n; j++)
a_x[j] = AD<double> (0);
// declare independent variables and starting recording
CppAD::Independent(a_x);
// range space vector
size_t m = 1;
a_vector a_y(m);
a_y[0] = a_x[0]*a_x[1];
for(j = 0; j < n; j++)
a_y[0] += a_x[j] * a_x[j] * a_x[j];
// create f: x -> y and stop tape recording
// (without executing zero order forward calculation)
CppAD::ADFun<double> f;
f.Dependent(a_x, a_y);
// new value for the independent variable vector, and weighting vector
d_vector w(m), x(n);
for(j = 0; j < n; j++)
x[j] = double(j);
w[0] = 1.0;
// vector used to check the value of the hessian
d_vector check(n * n);
for(ell = 0; ell < n * n; ell++)
check[ell] = 0.0;
ell = 0 * n + 1;
check[ell] = 1.0;
ell = 1 * n + 0;
check[ell] = 1.0 ;
for(j = 0; j < n; j++)
{ ell = j * n + j;
check[ell] = 6.0 * x[j];
}
// -------------------------------------------------------------------
// second derivative of y[0] w.r.t x
d_vector hes(n * n);
hes = f.SparseHessian(x, w);
for(ell = 0; ell < n * n; ell++)
ok &= NearEqual(w[0] * check[ell], hes[ell], eps, eps );
// --------------------------------------------------------------------
// example using vectors of bools to compute sparsity pattern for Hessian
b_vector r_bool(n * n);
for(i = 0; i < n; i++)
{ for(j = 0; j < n; j++)
r_bool[i * n + j] = false;
r_bool[i * n + i] = true;
}
f.ForSparseJac(n, r_bool);
//
b_vector s_bool(m);
for(i = 0; i < m; i++)
s_bool[i] = w[i] != 0;
b_vector p_bool = f.RevSparseHes(n, s_bool);
hes = f.SparseHessian(x, w, p_bool);
for(ell = 0; ell < n * n; ell++)
ok &= NearEqual(w[0] * check[ell], hes[ell], eps, eps );
// --------------------------------------------------------------------
// example using vectors of sets to compute sparsity pattern for Hessian
s_vector r_set(n);
for(i = 0; i < n; i++)
r_set[i].insert(i);
f.ForSparseJac(n, r_set);
//
s_vector s_set(m);
for(i = 0; i < m; i++)
if( w[i] != 0. )
s_set[0].insert(i);
s_vector p_set = f.RevSparseHes(n, s_set);
// example passing sparsity pattern to SparseHessian
hes = f.SparseHessian(x, w, p_set);
for(ell = 0; ell < n * n; ell++)
ok &= NearEqual(w[0] * check[ell], hes[ell], eps, eps );
// --------------------------------------------------------------------
// use row and column indices to specify upper triangle of
// non-zero elements of Hessian
size_t K = n + 1;
i_vector row(K), col(K);
hes.resize(K);
k = 0;
for(j = 0; j < n; j++)
{ // diagonal of Hessian
row[k] = j;
col[k] = j;
k++;
}
// only off diagonal non-zero elemenet in upper triangle
row[k] = 0;
col[k] = 1;
k++;
ok &= k == K;
CppAD::sparse_hessian_work work;
// can use p_set or p_bool.
size_t n_sweep = f.SparseHessian(x, w, p_set, row, col, hes, work);
for(k = 0; k < K; k++)
{ ell = row[k] * n + col[k];
ok &= NearEqual(w[0] * check[ell], hes[k], eps, eps );
}
ok &= n_sweep == 2;
// now recompute at a different x and w (using work from previous call
w[0] = 2.0;
x[1] = 0.5;
ell = 1 * n + 1;
check[ell] = 6.0 * x[1];
s_vector not_used;
n_sweep = f.SparseHessian(x, w, not_used, row, col, hes, work);
for(k = 0; k < K; k++)
{ ell = row[k] * n + col[k];
ok &= NearEqual(w[0] * check[ell], hes[k], eps, eps );
}
ok &= n_sweep == 2;
return ok;
}
/* %$$
$head Use Atomic Function$$
$srccode%cpp% */
bool norm_sq(void)
{ bool ok = true;
using CppAD::AD;
using CppAD::NearEqual;
double eps = 10. * CppAD::numeric_limits<double>::epsilon();
/* %$$
$subhead Constructor$$
$srccode%cpp% */
// --------------------------------------------------------------------
// Create the atomic reciprocal object
atomic_norm_sq afun("atomic_norm_sq");
/* %$$
$subhead Recording$$
$srccode%cpp% */
// Create the function f(x)
//
// domain space vector
size_t n = 2;
double x0 = 0.25;
double x1 = 0.75;
vector< AD<double> > ax(n);
ax[0] = x0;
ax[1] = x1;
// declare independent variables and start tape recording
CppAD::Independent(ax);
// range space vector
size_t m = 1;
vector< AD<double> > ay(m);
// call user function and store norm_sq(x) in au[0]
afun(ax, ay); // y_0 = x_0 * x_0 + x_1 * x_1
// create f: x -> y and stop tape recording
CppAD::ADFun<double> f;
f.Dependent (ax, ay);
/* %$$
$subhead forward$$
$srccode%cpp% */
// check function value
double check = x0 * x0 + x1 * x1;
ok &= NearEqual( Value(ay[0]) , check, eps, eps);
// check zero order forward mode
size_t q;
vector<double> x_q(n), y_q(m);
q = 0;
x_q[0] = x0;
x_q[1] = x1;
y_q = f.Forward(q, x_q);
ok &= NearEqual(y_q[0] , check, eps, eps);
// check first order forward mode
q = 1;
x_q[0] = 0.3;
x_q[1] = 0.7;
y_q = f.Forward(q, x_q);
check = 2.0 * x0 * x_q[0] + 2.0 * x1 * x_q[1];
ok &= NearEqual(y_q[0] , check, eps, eps);
/* %$$
$subhead reverse$$
$srccode%cpp% */
// first order reverse mode
q = 1;
vector<double> w(m), dw(n * q);
w[0] = 1.;
dw = f.Reverse(q, w);
check = 2.0 * x0;
ok &= NearEqual(dw[0] , check, eps, eps);
check = 2.0 * x1;
ok &= NearEqual(dw[1] , check, eps, eps);
/* %$$
$subhead for_sparse_jac$$
$srccode%cpp% */
// forward mode sparstiy pattern
size_t p = n;
CppAD::vectorBool r1(n * p), s1(m * p);
r1[0] = true; r1[1] = false; // sparsity pattern identity
r1[2] = false; r1[3] = true;
//
s1 = f.ForSparseJac(p, r1);
ok &= s1[0] == true; // f[0] depends on x[0]
ok &= s1[1] == true; // f[0] depends on x[1]
/* %$$
$subhead rev_sparse_jac$$
$srccode%cpp% */
// reverse mode sparstiy pattern
q = m;
CppAD::vectorBool s2(q * m), r2(q * n);
s2[0] = true; // compute sparsity pattern for f[0]
//
r2 = f.RevSparseJac(q, s2);
ok &= r2[0] == true; // f[0] depends on x[0]
ok &= r2[1] == true; // f[0] depends on x[1]
/* %$$
$subhead rev_sparse_hes$$
$srccode%cpp% */
// Hessian sparsity (using previous ForSparseJac call)
CppAD::vectorBool s3(m), h(p * n);
s3[0] = true; // compute sparsity pattern for f[0]
//
h = f.RevSparseHes(p, s3);
ok &= h[0] == true; // partial of f[0] w.r.t. x[0],x[0] is non-zero
ok &= h[1] == false; // partial of f[0] w.r.t. x[0],x[1] is zero
ok &= h[2] == false; // partial of f[0] w.r.t. x[1],x[0] is zero
ok &= h[3] == true; // partial of f[0] w.r.t. x[1],x[1] is non-zero
//
return ok;
}
/* %$$
$head Test Atomic Function$$
$srccode%cpp% */
bool set_sparsity(void)
{ bool ok = true;
using CppAD::AD;
using CppAD::NearEqual;
double eps = 10. * std::numeric_limits<double>::epsilon();
/* %$$
$subhead Constructor$$
$srccode%cpp% */
// Create the atomic get_started object
atomic_set_sparsity afun("atomic_set_sparsity");
/* %$$
$subhead Recording$$
$srccode%cpp% */
size_t n = 3;
size_t m = 2;
vector< AD<double> > ax(n), ay(m);
for(size_t j = 0; j < n; j++)
ax[j] = double(j + 1);
// declare independent variables and start tape recording
CppAD::Independent(ax);
// call atomic function
afun(ax, ay);
// create f: x -> y and stop tape recording
CppAD::ADFun<double> f;
f.Dependent (ax, ay); // f(x) = x
// check function value
ok &= NearEqual(ay[0] , ax[2], eps, eps);
ok &= NearEqual(ay[1] , ax[0] * ax[1], eps, eps);
/* %$$
$subhead for_sparse_jac$$
$srccode%cpp% */
// correct Jacobian result
set_vector check_s(m);
check_s[0].insert(2);
check_s[1].insert(0);
check_s[1].insert(1);
// compute and test forward mode
{ set_vector r(n), s(m);
for(size_t i = 0; i < n; i++)
r[i].insert(i);
s = f.ForSparseJac(n, r);
for(size_t i = 0; i < m; i++)
ok &= s[i] == check_s[i];
}
/* %$$
$subhead rev_sparse_jac$$
$srccode%cpp% */
// compute and test reverse mode
{ set_vector r(m), s(m);
for(size_t i = 0; i < m; i++)
r[i].insert(i);
s = f.RevSparseJac(m, r);
for(size_t i = 0; i < m; i++)
ok &= s[i] == check_s[i];
}
/* %$$
$subhead for_sparse_hes$$
$srccode%cpp% */
// correct Hessian result
set_vector check_h(n);
check_h[0].insert(1);
check_h[1].insert(0);
// compute and test forward mode
{ set_vector r(1), s(1), h(n);
for(size_t i = 0; i < m; i++)
s[0].insert(i);
for(size_t j = 0; j < n; j++)
r[0].insert(j);
h = f.ForSparseHes(r, s);
for(size_t i = 0; i < n; i++)
ok &= h[i] == check_h[i];
}
/* %$$
$subhead rev_sparse_hes$$
Note the previous call to $code ForSparseJac$$ above
stored its results in $icode f$$.
$srccode%cpp% */
// compute and test reverse mode
{ set_vector s(1), h(n);
for(size_t i = 0; i < m; i++)
s[0].insert(i);
h = f.RevSparseHes(n, s);
for(size_t i = 0; i < n; i++)
ok &= h[i] == check_h[i];
}
/* %$$
$subhead Test Result$$
$srccode%cpp% */
return ok;
}
bool sub_sparse_hes(void)
{ bool ok = true;
using CppAD::AD;
typedef AD<double> adouble;
typedef AD<adouble> a2double;
typedef vector< std::set<size_t> > pattern;
double eps = 10. * std::numeric_limits<double>::epsilon();
size_t i, j;
// start recording with x = (u , v)
size_t nu = 10;
size_t nv = 5;
size_t n = nu + nv;
vector<adouble> ax(n);
for(j = 0; j < n; j++)
ax[j] = adouble(j + 2);
CppAD::Independent(ax);
// extract u as independent variables
vector<a2double> a2u(nu);
for(j = 0; j < nu; j++)
a2u[j] = a2double(j + 2);
CppAD::Independent(a2u);
// extract v as parameters
vector<a2double> a2v(nv);
for(j = 0; j < nv; j++)
a2v[j] = ax[nu+j];
// record g(u)
vector<a2double> a2y(1);
a2y[0] = f(a2u, a2v);
CppAD::ADFun<adouble> g;
g.Dependent(a2u, a2y);
// compue sparsity pattern for Hessian of g(u)
pattern r(nu), s(1);
for(j = 0; j < nu; j++)
r[j].insert(j);
g.ForSparseJac(nu, r);
s[0].insert(0);
pattern p = g.RevSparseHes(nu, s);
// Row and column indices for non-zeros in lower triangle of Hessian
vector<size_t> row, col;
for(i = 0; i < nu; i++)
{ std::set<size_t>::const_iterator itr;
for(itr = p[i].begin(); itr != p[i].end(); itr++)
{ j = *itr;
if( j <= i )
{ row.push_back(i);
col.push_back(j);
}
}
}
size_t K = row.size();
CppAD::sparse_hessian_work work;
vector<adouble> au(nu), ahes(K), aw(1);
aw[0] = 1.0;
for(j = 0; j < nu; j++)
au[j] = ax[j];
size_t n_sweep = g.SparseHessian(au, aw, p, row, col, ahes, work);
// The Hessian w.r.t u is diagonal
ok &= n_sweep == 1;
// record H(u, v) = Hessian of f w.r.t u
CppAD::ADFun<double> H(ax, ahes);
// remove unecessary operations
H.optimize();
// Now evaluate the Hessian at a particular value for u, v
vector<double> u(nu), v(nv), x(n);
for(j = 0; j < n; j++)
x[j] = double(j + 2);
vector<double> hes = H.Forward(0, x);
// Now check the Hessian
double sum_v = 0.0;
for(j = 0; j < nv; j++)
sum_v += x[nu + j];
for(size_t k = 0; k < K; k++)
{ i = row[k];
j = col[k];
ok &= i == j;
double check = sum_v * x[i];
ok &= CppAD::NearEqual(hes[k], check, eps, eps);
}
return ok;
}
void mexFunction(int nlhs, mxArray *plhs[],int nrhs, const mxArray *prhs[])
{
double *v, *x, sigma, *lambda, *pr;
char *mode; int imode;
//Check Inputs
if(nrhs < 1) {
printInfo();
return;
}
if(mxIsEmpty(prhs[0]) || !mxIsChar(prhs[0])) {
mexErrMsgTxt("The mode must be a string!");
return;
}
//If we have x, check it
if(nrhs > 1) {
if(!mxIsEmpty(prhs[1])) {
if(mxIsClass(prhs[1],"scipvar") || mxIsClass(prhs[1],"barvec")) {
mexErrMsgTxt("SCIP and BARON cannot be used with this callback function - please specify 'mcode' via symbset as the cbmode.");
return;
}
if(!mxIsDouble(prhs[1]) || mxIsComplex(prhs[1]) || mxIsSparse(prhs[1])) {
mexErrMsgTxt("The input vector must be a dense real double vector!");
return;
}
}
else {
mexErrMsgTxt("The input vector must be a dense real double vector!");
return;
}
//Check x input size
if(mxGetNumberOfElements(prhs[1]) != getNoVar()) {
mexErrMsgTxt("The input vector is not the right size!");
}
//Allocate memory, if required
if(xvec.empty())
xvec.resize(getNoVar());
//Get x and copy to xvec
x = mxGetPr(prhs[1]);
memcpy(&xvec[0],x,getNoVar()*sizeof(double));
}
//Determine input mode and setup return variable
mode = mxArrayToString(prhs[0]);
lower(mode);
if(!strcmp(mode,"obj")) {
imode = 0;
plhs[0] = mxCreateDoubleMatrix(1,1, mxREAL);
v = mxGetPr(plhs[0]);
}
else if(!strcmp(mode,"grad")) {
imode = 1;
plhs[0] = mxCreateDoubleMatrix(1,getNoVar(), mxREAL);
v = mxGetPr(plhs[0]);
}
else if(!strcmp(mode,"con")) {
imode = 2;
plhs[0] = mxCreateDoubleMatrix(getNoCon(),1, mxREAL);
v = mxGetPr(plhs[0]);
}
else if(!strcmp(mode,"jac")) {
imode = 3;
//Can't allocate here until we know sparsity pattern
}
else if(!strcmp(mode,"jacstr")) {
imode = 4;
//Can't allocate here until we know sparsity pattern
}
else if(!strcmp(mode,"hess")) {
if(nrhs < 4) {
mexErrMsgTxt("You must supply the callback mode, input vector, sigma and lambda for Hessian Evaluations.");
return;
}
//Check length of Sigma
if(mxIsEmpty(prhs[2]) || !mxIsDouble(prhs[2]) || mxIsComplex(prhs[2]) || mxGetNumberOfElements(prhs[2]) != 1)
mexErrMsgTxt("Sigma must be a real, double scalar.");
//Check length of Lambda
if(!mxIsDouble(prhs[3]) || mxIsComplex(prhs[3]) || mxIsSparse(prhs[3]) || mxGetNumberOfElements(prhs[3]) != getNoCon())
mexErrMsgTxt("Lambda must be a real, double, dense vector with ncon elements.");
//Get Sigma, Lambda
sigma = *mxGetPr(prhs[2]);
lambda = mxGetPr(prhs[3]);
imode = 5;
//Can't allocate here until we know sparsity pattern
}
else if(!strcmp(mode,"hstr")) {
imode = 6;
//Can't allocate here until we know sparsity pattern
}
else
mexErrMsgTxt("Unknown mode - options are 'obj', 'grad', 'con', 'jac', 'jacstr', 'hess' or 'hstr'");
mxFree(mode);
//Ensure we did have x for normal callbacks
if(imode != 4 && imode != 6 && nrhs < 2)
mexErrMsgTxt("You must supply the callback mode and input vector.");
//Call Req Callback
switch(imode)
{
case 0: //objective
*v = objective(xvec);
break;
case 1: //gradient
//Check if we have recorded the objective yet
if(obj.Memory()==0) { //new, tape operations
vector< CppAD::AD<double> > X(getNoVar());
memcpy(&X[0],x,getNoVar()*sizeof(double));
CppAD::Independent(X);
vector< CppAD::AD<double> > Y(1);
Y[0] = objective(X);
obj = CppAD::ADFun<double>(X, Y);
//obj.optimize();
mexAtExit(mexExit); //also register memory clear function
//mexPrintf("Evaluated Tape for Gradient\n");
}
//Evaluate "Jacobian" for gradient
memcpy(v,&(obj.Jacobian(xvec)[0]),getNoVar()*sizeof(double));
break;
case 2: //constraints
//Check if we have constraint memory yet
if(cvec.empty())
cvec.resize(getNoCon()); //allocate it
//Evaluate Constraints
constraints(xvec,cvec);
//Copy Out
memcpy(v,&cvec[0],getNoCon()*sizeof(double));
break;
case 3: //jacobian
case 4: //jacobian structure
//Check if we have recorded the constraints yet
if(con.Memory()==0){ //new, tape operations
vector< CppAD::AD<double> > X(getNoVar());
memcpy(&X[0],x,getNoVar()*sizeof(double));
CppAD::Independent(X);
vector< CppAD::AD<double> > Y(getNoCon());
constraints(X,Y);
con = CppAD::ADFun<double>(X, Y);
//con.optimize();
mexAtExit(mexExit); //also register memory clear function
//mexPrintf("Evaluated Tape for Jacobian\n");
}
//Check if we have the sparsity pattern yet
if(jstr.empty()) {
vector<set<size_t>> r(getNoVar());
for(size_t i = 0; i < getNoVar(); i++)
r[i].insert(i); //identity matrix
jstr.resize(getNoCon());
jstr = con.ForSparseJac(getNoVar(),r,true); //note transpose
//Determine nnzs
for(int i = 0; i < jstr.size(); i++)
nnzJac += jstr[i].size();
//Save ir, jc for jac
jir = (mwIndex*)mxCalloc(nnzJac,sizeof(mwIndex));
jjc = (mwIndex*)mxCalloc(getNoVar()+1,sizeof(mwIndex));
mexMakeMemoryPersistent(jir);
mexMakeMemoryPersistent(jjc);
jwork.clear(); //reset jacobian calculations
//Col starts
jjc[0] = 0;
for(int i = 1; i <= getNoVar(); i++)
jjc[i] = (mwIndex)(jjc[i-1] + jstr[i-1].size());
//Rows
size_t idx = 0;
for(int i = 0; i < jstr.size(); i++)
for (set<size_t>::iterator it=jstr[i].begin(); it!=jstr[i].end(); ++it)
jir[idx++] = (mwIndex)*it;
//Build missing triple so we can eval just sparse elements of Jac
jrow.resize(nnzJac);
jcol.resize(nnzJac);
idx = 0;
for(size_t i = 0; i < nnzJac; i++)
jrow[i] = jir[i];
for(size_t i = 0; i < getNoVar();i++)
for(size_t j = jjc[i]; j < jjc[i+1]; j++)
jcol[idx++] = i;
//Re-do with no transpose... (bad really...)
jstr = con.ForSparseJac(getNoVar(),r,false);
//mexPrintf("Determined Jac Sparsity Structure (%d nzs)\n",nnzJac);
}
//Create Sparse Return Matrix
plhs[0] = mxCreateSparse(getNoCon(),getNoVar(),nnzJac,mxREAL);
pr = mxGetPr(plhs[0]);
memcpy(mxGetIr(plhs[0]),jir,nnzJac*sizeof(mwIndex));
memcpy(mxGetJc(plhs[0]),jjc,(getNoVar()+1)*sizeof(mwIndex));
//If we want the sparsity pattern only, fill in return matrix with 1s
if(imode==4) {
for(int i = 0; i < nnzJac; i++)
pr[i] = 1.0;
}
//Else, evaluate sparse jacobian and return as sparse matrix
else {
//Check if we have jacobian memory yet
if(jac.empty())
jac.resize(nnzJac); //allocate it
//If ndec > ncon, use reverse mode
if(getNoVar() > getNoCon())
con.SparseJacobianReverse(xvec,jstr,jrow,jcol,jac,jwork);
//else use forward
else
con.SparseJacobianForward(xvec,jstr,jrow,jcol,jac,jwork);
//Copy out
memcpy(pr,&jac[0],nnzJac*sizeof(double));
}
break;
case 5: //hessian of the lagrangian
case 6: //hessian structure
//Check if we have recorded the objective+constraints yet
//Not sure if we can reuse ones we have done above??
if(lag.Memory()==0){ //new, tape operations
vector< CppAD::AD<double> > X(getNoVar());
memcpy(&X[0],x,getNoVar()*sizeof(double));
CppAD::Independent(X);
//Output Array
vector< CppAD::AD<double> > Y(1);
vector< CppAD::AD<double> > Yc(getNoCon());
Y[0] = objective(X); //eval objective
if(getNoCon() > 0)
constraints(X,Yc); //eval constraints
Yc.insert(Yc.begin(),Y.begin(),Y.end());
//Create ADFun
lag.Dependent(Yc);
//lag.optimize();
mexAtExit(mexExit); //also register memory clear function
//mexPrintf("Evaluated Tape for Hessian\n");
}
//Check if we have the sparsity pattern yet
if(hstr.empty()) {
//First eval jac structure (not sure why)
vector< std::set<size_t> > r(getNoVar());
for(size_t i = 0; i < getNoVar(); i++)
r[i].insert(i);
lag.ForSparseJac(getNoVar(), r);
//Now do Hessian structure
vector<set<size_t>> s(1);
for(size_t i = 0; i < getNoCon()+1; i++)
s[0].insert(i); //identity matrix
hstr.resize(getNoVar());
hstr = lag.RevSparseHes(getNoVar(),s);
//Determine total nnzs
for(int i = 0; i < hstr.size(); i++)
nnzHess += hstr[i].size();
//Determine nnzs in lower tri
for(int i = 0; i < hstr.size(); i++)
for (set<size_t>::iterator it=hstr[i].begin(); it!=hstr[i].end(); ++it)
if(*it >= i)
nnzHessLT++;
//Save ir, jc for jac
hir = (mwIndex*)mxCalloc(nnzHessLT,sizeof(mwIndex));
hjc = (mwIndex*)mxCalloc(getNoVar()+1,sizeof(mwIndex));
mexMakeMemoryPersistent(hir);
mexMakeMemoryPersistent(hjc);
hwork.clear(); //reset hessian calculations
//Col & Row Starts
size_t idx = 0;
for(int i = 0; i < hstr.size(); i++) {
hjc[i] = idx;
for (set<size_t>::iterator it=hstr[i].begin(); it!=hstr[i].end(); ++it)
if(*it >= i)
hir[idx++] = (mwIndex)*it;
}
hjc[getNoVar()] = nnzHessLT;
//Build missing triple so we can eval just sparse elements of Jac
hrow.resize(nnzHessLT);
hcol.resize(nnzHessLT);
idx = 0;
for(size_t i = 0; i < nnzHessLT; i++)
hrow[i] = hir[i];
for(size_t i = 0; i < getNoVar();i++)
for(size_t j = hjc[i]; j < hjc[i+1]; j++)
hcol[idx++] = i;
//mexPrintf("Determined Hess Sparsity Structure (%d nzs in tril)\n",nnzHessLT);
}
//Create Sparse Return Matrix
plhs[0] = mxCreateSparse(getNoVar(),getNoVar(),nnzHessLT,mxREAL);
pr = mxGetPr(plhs[0]);
memcpy(mxGetIr(plhs[0]),hir,nnzHessLT*sizeof(mwIndex));
memcpy(mxGetJc(plhs[0]),hjc,(getNoVar()+1)*sizeof(mwIndex));
//If we want the sparsity pattern only, fill in return matrix with 1s
if(imode==6) {
for(int i = 0; i < nnzHessLT; i++)
pr[i] = 1.0;
}
//Else, evaluate sparse hessian and return as sparse matrix
else {
//Check if we have hessian memory yet
if(hes.empty())
hes.resize(nnzHessLT); //allocate it
if(w.empty())
w.resize(1+getNoCon()); //allocate it
//Copy in Weights
w[0] = sigma;
for(int i = 0; i < getNoCon(); i++)
w[i+1] = lambda[i];
//If ndec > ncon, use reverse mode
lag.SparseHessian(xvec,w,hstr,hrow,hcol,hes,hwork);
//Copy out elements
memcpy(pr,&hes[0],nnzHessLT*sizeof(double));
}
break;
}
}
bool link_sparse_jacobian(
size_t size ,
size_t repeat ,
size_t m ,
const CppAD::vector<size_t>& row ,
const CppAD::vector<size_t>& col ,
CppAD::vector<double>& x ,
CppAD::vector<double>& jacobian ,
size_t& n_sweep )
{
if( global_atomic )
return false;
# ifndef CPPAD_COLPACK_SPEED
if( global_colpack )
return false;
# endif
// -----------------------------------------------------
// setup
typedef vector< std::set<size_t> > SetVector;
typedef CppAD::AD<double> ADScalar;
typedef CppAD::vector<ADScalar> ADVector;
size_t j;
size_t order = 0; // derivative order corresponding to function
size_t n = size; // number of independent variables
ADVector a_x(n); // AD domain space vector
ADVector a_y(m); // AD range space vector y = g(x)
CppAD::ADFun<double> f; // AD function object
// declare sparsity pattern
SetVector set_sparsity(m);
BoolVector bool_sparsity(m * n);
// ------------------------------------------------------
if( ! global_onetape ) while(repeat--)
{ // choose a value for x
CppAD::uniform_01(n, x);
for(j = 0; j < n; j++)
a_x[j] = x[j];
// declare independent variables
Independent(a_x);
// AD computation of f (x)
CppAD::sparse_jac_fun<ADScalar>(m, n, a_x, row, col, order, a_y);
// create function object f : X -> Y
f.Dependent(a_x, a_y);
if( global_optimize )
f.optimize();
// skip comparison operators
f.compare_change_count(0);
// calculate the Jacobian sparsity pattern for this function
if( global_boolsparsity )
calc_sparsity(bool_sparsity, f);
else
calc_sparsity(set_sparsity, f);
// structure that holds some of the work done by SparseJacobian
CppAD::sparse_jacobian_work work;
# ifdef CPPAD_COLPACK_SPEED
if( global_colpack )
work.color_method = "colpack";
# endif
// calculate the Jacobian at this x
// (use forward mode because m > n ?)
if( global_boolsparsity) n_sweep = f.SparseJacobianForward(
x, bool_sparsity, row, col, jacobian, work
);
else n_sweep = f.SparseJacobianForward(
x, set_sparsity, row, col, jacobian, work
);
}
else
{ // choose a value for x
CppAD::uniform_01(n, x);
for(j = 0; j < n; j++)
a_x[j] = x[j];
// declare independent variables
Independent(a_x);
// AD computation of f (x)
CppAD::sparse_jac_fun<ADScalar>(m, n, a_x, row, col, order, a_y);
// create function object f : X -> Y
f.Dependent(a_x, a_y);
if( global_optimize )
f.optimize();
// skip comparison operators
f.compare_change_count(0);
// calculate the Jacobian sparsity pattern for this function
if( global_boolsparsity )
calc_sparsity(bool_sparsity, f);
else
calc_sparsity(set_sparsity, f);
// structure that holds some of the work done by SparseJacobian
CppAD::sparse_jacobian_work work;
# ifdef CPPAD_COLPACK_SPEED
if( global_colpack )
work.color_method = "colpack";
# endif
while(repeat--)
{ // choose a value for x
CppAD::uniform_01(n, x);
// calculate the Jacobian at this x
// (use forward mode because m > n ?)
if( global_boolsparsity ) n_sweep = f.SparseJacobianForward(
x, bool_sparsity, row, col, jacobian, work
);
else n_sweep = f.SparseJacobianForward(
x, set_sparsity, row, col, jacobian, work
);
}
}
return true;
}
/* $$
$head Use Atomic Function$$
$codep */
bool reciprocal(void)
{ bool ok = true;
using CppAD::AD;
using CppAD::NearEqual;
double eps = 10. * CppAD::numeric_limits<double>::epsilon();
/* $$
$subhead Constructor$$
$codep */
// --------------------------------------------------------------------
// Create the atomic reciprocal object
atomic_reciprocal afun("atomic_reciprocal");
/* $$
$subhead Recording$$
$codep */
// Create the function f(x)
//
// domain space vector
size_t n = 1;
double x0 = 0.5;
vector< AD<double> > ax(n);
ax[0] = x0;
// declare independent variables and start tape recording
CppAD::Independent(ax);
// range space vector
size_t m = 1;
vector< AD<double> > ay(m);
// call user function and store reciprocal(x) in au[0]
vector< AD<double> > au(m);
afun(ax, au); // u = 1 / x
// now use AD division to invert to invert the operation
ay[0] = 1.0 / au[0]; // y = 1 / u = x
// create f: x -> y and stop tape recording
CppAD::ADFun<double> f;
f.Dependent (ax, ay); // f(x) = x
/* $$
$subhead forward$$
$codep */
// check function value
double check = x0;
ok &= NearEqual( Value(ay[0]) , check, eps, eps);
// check zero order forward mode
size_t p;
vector<double> x_p(n), y_p(m);
p = 0;
x_p[0] = x0;
y_p = f.Forward(p, x_p);
ok &= NearEqual(y_p[0] , check, eps, eps);
// check first order forward mode
p = 1;
x_p[0] = 1;
y_p = f.Forward(p, x_p);
check = 1.;
ok &= NearEqual(y_p[0] , check, eps, eps);
// check second order forward mode
p = 2;
x_p[0] = 0;
y_p = f.Forward(p, x_p);
check = 0.;
ok &= NearEqual(y_p[0] , check, eps, eps);
/* $$
$subhead reverse$$
$codep */
// third order reverse mode
p = 3;
vector<double> w(m), dw(n * p);
w[0] = 1.;
dw = f.Reverse(p, w);
check = 1.;
ok &= NearEqual(dw[0] , check, eps, eps);
check = 0.;
ok &= NearEqual(dw[1] , check, eps, eps);
ok &= NearEqual(dw[2] , check, eps, eps);
/* $$
$subhead for_sparse_jac$$
$codep */
// forward mode sparstiy pattern
size_t q = n;
CppAD::vectorBool r1(n * q), s1(m * q);
r1[0] = true; // compute sparsity pattern for x[0]
//
afun.option( CppAD::atomic_base<double>::bool_sparsity_enum );
s1 = f.ForSparseJac(q, r1);
ok &= s1[0] == true; // f[0] depends on x[0]
//
afun.option( CppAD::atomic_base<double>::set_sparsity_enum );
s1 = f.ForSparseJac(q, r1);
ok &= s1[0] == true; // f[0] depends on x[0]
/* $$
$subhead rev_sparse_jac$$
$codep */
// reverse mode sparstiy pattern
p = m;
CppAD::vectorBool s2(p * m), r2(p * n);
s2[0] = true; // compute sparsity pattern for f[0]
//
afun.option( CppAD::atomic_base<double>::bool_sparsity_enum );
r2 = f.RevSparseJac(p, s2);
ok &= r2[0] == true; // f[0] depends on x[0]
//
afun.option( CppAD::atomic_base<double>::set_sparsity_enum );
r2 = f.RevSparseJac(p, s2);
ok &= r2[0] == true; // f[0] depends on x[0]
/* $$
$subhead rev_sparse_hes$$
$codep */
// Hessian sparsity (using previous ForSparseJac call)
CppAD::vectorBool s3(m), h(q * n);
s3[0] = true; // compute sparsity pattern for f[0]
//
afun.option( CppAD::atomic_base<double>::bool_sparsity_enum );
h = f.RevSparseHes(q, s3);
ok &= h[0] == true; // second partial of f[0] w.r.t. x[0] may be non-zero
//
afun.option( CppAD::atomic_base<double>::set_sparsity_enum );
h = f.RevSparseHes(q, s3);
ok &= h[0] == true; // second partial of f[0] w.r.t. x[0] may be non-zero
return ok;
}
bool link_ode(
size_t size ,
size_t repeat ,
CppAD::vector<double> &x ,
CppAD::vector<double> &jacobian
)
{
// speed test global option values
if( global_option["atomic"] )
return false;
// optimization options: no conditional skips or compare operators
std::string options="no_compare_op";
// --------------------------------------------------------------------
// setup
assert( x.size() == size );
assert( jacobian.size() == size * size );
typedef CppAD::AD<double> ADScalar;
typedef CppAD::vector<ADScalar> ADVector;
size_t j;
size_t p = 0; // use ode to calculate function values
size_t n = size; // number of independent variables
size_t m = n; // number of dependent variables
ADVector X(n), Y(m); // independent and dependent variables
CppAD::ADFun<double> f; // AD function
// -------------------------------------------------------------
if( ! global_option["onetape"] ) while(repeat--)
{ // choose next x value
uniform_01(n, x);
for(j = 0; j < n; j++)
X[j] = x[j];
// declare the independent variable vector
Independent(X);
// evaluate function
CppAD::ode_evaluate(X, p, Y);
// create function object f : X -> Y
f.Dependent(X, Y);
if( global_option["optimize"] )
f.optimize(options);
// skip comparison operators
f.compare_change_count(0);
jacobian = f.Jacobian(x);
}
else
{ // an x value
uniform_01(n, x);
for(j = 0; j < n; j++)
X[j] = x[j];
// declare the independent variable vector
Independent(X);
// evaluate function
CppAD::ode_evaluate(X, p, Y);
// create function object f : X -> Y
f.Dependent(X, Y);
if( global_option["optimize"] )
f.optimize(options);
// skip comparison operators
f.compare_change_count(0);
while(repeat--)
{ // get next argument value
uniform_01(n, x);
// evaluate jacobian
jacobian = f.Jacobian(x);
}
}
return true;
}
bool mul_cond_rev(void)
{
bool ok = true;
using CppAD::vector;
using CppAD::NearEqual;
double eps = 10. * std::numeric_limits<double>::epsilon();
//
typedef CppAD::AD<double> a1double;
typedef CppAD::AD<a1double> a2double;
//
a1double a1zero = 0.0;
a2double a2zero = a1zero;
a1double a1one = 1.0;
a2double a2one = a1one;
//
// --------------------------------------------------------------------
// create a1f = f(x)
size_t n = 1;
size_t m = 25;
//
vector<a2double> a2x(n), a2y(m);
a2x[0] = a2double( 5.0 );
Independent(a2x);
//
size_t i = 0;
// variable that is greater than one when x[0] is zero
// and less than one when x[0] is 1.0 or greater
a2double a2switch = a2one / (a2x[0] + a2double(0.5));
// variable that is infinity when x[0] is zero
// and a normal number when x[0] is 1.0 or greater
a2double a2inf_var = a2one / a2x[0];
// variable that is nan when x[0] is zero
// and a normal number when x[0] is 1.0 or greater
a2double a2nan_var = ( a2one / a2inf_var ) / a2x[0];
// variable that is one when x[0] is zero
// and less then one when x[0] is 1.0 or greater
a2double a2one_var = a2one / ( a2one + a2x[0] );
// div
a2y[i++] = CondExpGt(a2x[0], a2zero, a2nan_var, a2zero);
// abs
a2y[i++] = CondExpGt(a2x[0], a2zero, abs( a2y[0] ), a2zero);
// add
a2y[i++] = CondExpGt(a2x[0], a2zero, a2nan_var + a2nan_var, a2zero);
// acos
a2y[i++] = CondExpGt(a2x[0], a2zero, acos(a2switch), a2zero);
// asin
a2y[i++] = CondExpGt(a2x[0], a2zero, asin(a2switch), a2zero);
// atan
a2y[i++] = CondExpGt(a2x[0], a2zero, atan(a2nan_var), a2zero);
// cos
a2y[i++] = CondExpGt(a2x[0], a2zero, cos(a2nan_var), a2zero);
// cosh
a2y[i++] = CondExpGt(a2x[0], a2zero, cosh(a2nan_var), a2zero);
// exp
a2y[i++] = CondExpGt(a2x[0], a2zero, exp(a2nan_var), a2zero);
// log
a2y[i++] = CondExpGt(a2x[0], a2zero, log(a2x[0]), a2zero);
// mul
a2y[i++] = CondExpGt(a2x[0], a2zero, a2x[0] * a2inf_var, a2zero);
// pow
a2y[i++] = CondExpGt(a2x[0], a2zero, pow(a2inf_var, a2x[0]), a2zero);
// sin
a2y[i++] = CondExpGt(a2x[0], a2zero, sin(a2nan_var), a2zero);
// sinh
a2y[i++] = CondExpGt(a2x[0], a2zero, sinh(a2nan_var), a2zero);
// sqrt
a2y[i++] = CondExpGt(a2x[0], a2zero, sqrt(a2x[0]), a2zero);
// sub
a2y[i++] = CondExpGt(a2x[0], a2zero, a2inf_var - a2nan_var, a2zero);
// tan
a2y[i++] = CondExpGt(a2x[0], a2zero, tan(a2nan_var), a2zero);
// tanh
a2y[i++] = CondExpGt(a2x[0], a2zero, tanh(a2nan_var), a2zero);
// azmul
a2y[i++] = CondExpGt(a2x[0], a2zero, azmul(a2x[0], a2inf_var), a2zero);
//
// Operations that are C+11 atomic
//
// acosh
a2y[i++] = CondExpGt(a2x[0], a2zero, acosh( a2x[0] ), a2zero);
// asinh
a2y[i++] = CondExpGt(a2x[0], a2zero, asinh( a2nan_var ), a2zero);
// atanh
a2y[i++] = CondExpGt(a2x[0], a2zero, atanh( a2one_var ), a2zero);
// erf
a2y[i++] = CondExpGt(a2x[0], a2zero, erf( a2nan_var ), a2zero);
// expm1
a2y[i++] = CondExpGt(a2x[0], a2zero, expm1(a2nan_var), a2zero);
// log1p
a2y[i++] = CondExpGt(a2x[0], a2zero, log1p(- a2one_var ), a2zero);
//
ok &= i == m;
CppAD::ADFun<a1double> a1f;
a1f.Dependent(a2x, a2y);
// --------------------------------------------------------------------
// create h = f(x)
vector<a1double> a1x(n), a1y(m);
a1x[0] = 5.0;
//
Independent(a1x);
i = 0;
a1double a1switch = a1one / (a1x[0] + a1double(0.5));
a1double a1inf_var = a1one / a1x[0];
a1double a1nan_var = ( a1one / a1inf_var ) / a1x[0];
a1double a1one_var = a1one / ( a1one + a1x[0] );
// div
a1y[i++] = CondExpGt(a1x[0], a1zero, a1nan_var, a1zero);
// abs
a1y[i++] = CondExpGt(a1x[0], a1zero, abs( a1y[0] ), a1zero);
// add
a1y[i++] = CondExpGt(a1x[0], a1zero, a1nan_var + a1nan_var, a1zero);
// acos
a1y[i++] = CondExpGt(a1x[0], a1zero, acos(a1switch), a1zero);
// asin
a1y[i++] = CondExpGt(a1x[0], a1zero, asin(a1switch), a1zero);
// atan
a1y[i++] = CondExpGt(a1x[0], a1zero, atan(a1nan_var), a1zero);
// cos
a1y[i++] = CondExpGt(a1x[0], a1zero, cos(a1nan_var), a1zero);
// cosh
a1y[i++] = CondExpGt(a1x[0], a1zero, cosh(a1nan_var), a1zero);
// exp
a1y[i++] = CondExpGt(a1x[0], a1zero, exp(a1nan_var), a1zero);
// log
a1y[i++] = CondExpGt(a1x[0], a1zero, log(a1x[0]), a1zero);
// mul
a1y[i++] = CondExpGt(a1x[0], a1zero, a1x[0] * a1inf_var, a1zero);
// pow
a1y[i++] = CondExpGt(a1x[0], a1zero, pow(a1inf_var, a1x[0]), a1zero);
// sin
a1y[i++] = CondExpGt(a1x[0], a1zero, sin(a1nan_var), a1zero);
// sinh
a1y[i++] = CondExpGt(a1x[0], a1zero, sinh(a1nan_var), a1zero);
// sqrt
a1y[i++] = CondExpGt(a1x[0], a1zero, sqrt(a1x[0]), a1zero);
// sub
a1y[i++] = CondExpGt(a1x[0], a1zero, a1inf_var - a1nan_var, a1zero);
// tan
a1y[i++] = CondExpGt(a1x[0], a1zero, tan(a1nan_var), a1zero);
// tanh
a1y[i++] = CondExpGt(a1x[0], a1zero, tanh(a1nan_var), a1zero);
// azmul
a1y[i++] = CondExpGt(a1x[0], a1zero, azmul(a1x[0], a1inf_var), a1zero);
//
// Operations that are C+11 atomic
//
// acosh
a1y[i++] = CondExpGt(a1x[0], a1zero, acosh( a1x[0] ), a1zero);
// asinh
a1y[i++] = CondExpGt(a1x[0], a1zero, asinh( a1nan_var ), a1zero);
// atanh
a1y[i++] = CondExpGt(a1x[0], a1zero, atanh( a1one_var ), a1zero);
// erf
a1y[i++] = CondExpGt(a1x[0], a1zero, erf( a1nan_var ), a1zero);
// expm1
a1y[i++] = CondExpGt(a1x[0], a1zero, expm1(a1nan_var), a1zero);
// log1p
a1y[i++] = CondExpGt(a1x[0], a1zero, log1p(- a1one_var ), a1zero);
//
ok &= i == m;
CppAD::ADFun<double> h;
h.Dependent(a1x, a1y);
// --------------------------------------------------------------------
// create g = f'(x)
vector<a1double> a1dy(m), a1w(m);
a1x[0] = 2.0;
for(i = 0; i < m; i++)
a1w[i] = 0.0;
//
Independent(a1x);
a1f.Forward(0, a1x);
//
for(i = 0; i < m; i++)
{ a1w[i] = 1.0;
vector<a1double> dyi_dx = a1f.Reverse(1, a1w);
a1dy[i] = dyi_dx[0];
a1w[i] = 0.0;
}
CppAD::ADFun<double> g; // g uses reverse mode derivatives
g.Dependent(a1x, a1dy);
// --------------------------------------------------------------------
// check case where x[0] > 0
vector<double> x(1), dx(1), dg(m), dh(m);
x[0] = 2.0;
dx[0] = 1.0;
h.Forward(0, x);
dh = h.Forward(1, dx); // dh uses forward mode derivatives
dg = g.Forward(0, x);
for(i = 0; i < m; i++)
ok &= NearEqual(dg[i], dh[i], eps, eps);
// --------------------------------------------------------------------
// check case where x[0] = 0
x[0] = 0.0;
dg = g.Forward(0, x);
h.Forward(0, x);
dh = h.Forward(1, dx);
for(i = 0; i < m; i++)
{ ok &= dg[i] == 0.0;
ok &= dh[i] == 0.0;
}
// --------------------------------------------------------------------
return ok;
}
bool old_mat_mul(void)
{ bool ok = true;
using CppAD::AD;
// matrix sizes for this test
size_t nr_result = 2;
size_t n_middle = 2;
size_t nc_result = 2;
// declare the AD<double> vectors ax and ay and X
size_t n = nr_result * n_middle + n_middle * nc_result;
size_t m = nr_result * nc_result;
CppAD::vector< AD<double> > X(4), ax(n), ay(m);
size_t i, j;
for(j = 0; j < X.size(); j++)
X[j] = (j + 1);
// X is the vector of independent variables
CppAD::Independent(X);
// left matrix
ax[0] = X[0]; // left[0,0] = x[0] = 1
ax[1] = X[1]; // left[0,1] = x[1] = 2
ax[2] = 5.; // left[1,0] = 5
ax[3] = 6.; // left[1,1] = 6
// right matrix
ax[4] = X[2]; // right[0,0] = x[2] = 3
ax[5] = 7.; // right[0,1] = 7
ax[6] = X[3]; // right[1,0] = x[3] = 4
ax[7] = 8.; // right[1,1] = 8
/*
[ x0 , x1 ] * [ x2 , 7 ] = [ x0*x2 + x1*x3 , x0*7 + x1*8 ]
[ 5 , 6 ] [ x3 , 8 ] [ 5*x2 + 6*x3 , 5*7 + 6*8 ]
*/
// The call back routines need to know the dimensions of the matrices.
// Store information about the matrix multiply for this call to mat_mul.
call_info info;
info.nr_result = nr_result;
info.n_middle = n_middle;
info.nc_result = nc_result;
// info.vx gets set by forward during call to mat_mul below
assert( info.vx.size() == 0 );
size_t id = info_.size();
info_.push_back(info);
// user defined AD<double> version of matrix multiply
mat_mul(id, ax, ay);
//----------------------------------------------------------------------
// check AD<double> results
ok &= ay[0] == (1*3 + 2*4); ok &= Variable( ay[0] );
ok &= ay[1] == (1*7 + 2*8); ok &= Variable( ay[1] );
ok &= ay[2] == (5*3 + 6*4); ok &= Variable( ay[2] );
ok &= ay[3] == (5*7 + 6*8); ok &= Parameter( ay[3] );
//----------------------------------------------------------------------
// use mat_mul to define a function g : X -> ay
CppAD::ADFun<double> G;
G.Dependent(X, ay);
// g(x) = [ x0*x2 + x1*x3 , x0*7 + x1*8 , 5*x2 + 6*x3 , 5*7 + 6*8 ]^T
//----------------------------------------------------------------------
// Test zero order forward mode evaluation of g(x)
CppAD::vector<double> x( X.size() ), y(m);
for(j = 0; j < X.size() ; j++)
x[j] = j + 2;
y = G.Forward(0, x);
ok &= y[0] == x[0] * x[2] + x[1] * x[3];
ok &= y[1] == x[0] * 7. + x[1] * 8.;
ok &= y[2] == 5. * x[2] + 6. * x[3];
ok &= y[3] == 5. * 7. + 6. * 8.;
//----------------------------------------------------------------------
// Test first order forward mode evaluation of g'(x) * [1, 2, 3, 4]^T
// g'(x) = [ x2, x3, x0, x1 ]
// [ 7 , 8, 0, 0 ]
// [ 0 , 0, 5, 6 ]
// [ 0 , 0, 0, 0 ]
CppAD::vector<double> dx( X.size() ), dy(m);
for(j = 0; j < X.size() ; j++)
dx[j] = j + 1;
dy = G.Forward(1, dx);
ok &= dy[0] == 1. * x[2] + 2. * x[3] + 3. * x[0] + 4. * x[1];
ok &= dy[1] == 1. * 7. + 2. * 8. + 3. * 0. + 4. * 0.;
ok &= dy[2] == 1. * 0. + 2. * 0. + 3. * 5. + 4. * 6.;
ok &= dy[3] == 1. * 0. + 2. * 0. + 3. * 0. + 4. * 0.;
//----------------------------------------------------------------------
// Test second order forward mode
// g_0^2 (x) = [ 0, 0, 1, 0 ], g_0^2 (x) * [1] = [3]
// [ 0, 0, 0, 1 ] [2] [4]
// [ 1, 0, 0, 0 ] [3] [1]
// [ 0, 1, 0, 0 ] [4] [2]
CppAD::vector<double> ddx( X.size() ), ddy(m);
for(j = 0; j < X.size() ; j++)
ddx[j] = 0.;
ddy = G.Forward(2, ddx);
// [1, 2, 3, 4] * g_0^2 (x) * [1, 2, 3, 4]^T = 1*3 + 2*4 + 3*1 + 4*2
ok &= 2. * ddy[0] == 1. * 3. + 2. * 4. + 3. * 1. + 4. * 2.;
// for i > 0, [1, 2, 3, 4] * g_i^2 (x) * [1, 2, 3, 4]^T = 0
ok &= ddy[1] == 0.;
ok &= ddy[2] == 0.;
ok &= ddy[3] == 0.;
//----------------------------------------------------------------------
// Test second order reverse mode
CppAD::vector<double> w(m), dw(2 * X.size() );
for(i = 0; i < m; i++)
w[i] = 0.;
w[0] = 1.;
dw = G.Reverse(2, w);
// g_0'(x) = [ x2, x3, x0, x1 ]
ok &= dw[0*2 + 0] == x[2];
ok &= dw[1*2 + 0] == x[3];
ok &= dw[2*2 + 0] == x[0];
ok &= dw[3*2 + 0] == x[1];
// g_0'(x) * [1, 2, 3, 4] = 1 * x2 + 2 * x3 + 3 * x0 + 4 * x1
// g_0^2 (x) * [1, 2, 3, 4] = [3, 4, 1, 2]
ok &= dw[0*2 + 1] == 3.;
ok &= dw[1*2 + 1] == 4.;
ok &= dw[2*2 + 1] == 1.;
ok &= dw[3*2 + 1] == 2.;
//----------------------------------------------------------------------
// Test forward and reverse Jacobian sparsity pattern
/*
[ x0 , x1 ] * [ x2 , 7 ] = [ x0*x2 + x1*x3 , x0*7 + x1*8 ]
[ 5 , 6 ] [ x3 , 8 ] [ 5*x2 + 6*x3 , 5*7 + 6*8 ]
so the sparsity pattern should be
s[0] = {0, 1, 2, 3}
s[1] = {0, 1}
s[2] = {2, 3}
s[3] = {}
*/
CppAD::vector< std::set<size_t> > r( X.size() ), s(m);
for(j = 0; j < X.size() ; j++)
{ assert( r[j].empty() );
r[j].insert(j);
}
s = G.ForSparseJac( X.size() , r);
for(j = 0; j < X.size() ; j++)
{ // s[0] = {0, 1, 2, 3}
ok &= s[0].find(j) != s[0].end();
// s[1] = {0, 1}
if( j == 0 || j == 1 )
ok &= s[1].find(j) != s[1].end();
else ok &= s[1].find(j) == s[1].end();
// s[2] = {2, 3}
if( j == 2 || j == 3 )
ok &= s[2].find(j) != s[2].end();
else ok &= s[2].find(j) == s[2].end();
}
// s[3] == {}
ok &= s[3].empty();
//----------------------------------------------------------------------
// Test reverse Jacobian sparsity pattern
/*
[ x0 , x1 ] * [ x2 , 7 ] = [ x0*x2 + x1*x3 , x0*7 + x1*8 ]
[ 5 , 6 ] [ x3 , 8 ] [ 5*x2 + 6*x3 , 5*7 + 6*8 ]
so the sparsity pattern should be
r[0] = {0, 1, 2, 3}
r[1] = {0, 1}
r[2] = {2, 3}
r[3] = {}
*/
for(i = 0; i < m; i++)
{ s[i].clear();
s[i].insert(i);
}
r = G.RevSparseJac(m, s);
for(j = 0; j < X.size() ; j++)
{ // r[0] = {0, 1, 2, 3}
ok &= r[0].find(j) != r[0].end();
// r[1] = {0, 1}
if( j == 0 || j == 1 )
ok &= r[1].find(j) != r[1].end();
else ok &= r[1].find(j) == r[1].end();
// r[2] = {2, 3}
if( j == 2 || j == 3 )
ok &= r[2].find(j) != r[2].end();
else ok &= r[2].find(j) == r[2].end();
}
// r[3] == {}
ok &= r[3].empty();
//----------------------------------------------------------------------
/* Test reverse Hessian sparsity pattern
g_0^2 (x) = [ 0, 0, 1, 0 ] and for i > 0, g_i^2 = 0
[ 0, 0, 0, 1 ]
[ 1, 0, 0, 0 ]
[ 0, 1, 0, 0 ]
so for the sparsity pattern for the first component of g is
h[0] = {2}
h[1] = {3}
h[2] = {0}
h[3] = {1}
*/
CppAD::vector< std::set<size_t> > h( X.size() ), t(1);
t[0].clear();
t[0].insert(0);
h = G.RevSparseHes(X.size() , t);
size_t check[] = {2, 3, 0, 1};
for(j = 0; j < X.size() ; j++)
{ // h[j] = { check[j] }
for(i = 0; i < n; i++)
{ if( i == check[j] )
ok &= h[j].find(i) != h[j].end();
else ok &= h[j].find(i) == h[j].end();
}
}
t[0].clear();
for( j = 1; j < X.size(); j++)
t[0].insert(j);
h = G.RevSparseHes(X.size() , t);
for(j = 0; j < X.size() ; j++)
{ // h[j] = { }
for(i = 0; i < X.size(); i++)
ok &= h[j].find(i) == h[j].end();
}
// --------------------------------------------------------------------
// Free temporary work space. (If there are future calls to
// old_mat_mul they would create new temporary work space.)
CppAD::user_atomic<double>::clear();
info_.clear();
return ok;
}
bool link_det_minor(
size_t size ,
size_t repeat ,
CppAD::vector<double> &matrix ,
CppAD::vector<double> &gradient )
{
// -----------------------------------------------------
// setup
// object for computing determinant
typedef CppAD::AD<double> ADScalar;
typedef CppAD::vector<ADScalar> ADVector;
CppAD::det_by_minor<ADScalar> Det(size);
size_t i; // temporary index
size_t m = 1; // number of dependent variables
size_t n = size * size; // number of independent variables
ADVector A(n); // AD domain space vector
ADVector detA(m); // AD range space vector
// vectors of reverse mode weights
CppAD::vector<double> w(1);
w[0] = 1.;
// the AD function object
CppAD::ADFun<double> f;
static bool printed = false;
bool print_this_time = (! printed) & (repeat > 1) & (size >= 3);
extern bool global_retape;
if( global_retape ) while(repeat--)
{
// choose a matrix
CppAD::uniform_01(n, matrix);
for( i = 0; i < size * size; i++)
A[i] = matrix[i];
// declare independent variables
Independent(A);
// AD computation of the determinant
detA[0] = Det(A);
// create function object f : A -> detA
f.Dependent(A, detA);
extern bool global_optimize;
if( global_optimize )
{ size_t before, after;
before = f.size_var();
f.optimize();
if( print_this_time )
{ after = f.size_var();
std::cout << "cppad_det_minor_optimize_size_"
<< int(size) << " = [ " << int(before)
<< ", " << int(after) << "]" << std::endl;
printed = true;
print_this_time = false;
}
}
// get the next matrix
CppAD::uniform_01(n, matrix);
// evaluate the determinant at the new matrix value
f.Forward(0, matrix);
// evaluate and return gradient using reverse mode
gradient = f.Reverse(1, w);
}
else
{
// choose a matrix
CppAD::uniform_01(n, matrix);
for( i = 0; i < size * size; i++)
A[i] = matrix[i];
// declare independent variables
Independent(A);
// AD computation of the determinant
detA[0] = Det(A);
// create function object f : A -> detA
CppAD::ADFun<double> f;
f.Dependent(A, detA);
extern bool global_optimize;
if( global_optimize )
{ size_t before, after;
before = f.size_var();
f.optimize();
if( print_this_time )
{ after = f.size_var();
std::cout << "optimize: size = " << size
<< ": size_var() = "
<< before << "(before) "
<< after << "(after) "
<< std::endl;
printed = true;
print_this_time = false;
}
}
// ------------------------------------------------------
while(repeat--)
{ // get the next matrix
CppAD::uniform_01(n, matrix);
// evaluate the determinant at the new matrix value
f.Forward(0, matrix);
// evaluate and return gradient using reverse mode
gradient = f.Reverse(1, w);
}
}
return true;
}
/* %$$
$head Use Atomic Function$$
$srccode%cpp% */
bool forward(void)
{ bool ok = true;
using CppAD::AD;
using CppAD::NearEqual;
double eps = 10. * CppAD::numeric_limits<double>::epsilon();
//
// Create the atomic_forward object corresponding to g(x)
atomic_forward afun("atomic_forward");
//
// Create the function f(u) = g(u) for this example.
//
// domain space vector
size_t n = 3;
double u_0 = 1.00;
double u_1 = 2.00;
double u_2 = 3.00;
vector< AD<double> > au(n);
au[0] = u_0;
au[1] = u_1;
au[2] = u_2;
// declare independent variables and start tape recording
CppAD::Independent(au);
// range space vector
size_t m = 2;
vector< AD<double> > ay(m);
// call atomic function
vector< AD<double> > ax = au;
afun(ax, ay);
// create f: u -> y and stop tape recording
CppAD::ADFun<double> f;
f.Dependent (au, ay); // y = f(u)
//
// check function value
double check = u_2 * u_2;
ok &= NearEqual( Value(ay[0]) , check, eps, eps);
check = u_0 * u_1;
ok &= NearEqual( Value(ay[1]) , check, eps, eps);
// --------------------------------------------------------------------
// zero order forward
//
vector<double> u0(n), y0(m);
u0[0] = u_0;
u0[1] = u_1;
u0[2] = u_2;
y0 = f.Forward(0, u0);
check = u_2 * u_2;
ok &= NearEqual(y0[0] , check, eps, eps);
check = u_0 * u_1;
ok &= NearEqual(y0[1] , check, eps, eps);
// --------------------------------------------------------------------
// first order forward
//
// value of Jacobian of f
double check_jac[] = {
0.0, 0.0, 2.0 * u_2,
u_1, u_0, 0.0
};
vector<double> u1(n), y1(m);
// check first order forward mode
for(size_t j = 0; j < n; j++)
u1[j] = 0.0;
for(size_t j = 0; j < n; j++)
{ // compute partial in j-th component direction
u1[j] = 1.0;
y1 = f.Forward(1, u1);
u1[j] = 0.0;
// check this direction
for(size_t i = 0; i < m; i++)
ok &= NearEqual(y1[i], check_jac[i * n + j], eps, eps);
}
// --------------------------------------------------------------------
// second order forward
//
// value of Hessian of g_0
double check_hes_0[] = {
0.0, 0.0, 0.0,
0.0, 0.0, 0.0,
0.0, 0.0, 2.0
};
//
// value of Hessian of g_1
double check_hes_1[] = {
0.0, 1.0, 0.0,
1.0, 0.0, 0.0,
0.0, 0.0, 0.0
};
vector<double> u2(n), y2(m);
for(size_t j = 0; j < n; j++)
u2[j] = 0.0;
// compute diagonal elements of the Hessian
for(size_t j = 0; j < n; j++)
{ // first order forward in j-th direction
u1[j] = 1.0;
f.Forward(1, u1);
y2 = f.Forward(2, u2);
// check this element of Hessian diagonal
ok &= NearEqual(y2[0], check_hes_0[j * n + j] / 2.0, eps, eps);
ok &= NearEqual(y2[1], check_hes_1[j * n + j] / 2.0, eps, eps);
//
for(size_t k = 0; k < n; k++) if( k != j )
{ u1[k] = 1.0;
f.Forward(1, u1);
y2 = f.Forward(2, u2);
//
// y2 = (H_jj + H_kk + H_jk + H_kj) / 2.0
// y2 = (H_jj + H_kk) / 2.0 + H_jk
//
double H_jj = check_hes_0[j * n + j];
double H_kk = check_hes_0[k * n + k];
double H_jk = y2[0] - (H_kk + H_jj) / 2.0;
ok &= NearEqual(H_jk, check_hes_0[j * n + k], eps, eps);
//
H_jj = check_hes_1[j * n + j];
H_kk = check_hes_1[k * n + k];
H_jk = y2[1] - (H_kk + H_jj) / 2.0;
ok &= NearEqual(H_jk, check_hes_1[j * n + k], eps, eps);
//
u1[k] = 0.0;
}
u1[j] = 0.0;
}
// --------------------------------------------------------------------
return ok;
}
bool base2ad(void)
{ bool ok = true;
using CppAD::NearEqual;
double eps = 10. * CppAD::numeric_limits<double>::epsilon();
//
// Create the atomic base2ad object
atomic_base2ad afun("atomic_base2ad");
//
// Create the function f(x) = 1 / g(x) = x
//
size_t n = 1;
double x0 = 0.5;
vector< AD<double> > ax(n);
ax[0] = x0;
// declare independent variables and start tape recording
CppAD::Independent(ax);
// range space vector
size_t m = 1;
vector< AD<double> > av(m);
// call atomic function and store g(x) in au[0]
vector< AD<double> > au(m);
afun(ax, au); // u = 1 / x
// now use AD division to invert to invert the operation
av[0] = 1.0 / au[0]; // v = 1 / u = x
// create f: x -> y and stop tape recording
CppAD::ADFun<double> f;
f.Dependent (ax, av); // g(x) = x
// check function value
double check = x0;
ok &= NearEqual( Value(av[0]) , check, eps, eps);
// check zero order forward mode
size_t q;
vector<double> x_q(n), v_q(m);
q = 0;
x_q[0] = x0;
v_q = f.Forward(q, x_q);
ok &= NearEqual(v_q[0] , check, eps, eps);
// check first order reverse
vector<double> dw(n), w(m);
w[0] = 1.0;
dw = f.Reverse(q+1, w);
check = 1.0;
ok &= NearEqual(dw[0] , check, eps, eps);
// create ag : x -> y
CppAD::ADFun< AD<double> , double > af;
af = f.base2ad();
// check zero order forward mode
vector< AD<double> > ax_q(n), av_q(m);
q = 0;
ax_q[0] = x0;
av_q = af.Forward(q, ax_q);
check = x0;
ok &= NearEqual( Value(av_q[0]) , check, eps, eps);
// check first order reverse
vector< AD<double> > adw(n), aw(m);
aw[0] = 1.0;
adw = af.Reverse(q+1, aw);
check = 1.0;
ok &= NearEqual( Value(adw[0]) , check, eps, eps);
return ok;
}
bool old_tan(void)
{ bool ok = true;
using CppAD::AD;
using CppAD::NearEqual;
float eps = 10.f * CppAD::numeric_limits<float>::epsilon();
// domain space vector
size_t n = 1;
float x0 = 0.5;
CppAD::vector< AD<float> > ax(n);
ax[0] = x0;
// declare independent variables and start tape recording
CppAD::Independent(ax);
// range space vector
size_t m = 3;
CppAD::vector< AD<float> > af(m);
// temporary vector for old_tan computations
// (old_tan computes tan or tanh and its square)
CppAD::vector< AD<float> > az(2);
// call user tan function and store tan(x) in f[0] (ignore tan(x)^2)
size_t id = 0;
old_tan(id, ax, az);
af[0] = az[0];
// call user tanh function and store tanh(x) in f[1] (ignore tanh(x)^2)
id = 1;
old_tan(id, ax, az);
af[1] = az[0];
// put a constant in f[2] = tanh(1.) (for sparsity pattern testing)
CppAD::vector< AD<float> > one(1);
one[0] = 1.;
old_tan(id, one, az);
af[2] = az[0];
// create f: x -> f and stop tape recording
CppAD::ADFun<float> F;
F.Dependent(ax, af);
// check function value
float tan = std::tan(x0);
ok &= NearEqual(af[0] , tan, eps, eps);
float tanh = std::tanh(x0);
ok &= NearEqual(af[1] , tanh, eps, eps);
// check zero order forward
CppAD::vector<float> x(n), f(m);
x[0] = x0;
f = F.Forward(0, x);
ok &= NearEqual(f[0] , tan, eps, eps);
ok &= NearEqual(f[1] , tanh, eps, eps);
// compute first partial of f w.r.t. x[0] using forward mode
CppAD::vector<float> dx(n), df(m);
dx[0] = 1.;
df = F.Forward(1, dx);
// compute derivative of tan - tanh using reverse mode
CppAD::vector<float> w(m), dw(n);
w[0] = 1.;
w[1] = 1.;
w[2] = 0.;
dw = F.Reverse(1, w);
// tan'(x) = 1 + tan(x) * tan(x)
// tanh'(x) = 1 - tanh(x) * tanh(x)
float tanp = 1.f + tan * tan;
float tanhp = 1.f - tanh * tanh;
ok &= NearEqual(df[0], tanp, eps, eps);
ok &= NearEqual(df[1], tanhp, eps, eps);
ok &= NearEqual(dw[0], w[0]*tanp + w[1]*tanhp, eps, eps);
// compute second partial of f w.r.t. x[0] using forward mode
CppAD::vector<float> ddx(n), ddf(m);
ddx[0] = 0.;
ddf = F.Forward(2, ddx);
// compute second derivative of tan - tanh using reverse mode
CppAD::vector<float> ddw(2);
ddw = F.Reverse(2, w);
// tan''(x) = 2 * tan(x) * tan'(x)
// tanh''(x) = - 2 * tanh(x) * tanh'(x)
// Note that second order Taylor coefficient for u half the
// corresponding second derivative.
float two = 2;
float tanpp = two * tan * tanp;
float tanhpp = - two * tanh * tanhp;
ok &= NearEqual(two * ddf[0], tanpp, eps, eps);
ok &= NearEqual(two * ddf[1], tanhpp, eps, eps);
ok &= NearEqual(ddw[0], w[0]*tanp + w[1]*tanhp , eps, eps);
ok &= NearEqual(ddw[1], w[0]*tanpp + w[1]*tanhpp, eps, eps);
// Forward mode computation of sparsity pattern for F.
size_t p = n;
// user vectorBool because m and n are small
CppAD::vectorBool r1(p), s1(m * p);
r1[0] = true; // propagate sparsity for x[0]
s1 = F.ForSparseJac(p, r1);
ok &= (s1[0] == true); // f[0] depends on x[0]
ok &= (s1[1] == true); // f[1] depends on x[0]
ok &= (s1[2] == false); // f[2] does not depend on x[0]
// Reverse mode computation of sparsity pattern for F.
size_t q = m;
CppAD::vectorBool s2(q * m), r2(q * n);
// Sparsity pattern for identity matrix
size_t i, j;
for(i = 0; i < q; i++)
{ for(j = 0; j < m; j++)
s2[i * q + j] = (i == j);
}
r2 = F.RevSparseJac(q, s2);
ok &= (r2[0] == true); // f[0] depends on x[0]
ok &= (r2[1] == true); // f[1] depends on x[0]
ok &= (r2[2] == false); // f[2] does not depend on x[0]
// Hessian sparsity for f[0]
CppAD::vectorBool s3(m), h(p * n);
s3[0] = true;
s3[1] = false;
s3[2] = false;
h = F.RevSparseHes(p, s3);
ok &= (h[0] == true); // Hessian is non-zero
// Hessian sparsity for f[2]
s3[0] = false;
s3[2] = true;
h = F.RevSparseHes(p, s3);
ok &= (h[0] == false); // Hessian is zero
// check tanh results for a large value of x
x[0] = std::numeric_limits<float>::max() / two;
f = F.Forward(0, x);
tanh = 1.;
ok &= NearEqual(f[1], tanh, eps, eps);
df = F.Forward(1, dx);
tanhp = 0.;
ok &= NearEqual(df[1], tanhp, eps, eps);
// --------------------------------------------------------------------
// Free all temporary work space associated with old_atomic objects.
// (If there are future calls to user atomic functions, they will
// create new temporary work space.)
CppAD::user_atomic<float>::clear();
return ok;
}
bool old_reciprocal(void)
{ bool ok = true;
using CppAD::AD;
using CppAD::NearEqual;
double eps = 10. * CppAD::numeric_limits<double>::epsilon();
// --------------------------------------------------------------------
// Create the function f(x)
//
// domain space vector
size_t n = 1;
double x0 = 0.5;
vector< AD<double> > ax(n);
ax[0] = x0;
// declare independent variables and start tape recording
CppAD::Independent(ax);
// range space vector
size_t m = 1;
vector< AD<double> > ay(m);
// call atomic function and store reciprocal(x) in au[0]
vector< AD<double> > au(m);
size_t id = 0; // not used
reciprocal(id, ax, au); // u = 1 / x
// call atomic function and store reciprocal(u) in ay[0]
reciprocal(id, au, ay); // y = 1 / u = x
// create f: x -> y and stop tape recording
CppAD::ADFun<double> f;
f.Dependent (ax, ay); // f(x) = x
// --------------------------------------------------------------------
// Check forward mode results
//
// check function value
double check = x0;
ok &= NearEqual( Value(ay[0]) , check, eps, eps);
// check zero order forward mode
size_t q;
vector<double> x_q(n), y_q(m);
q = 0;
x_q[0] = x0;
y_q = f.Forward(q, x_q);
ok &= NearEqual(y_q[0] , check, eps, eps);
// check first order forward mode
q = 1;
x_q[0] = 1;
y_q = f.Forward(q, x_q);
check = 1.;
ok &= NearEqual(y_q[0] , check, eps, eps);
// check second order forward mode
q = 2;
x_q[0] = 0;
y_q = f.Forward(q, x_q);
check = 0.;
ok &= NearEqual(y_q[0] , check, eps, eps);
// --------------------------------------------------------------------
// Check reverse mode results
//
// third order reverse mode
q = 3;
vector<double> w(m), dw(n * q);
w[0] = 1.;
dw = f.Reverse(q, w);
check = 1.;
ok &= NearEqual(dw[0] , check, eps, eps);
check = 0.;
ok &= NearEqual(dw[1] , check, eps, eps);
ok &= NearEqual(dw[2] , check, eps, eps);
// --------------------------------------------------------------------
// forward mode sparstiy pattern
size_t p = n;
CppAD::vectorBool r1(n * p), s1(m * p);
r1[0] = true; // compute sparsity pattern for x[0]
s1 = f.ForSparseJac(p, r1);
ok &= s1[0] == true; // f[0] depends on x[0]
// --------------------------------------------------------------------
// reverse mode sparstiy pattern
q = m;
CppAD::vectorBool s2(q * m), r2(q * n);
s2[0] = true; // compute sparsity pattern for f[0]
r2 = f.RevSparseJac(q, s2);
ok &= r2[0] == true; // f[0] depends on x[0]
// --------------------------------------------------------------------
// Hessian sparsity (using previous ForSparseJac call)
CppAD::vectorBool s3(m), h(p * n);
s3[0] = true; // compute sparsity pattern for f[0]
h = f.RevSparseHes(p, s3);
ok &= h[0] == true; // second partial of f[0] w.r.t. x[0] may be non-zero
// -----------------------------------------------------------------
// Free all temporary work space associated with atomic_one objects.
// (If there are future calls to atomic functions, they will
// create new temporary work space.)
CppAD::user_atomic<double>::clear();
return ok;
}
bool link_poly(
size_t size ,
size_t repeat ,
CppAD::vector<double> &a , // coefficients of polynomial
CppAD::vector<double> &z , // polynomial argument value
CppAD::vector<double> &ddp ) // second derivative w.r.t z
{
// speed test global option values
if( global_atomic )
return false;
// -----------------------------------------------------
// setup
typedef CppAD::AD<double> ADScalar;
typedef CppAD::vector<ADScalar> ADVector;
size_t i; // temporary index
size_t m = 1; // number of dependent variables
size_t n = 1; // number of independent variables
ADVector Z(n); // AD domain space vector
ADVector P(m); // AD range space vector
// choose the polynomial coefficients
CppAD::uniform_01(size, a);
// AD copy of the polynomial coefficients
ADVector A(size);
for(i = 0; i < size; i++)
A[i] = a[i];
// forward mode first and second differentials
CppAD::vector<double> p(1), dp(1), dz(1), ddz(1);
dz[0] = 1.;
ddz[0] = 0.;
// AD function object
CppAD::ADFun<double> f;
// --------------------------------------------------------------------
if( ! global_onetape ) while(repeat--)
{
// choose an argument value
CppAD::uniform_01(1, z);
Z[0] = z[0];
// declare independent variables
Independent(Z);
// AD computation of the function value
P[0] = CppAD::Poly(0, A, Z[0]);
// create function object f : A -> detA
f.Dependent(Z, P);
if( global_optimize )
f.optimize();
// skip comparison operators
f.compare_change_count(0);
// pre-allocate memory for three forward mode calculations
f.capacity_order(3);
// evaluate the polynomial
p = f.Forward(0, z);
// evaluate first order Taylor coefficient
dp = f.Forward(1, dz);
// second derivative is twice second order Taylor coef
ddp = f.Forward(2, ddz);
ddp[0] *= 2.;
}
else
{
// choose an argument value
CppAD::uniform_01(1, z);
Z[0] = z[0];
// declare independent variables
Independent(Z);
// AD computation of the function value
P[0] = CppAD::Poly(0, A, Z[0]);
// create function object f : A -> detA
f.Dependent(Z, P);
if( global_optimize )
f.optimize();
// skip comparison operators
f.compare_change_count(0);
while(repeat--)
{ // sufficient memory is allocated by second repetition
// get the next argument value
CppAD::uniform_01(1, z);
// evaluate the polynomial at the new argument value
p = f.Forward(0, z);
// evaluate first order Taylor coefficient
dp = f.Forward(1, dz);
// second derivative is twice second order Taylor coef
ddp = f.Forward(2, ddz);
ddp[0] *= 2.;
}
}
return true;
}
bool forward_order(void)
{ bool ok = true;
using CppAD::AD;
using CppAD::NearEqual;
size_t j, k;
double eps = 10. * CppAD::numeric_limits<double>::epsilon();
// domain space vector
size_t n = 23, m = n;
CPPAD_TESTVECTOR(AD<double>) X(n), Y(m);
for(j = 0; j < n; j++)
X[j] = 0.0;
// declare independent variables and starting recording
CppAD::Independent(X);
// identity function values
size_t i = 0;
size_t identity_begin = i;
Y[i] = cos( acos( X[i] ) ); i++; // AcosOp, CosOp
Y[i] = sin( asin( X[i] ) ); i++; // AsinOp, SinOp
Y[i] = tan( atan( X[i] ) ); i++; // AtanOp, TanOp
Y[i] = CondExpGt(X[i], X[i-1], X[i], X[i-2]); i++; // CExpOp
Y[i] = X[i-1] * X[i] / X[i-1]; i++; // DivvvOp, MulvvOp
Y[i] = X[i] * X[i] * 1.0 / X[i]; i++; // DivpvOp
Y[i] = 5.0 * X[i] / 5.0; i++; // DivvpOp, MulpvOp
Y[i] = exp( log( X[i] ) ); i++; // ExpOp, LogOp
Y[i] = pow( sqrt( X[i] ), 2.0); i++; // PowvpOp, SqrtOp
Y[i] = log( pow( std::exp(1.), X[i] ) ); i++; // PowpvOp
Y[i] = log( pow( X[i], X[i] ) ) / log( X[i]); i++; // PowvvOp
Y[i] = -2. - ((X[i-1] - X[i]) - 2.) + X[i-1]; i++; // Sub*Op: pv, vv, vp
size_t identity_end = i;
// other functions
Y[i] = abs( X[i] ); i++; // AbsOp
Y[i] = X[i-1] + X[i] + 2.0; i++; // AddvvOp, AddvpOp
Y[i] = cosh( X[i] ); i++; // CoshOp
Y[i] = my_discrete( X[i] ); i++; // DisOp
Y[i] = 4.0; i++; // ParOp
Y[i] = sign( X[i] ); i++; // SignOp
Y[i] = sinh( X[i] ); i++; // SinhOp
Y[i] = tanh(X[i]); i++; // TanhOp
// VecAD operations
CppAD::VecAD<double> V(n);
AD<double> index = 1.;
V[index] = 3.0;
Y[i] = V[index]; i++; // StppOp, LdpOp
V[index] = X[0];
Y[i] = V[index]; i++; // StpvOp, LdpOp
index = double(n) * X[3];
V[index] = X[1];
Y[i] = V[index]; i++; // StvvOp, LdvOp
// create f: X -> Y and stop tape recording
assert( i == m );
CppAD::ADFun<double> f;
f.Dependent(X, Y);
// initially, no values stored in f
ok &= f.size_order() == 0;
// Set X_j (t) = x + t
size_t p = 2, p1 = p+1;
CPPAD_TESTVECTOR(double) x(n), x_p(n * p1), y_p(m * p1);
for(j = 0; j < n; j++)
{ x[j] = double(j) / double(n);
x_p[j * p1 + 0] = x[j]; // order 0
x_p[j * p1 + 1] = 1.; // order 1
x_p[j * p1 + 2] = 0.; // order 2
}
// compute orders 0, 1, 2
y_p = f.Forward(p, x_p);
// identity functions
CPPAD_TESTVECTOR(double) y(p1);
i = 0;
for(j = identity_begin; j != identity_end; j++)
{ y[0] = x[j];
y[1] = 1.0;
y[2] = 0.0;
for(k = 0; k < p1; k++)
ok &= NearEqual(y[k] , y_p[i * p1 + k], eps, eps);
i++;
}
// y_i = abs( x_i )
y[0] = CppAD::abs( x[i] );
y[1] = CppAD::sign( x[i] );
y[2] = 0.0;
for(k = 0; k < p1; k++)
ok &= NearEqual(y[k] , y_p[i * p1 + k], eps, eps);
// y_i = x_[i-1] + x_i + 2
i++;
y[0] = x[i-1] + x[i] + 2.0;
y[1] = 2.0;
y[2] = 0.0;
for(k = 0; k < p1; k++)
ok &= NearEqual(y[k] , y_p[i * p1 + k], eps, eps);
// y_i = cosh( x_i )
i++;
y[0] = CppAD::cosh( x[i] );
y[1] = CppAD::sinh( x[i] );
y[2] = CppAD::cosh( x[i] ) / 2.0;
for(k = 0; k < p1; k++)
ok &= NearEqual(y[k] , y_p[i * p1 + k], eps, eps);
// y_i = my_discrete( x_i )
i++;
y[0] = my_discrete( x[i] );
y[1] = 0.0;
y[2] = 0.0;
for(k = 0; k < p1; k++)
ok &= NearEqual(y[k] , y_p[i * p1 + k], eps, eps);
// y_i = 4
i++;
y[0] = 4.0;
y[1] = 0.0;
y[2] = 0.0;
for(k = 0; k < p1; k++)
ok &= NearEqual(y[k] , y_p[i * p1 + k], eps, eps);
// y_i = sign( x_i )
i++;
y[0] = CppAD::sign( x[i] );
y[1] = 0.0;
y[2] = 0.0;
for(k = 0; k < p1; k++)
ok &= NearEqual(y[k] , y_p[i * p1 + k], eps, eps);
// y_i = sinh( x_i )
i++;
y[0] = CppAD::sinh( x[i] );
y[1] = CppAD::cosh( x[i] );
y[2] = CppAD::sinh( x[i] ) / 2.0;
for(k = 0; k < p1; k++)
ok &= NearEqual(y[k] , y_p[i * p1 + k], eps, eps);
// y_i = tanh( x_i )
i++;
y[0] = CppAD::tanh( x[i] );
y[1] = 1.0 - y[0] * y[0];
y[2] = - 2.0 * y[0] * y[1] / 2.0;
for(k = 0; k < p1; k++)
ok &= NearEqual(y[k] , y_p[i * p1 + k], eps, eps);
// y_i = 3.0;
i++;
y[0] = 3.0;
y[1] = 0.0;
y[2] = 0.0;
for(k = 0; k < p1; k++)
ok &= NearEqual(y[k] , y_p[i * p1 + k], eps, eps);
// y_i = x_0
i++;
y[0] = x[0];
y[1] = 1.0;
y[2] = 0.0;
for(k = 0; k < p1; k++)
ok &= NearEqual(y[k] , y_p[i * p1 + k], eps, eps);
// y_i = x_1
i++;
y[0] = x[1];
y[1] = 1.0;
y[2] = 0.0;
for(k = 0; k < p1; k++)
ok &= NearEqual(y[k] , y_p[i * p1 + k], eps, eps);
return ok;
}
/* %$$
$head Use Atomic Function$$
$srccode%cpp% */
bool sparsity(void)
{ bool ok = true;
using CppAD::AD;
using CppAD::NearEqual;
double eps = 10. * std::numeric_limits<double>::epsilon();
/* %$$
$subhead Constructor$$
$srccode%cpp% */
// Create the atomic get_started object
atomic_sparsity afun("atomic_sparsity");
/* %$$
$subhead Recording$$
$srccode%cpp% */
size_t n = 3;
size_t m = 2;
vector< AD<double> > ax(n), ay(m);
for(size_t j = 0; j < n; j++)
ax[j] = double(j + 1);
// declare independent variables and start tape recording
CppAD::Independent(ax);
// call user function
afun(ax, ay);
// create f: x -> y and stop tape recording
CppAD::ADFun<double> f;
f.Dependent (ax, ay); // f(x) = x
// check function value
ok &= NearEqual(ay[0] , ax[2], eps, eps);
ok &= NearEqual(ay[1] , ax[0] * ax[1], eps, eps);
/* %$$
$subhead forsparse_jac and rev_sparse_jac$$
$srccode%cpp% */
for(size_t dir = 0; dir < 2; dir++)
{ size_t ell;
if( dir == 0 )
ell = n;
else
ell = m;
// identity martrix
vectorBool r(ell * ell);
for(size_t i = 0; i < ell; i++)
for(size_t j = 0; j < ell; j++)
r[i * ell + j] = (i == j);
vectorBool s;
if( dir == 0 )
s = f.ForSparseJac(n, r);
else
s = f.RevSparseJac(m, r);
// check Jacobian result
ok &= s.size() == m * n;
ok &= s[0 * n + 0] == false;
ok &= s[0 * n + 1] == false;
ok &= s[0 * n + 2] == true;
ok &= s[1 * n + 0] == true;
ok &= s[1 * n + 1] == true;
ok &= s[1 * n + 2] == false;
}
/* %$$
$subhead rev_sparse_hes$$
$srccode%cpp% */
vectorBool s(m), h(n * n);
s[0] = true;
s[1] = true;
h = f.RevSparseHes(n, s);
for(size_t i = 0; i < n; i++)
{ for(size_t j = 0; j < n; j++)
{ bool check = false;
check |= (i == 0) && (j == 1);
check |= (j == 0) && (i == 1);
ok &= h[ i * n + j] == check;
}
}
//
return ok;
}
bool change_param(void)
{ bool ok = true; // initialize test result
typedef CppAD::AD<double> a1type; // for first level of taping
typedef CppAD::AD<a1type> a2type; // for second level of taping
size_t nu = 3; // number components in u
size_t nx = 2; // number components in x
size_t ny = 2; // num components in f(x)
size_t nJ = ny * nx; // number components in Jacobian of f(x)
// temporary indices
size_t j;
// declare first level of independent variables
// (Start taping now so can record dependency of a1f on a1p.)
CPPAD_TESTVECTOR(a1type) a1u(nu);
for(j = 0; j < nu; j++)
a1u[j] = 0.;
CppAD::Independent(a1u);
// parameter in computation of Jacobian
a1type a1p = a1u[2];
// declare second level of independent variables
CPPAD_TESTVECTOR(a2type) a2x(nx);
for(j = 0; j < nx; j++)
a2x[j] = 0.;
CppAD::Independent(a2x);
// compute dependent variables at second level
CPPAD_TESTVECTOR(a2type) a2y(ny);
a2y[0] = sin( a2x[0] ) * a1p;
a2y[1] = sin( a2x[1] ) * a1p;
// declare function object that computes values at the first level
// (make sure we do not run zero order forward during constructor)
CppAD::ADFun<a1type> a1f;
a1f.Dependent(a2x, a2y);
// compute the Jacobian of a1f at a1u[0], a1u[1]
CPPAD_TESTVECTOR(a1type) a1x(nx);
a1x[0] = a1u[0];
a1x[1] = a1u[1];
CPPAD_TESTVECTOR(a1type) a1J(nJ);
a1J = a1f.Jacobian( a1x );
// declare function object that maps u = (x, p) to Jacobian of f
// (make sure we do not run zero order forward during constructor)
CppAD::ADFun<double> g;
g.Dependent(a1u, a1J);
// remove extra variables used during the reconding of a1f,
// but not needed any more.
g.optimize();
// compute the Jacobian of f using zero order forward
// sweep with double values
CPPAD_TESTVECTOR(double) J(nJ), u(nu);
for(j = 0; j < nu; j++)
u[j] = double(j+1);
J = g.Forward(0, u);
// accuracy for tests
double eps = 100. * CppAD::numeric_limits<double>::epsilon();
// y[0] = sin( x[0] ) * p
// y[1] = sin( x[1] ) * p
CPPAD_TESTVECTOR(double) x(nx);
x[0] = u[0];
x[1] = u[1];
double p = u[2];
// J[0] = partial y[0] w.r.t x[0] = cos( x[0] ) * p
double check = cos( x[0] ) * p;
ok &= fabs( check - J[0] ) <= eps;
// J[1] = partial y[0] w.r.t x[1] = 0.;
check = 0.;
ok &= fabs( check - J[1] ) <= eps;
// J[2] = partial y[1] w.r.t. x[0] = 0.
check = 0.;
ok &= fabs( check - J[2] ) <= eps;
// J[3] = partial y[1] w.r.t x[1] = cos( x[1] ) * p
check = cos( x[1] ) * p;
ok &= fabs( check - J[3] ) <= eps;
return ok;
}
bool link_det_minor(
size_t size ,
size_t repeat ,
CppAD::vector<double> &matrix ,
CppAD::vector<double> &gradient )
{
// speed test global option values
if( global_option["atomic"] )
return false;
// -----------------------------------------------------
// setup
// object for computing determinant
typedef CppAD::AD<double> ADScalar;
typedef CppAD::vector<ADScalar> ADVector;
CppAD::det_by_minor<ADScalar> Det(size);
size_t i; // temporary index
size_t m = 1; // number of dependent variables
size_t n = size * size; // number of independent variables
ADVector A(n); // AD domain space vector
ADVector detA(m); // AD range space vector
// vectors of reverse mode weights
CppAD::vector<double> w(1);
w[0] = 1.;
// the AD function object
CppAD::ADFun<double> f;
// ---------------------------------------------------------------------
if( ! global_option["onetape"] ) while(repeat--)
{
// choose a matrix
CppAD::uniform_01(n, matrix);
for( i = 0; i < size * size; i++)
A[i] = matrix[i];
// declare independent variables
Independent(A);
// AD computation of the determinant
detA[0] = Det(A);
// create function object f : A -> detA
f.Dependent(A, detA);
if( global_option["optimize"] )
f.optimize();
// skip comparison operators
f.compare_change_count(0);
// evaluate the determinant at the new matrix value
f.Forward(0, matrix);
// evaluate and return gradient using reverse mode
gradient = f.Reverse(1, w);
}
else
{
// choose a matrix
CppAD::uniform_01(n, matrix);
for( i = 0; i < size * size; i++)
A[i] = matrix[i];
// declare independent variables
Independent(A);
// AD computation of the determinant
detA[0] = Det(A);
// create function object f : A -> detA
f.Dependent(A, detA);
if( global_option["optimize"] )
f.optimize();
// skip comparison operators
f.compare_change_count(0);
// ------------------------------------------------------
while(repeat--)
{ // get the next matrix
CppAD::uniform_01(n, matrix);
// evaluate the determinant at the new matrix value
f.Forward(0, matrix);
// evaluate and return gradient using reverse mode
gradient = f.Reverse(1, w);
}
}
return true;
}