/** @file exam_matrices.cpp
*
* Here we examine manipulations on GiNaC's symbolic matrices. */
/*
* GiNaC Copyright (C) 1999-2007 Johannes Gutenberg University Mainz, Germany
*
* This program is free software; you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* the Free Software Foundation; either version 2 of the License, or
* (at your option) any later version.
*
* This program is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU General Public License for more details.
*
* You should have received a copy of the GNU General Public License
* along with this program; if not, write to the Free Software
* Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA
*/
#include <stdexcept>
#include "exams.h"
static unsigned matrix_determinants()
{
unsigned result = 0;
ex det;
matrix m1(1,1), m2(2,2), m3(3,3), m4(4,4);
symbol a("a"), b("b"), c("c");
symbol d("d"), e("e"), f("f");
symbol g("g"), h("h"), i("i");
// check symbolic trivial matrix determinant
m1.set(0,0,a);
det = m1.determinant();
if (det != a) {
clog << "determinant of 1x1 matrix " << m1
<< " erroneously returned " << det << endl;
++result;
}
// check generic dense symbolic 2x2 matrix determinant
m2.set(0,0,a).set(0,1,b);
m2.set(1,0,c).set(1,1,d);
det = m2.determinant();
if (det != (a*d-b*c)) {
clog << "determinant of 2x2 matrix " << m2
<< " erroneously returned " << det << endl;
++result;
}
// check generic dense symbolic 3x3 matrix determinant
m3.set(0,0,a).set(0,1,b).set(0,2,c);
m3.set(1,0,d).set(1,1,e).set(1,2,f);
m3.set(2,0,g).set(2,1,h).set(2,2,i);
det = m3.determinant();
if (det != (a*e*i - a*f*h - d*b*i + d*c*h + g*b*f - g*c*e)) {
clog << "determinant of 3x3 matrix " << m3
<< " erroneously returned " << det << endl;
++result;
}
// check dense numeric 3x3 matrix determinant
m3.set(0,0,numeric(0)).set(0,1,numeric(-1)).set(0,2,numeric(3));
m3.set(1,0,numeric(3)).set(1,1,numeric(-2)).set(1,2,numeric(2));
m3.set(2,0,numeric(3)).set(2,1,numeric(4)).set(2,2,numeric(-2));
det = m3.determinant();
if (det != 42) {
clog << "determinant of 3x3 matrix " << m3
<< " erroneously returned " << det << endl;
++result;
}
// check dense symbolic 2x2 matrix determinant
m2.set(0,0,a/(a-b)).set(0,1,1);
m2.set(1,0,b/(a-b)).set(1,1,1);
det = m2.determinant();
if (det != 1) {
if (det.normal() == 1) // only half wrong
clog << "determinant of 2x2 matrix " << m2
<< " was returned unnormalized as " << det << endl;
else // totally wrong
clog << "determinant of 2x2 matrix " << m2
<< " erroneously returned " << det << endl;
++result;
}
// check sparse symbolic 4x4 matrix determinant
m4.set(0,1,a).set(1,0,b).set(3,2,c).set(2,3,d);
det = m4.determinant();
if (det != a*b*c*d) {
clog << "determinant of 4x4 matrix " << m4
<< " erroneously returned " << det << endl;
++result;
}
// check characteristic polynomial
m3.set(0,0,a).set(0,1,-2).set(0,2,2);
m3.set(1,0,3).set(1,1,a-1).set(1,2,2);
m3.set(2,0,3).set(2,1,4).set(2,2,a-3);
ex p = m3.charpoly(a);
if (p != 0) {
clog << "charpoly of 3x3 matrix " << m3
<< " erroneously returned " << p << endl;
++result;
}
return result;
}
static unsigned matrix_invert1()
{
unsigned result = 0;
matrix m(1,1);
symbol a("a");
m.set(0,0,a);
matrix m_i = m.inverse();
if (m_i(0,0) != pow(a,-1)) {
clog << "inversion of 1x1 matrix " << m
<< " erroneously returned " << m_i << endl;
++result;
}
return result;
}
static unsigned matrix_invert2()
{
unsigned result = 0;
matrix m(2,2);
symbol a("a"), b("b"), c("c"), d("d");
m.set(0,0,a).set(0,1,b);
m.set(1,0,c).set(1,1,d);
matrix m_i = m.inverse();
ex det = m.determinant();
if ((normal(m_i(0,0)*det) != d) ||
(normal(m_i(0,1)*det) != -b) ||
(normal(m_i(1,0)*det) != -c) ||
(normal(m_i(1,1)*det) != a)) {
clog << "inversion of 2x2 matrix " << m
<< " erroneously returned " << m_i << endl;
++result;
}
return result;
}
static unsigned matrix_invert3()
{
unsigned result = 0;
matrix m(3,3);
symbol a("a"), b("b"), c("c");
symbol d("d"), e("e"), f("f");
symbol g("g"), h("h"), i("i");
m.set(0,0,a).set(0,1,b).set(0,2,c);
m.set(1,0,d).set(1,1,e).set(1,2,f);
m.set(2,0,g).set(2,1,h).set(2,2,i);
matrix m_i = m.inverse();
ex det = m.determinant();
if ((normal(m_i(0,0)*det) != (e*i-f*h)) ||
(normal(m_i(0,1)*det) != (c*h-b*i)) ||
(normal(m_i(0,2)*det) != (b*f-c*e)) ||
(normal(m_i(1,0)*det) != (f*g-d*i)) ||
(normal(m_i(1,1)*det) != (a*i-c*g)) ||
(normal(m_i(1,2)*det) != (c*d-a*f)) ||
(normal(m_i(2,0)*det) != (d*h-e*g)) ||
(normal(m_i(2,1)*det) != (b*g-a*h)) ||
(normal(m_i(2,2)*det) != (a*e-b*d))) {
clog << "inversion of 3x3 matrix " << m
<< " erroneously returned " << m_i << endl;
++result;
}
return result;
}
static unsigned matrix_solve2()
{
// check the solution of the multiple system A*X = B:
// [ 1 2 -1 ] [ x0 y0 ] [ 4 0 ]
// [ 1 4 -2 ]*[ x1 y1 ] = [ 7 0 ]
// [ a -2 2 ] [ x2 y2 ] [ a 4 ]
unsigned result = 0;
symbol a("a");
symbol x0("x0"), x1("x1"), x2("x2");
symbol y0("y0"), y1("y1"), y2("y2");
matrix A(3,3);
A.set(0,0,1).set(0,1,2).set(0,2,-1);
A.set(1,0,1).set(1,1,4).set(1,2,-2);
A.set(2,0,a).set(2,1,-2).set(2,2,2);
matrix B(3,2);
B.set(0,0,4).set(1,0,7).set(2,0,a);
B.set(0,1,0).set(1,1,0).set(2,1,4);
matrix X(3,2);
X.set(0,0,x0).set(1,0,x1).set(2,0,x2);
X.set(0,1,y0).set(1,1,y1).set(2,1,y2);
matrix cmp(3,2);
cmp.set(0,0,1).set(1,0,3).set(2,0,3);
cmp.set(0,1,0).set(1,1,2).set(2,1,4);
matrix sol(A.solve(X, B));
for (unsigned ro=0; ro<3; ++ro)
for (unsigned co=0; co<2; ++co)
if (cmp(ro,co) != sol(ro,co))
result = 1;
if (result) {
clog << "Solving " << A << " * " << X << " == " << B << endl
<< "erroneously returned " << sol << endl;
}
return result;
}
static unsigned matrix_evalm()
{
unsigned result = 0;
matrix S(2, 2, lst(
1, 2,
3, 4
)), T(2, 2, lst(
1, 1,
2, -1
)), R(2, 2, lst(
27, 14,
36, 26
));
ex e = ((S + T) * (S + 2*T));
ex f = e.evalm();
if (!f.is_equal(R)) {
clog << "Evaluating " << e << " erroneously returned " << f << " instead of " << R << endl;
result++;
}
return result;
}
static unsigned matrix_rank()
{
unsigned result = 0;
symbol x("x"), y("y");
matrix m(3,3);
// the zero matrix always has rank 0
if (m.rank() != 0) {
clog << "The rank of " << m << " was not computed correctly." << endl;
++result;
}
// a trivial rank one example
m = 1, 0, 0,
2, 0, 0,
3, 0, 0;
if (m.rank() != 1) {
clog << "The rank of " << m << " was not computed correctly." << endl;
++result;
}
// an example from Maple's help with rank two
m = x, 1, 0,
0, 0, 1,
x*y, y, 1;
if (m.rank() != 2) {
clog << "The rank of " << m << " was not computed correctly." << endl;
++result;
}
// the 3x3 unit matrix has rank 3
m = ex_to<matrix>(unit_matrix(3,3));
if (m.rank() != 3) {
clog << "The rank of " << m << " was not computed correctly." << endl;
++result;
}
return result;
}
static unsigned matrix_misc()
{
unsigned result = 0;
matrix m1(2,2);
symbol a("a"), b("b"), c("c"), d("d"), e("e"), f("f");
m1.set(0,0,a).set(0,1,b);
m1.set(1,0,c).set(1,1,d);
ex tr = trace(m1);
// check a simple trace
if (tr.compare(a+d)) {
clog << "trace of 2x2 matrix " << m1
<< " erroneously returned " << tr << endl;
++result;
}
// and two simple transpositions
matrix m2 = transpose(m1);
if (m2(0,0) != a || m2(0,1) != c || m2(1,0) != b || m2(1,1) != d) {
clog << "transpose of 2x2 matrix " << m1
<< " erroneously returned " << m2 << endl;
++result;
}
matrix m3(3,2);
m3.set(0,0,a).set(0,1,b);
m3.set(1,0,c).set(1,1,d);
m3.set(2,0,e).set(2,1,f);
if (transpose(transpose(m3)) != m3) {
clog << "transposing 3x2 matrix " << m3 << " twice"
<< " erroneously returned " << transpose(transpose(m3)) << endl;
++result;
}
// produce a runtime-error by inverting a singular matrix and catch it
matrix m4(2,2);
matrix m5;
bool caught = false;
try {
m5 = inverse(m4);
} catch (std::runtime_error err) {
caught = true;
}
if (!caught) {
cerr << "singular 2x2 matrix " << m4
<< " erroneously inverted to " << m5 << endl;
++result;
}
return result;
}
unsigned exam_matrices()
{
unsigned result = 0;
cout << "examining symbolic matrix manipulations" << flush;
clog << "----------symbolic matrix manipulations:" << endl;
result += matrix_determinants(); cout << '.' << flush;
result += matrix_invert1(); cout << '.' << flush;
result += matrix_invert2(); cout << '.' << flush;
result += matrix_invert3(); cout << '.' << flush;
result += matrix_solve2(); cout << '.' << flush;
result += matrix_evalm(); cout << "." << flush;
result += matrix_rank(); cout << "." << flush;
result += matrix_misc(); cout << '.' << flush;
if (!result) {
cout << " passed " << endl;
clog << "(no output)" << endl;
} else {
cout << " failed " << endl;
}
return result;
}
syntax highlighted by Code2HTML, v. 0.9.1