/***************************************************************************** * deriv.cpp Blitz++ Array<T,1> example, illustrating expressions, * ranges, and subvectors. *****************************************************************************/ #include <blitz/array.h> BZ_USING_NAMESPACE(blitz) BZ_USING_NAMESPACE(blitz::tensor) typedef Array<double,1> Array1D; int main() { // In this example, the function cos(x)^2 and its second derivative // 2 (sin(x)^2 - cos(x)^2) are sampled over the range [0,1). // The second derivative is approximated numerically using a // [ 1 -2 1 ] mask, and the approximation error is computed. const int numSamples = 100; // Number of samples double delta = 1. / numSamples; // Spacing of samples Range R(0, numSamples - 1); // Index set of the vector // Sample the function y = cos(x)^2 over [0,1) // // The initialization for y (below) will be translated via expression // templates into something of the flavour // // for (unsigned i=0; i < 99; ++i) // { // double _t1 = cos(i * delta); // y[i] = _t1 * _t1; // } // // The variable i comes from the blitz::tensor namespace, and is // an "index placeholder" which represents the array index. Array1D y(R); y = sqr(cos(i * delta)); // Sample the exact second derivative Array1D y2exact(R); y2exact = 2.0 * (sqr(sin(i * delta)) - sqr(cos(i * delta))); // Approximate the 2nd derivative using a [ 1 -2 1 ] mask // We can only apply this mask to the elements 1 .. 98, since // we need one element on either side to apply the mask. Range I(1,numSamples-2); Array1D y2(numSamples); y2(I) = (y(I-1) - 2 * y(I) + y(I+1)) / (delta*delta); // The above difference equation will be transformed into // something along the lines of // // double _t2 = delta*delta; // for (int i=1; i < 99; ++i) // y2[i] = (y[i-1] - 2 * y[i] + y[i+1]) / _t2; // Now calculate the root mean square approximation error: double error = sqrt(mean(sqr(y2(I) - y2exact(I)))); // Display a few elements from the vectors. // This range constructor means elements 1 to 91 in increments // of 15. Range displayRange(1, 91, 15); cout << "Exact derivative:" << y2exact(displayRange) << endl << "Approximation: " << y2(Range(displayRange)) << endl << "RMS Error: " << error << endl; return 0; }