1 /** @file exam_matrices.cpp
3 * Here we examine manipulations on GiNaC's symbolic matrices. */
6 * GiNaC Copyright (C) 1999-2008 Johannes Gutenberg University Mainz, Germany
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14 * but WITHOUT ANY WARRANTY; without even the implied warranty of
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20 * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA
27 using namespace GiNaC;
29 static unsigned matrix_determinants()
33 matrix m1(1,1), m2(2,2), m3(3,3), m4(4,4);
34 symbol a("a"), b("b"), c("c");
35 symbol d("d"), e("e"), f("f");
36 symbol g("g"), h("h"), i("i");
38 // check symbolic trivial matrix determinant
40 det = m1.determinant();
42 clog << "determinant of 1x1 matrix " << m1
43 << " erroneously returned " << det << endl;
47 // check generic dense symbolic 2x2 matrix determinant
48 m2.set(0,0,a).set(0,1,b);
49 m2.set(1,0,c).set(1,1,d);
50 det = m2.determinant();
51 if (det != (a*d-b*c)) {
52 clog << "determinant of 2x2 matrix " << m2
53 << " erroneously returned " << det << endl;
57 // check generic dense symbolic 3x3 matrix determinant
58 m3.set(0,0,a).set(0,1,b).set(0,2,c);
59 m3.set(1,0,d).set(1,1,e).set(1,2,f);
60 m3.set(2,0,g).set(2,1,h).set(2,2,i);
61 det = m3.determinant();
62 if (det != (a*e*i - a*f*h - d*b*i + d*c*h + g*b*f - g*c*e)) {
63 clog << "determinant of 3x3 matrix " << m3
64 << " erroneously returned " << det << endl;
68 // check dense numeric 3x3 matrix determinant
69 m3.set(0,0,numeric(0)).set(0,1,numeric(-1)).set(0,2,numeric(3));
70 m3.set(1,0,numeric(3)).set(1,1,numeric(-2)).set(1,2,numeric(2));
71 m3.set(2,0,numeric(3)).set(2,1,numeric(4)).set(2,2,numeric(-2));
72 det = m3.determinant();
74 clog << "determinant of 3x3 matrix " << m3
75 << " erroneously returned " << det << endl;
79 // check dense symbolic 2x2 matrix determinant
80 m2.set(0,0,a/(a-b)).set(0,1,1);
81 m2.set(1,0,b/(a-b)).set(1,1,1);
82 det = m2.determinant();
84 if (det.normal() == 1) // only half wrong
85 clog << "determinant of 2x2 matrix " << m2
86 << " was returned unnormalized as " << det << endl;
88 clog << "determinant of 2x2 matrix " << m2
89 << " erroneously returned " << det << endl;
93 // check sparse symbolic 4x4 matrix determinant
94 m4.set(0,1,a).set(1,0,b).set(3,2,c).set(2,3,d);
95 det = m4.determinant();
97 clog << "determinant of 4x4 matrix " << m4
98 << " erroneously returned " << det << endl;
102 // check characteristic polynomial
103 m3.set(0,0,a).set(0,1,-2).set(0,2,2);
104 m3.set(1,0,3).set(1,1,a-1).set(1,2,2);
105 m3.set(2,0,3).set(2,1,4).set(2,2,a-3);
106 ex p = m3.charpoly(a);
108 clog << "charpoly of 3x3 matrix " << m3
109 << " erroneously returned " << p << endl;
116 static unsigned matrix_invert1()
123 matrix m_i = m.inverse();
125 if (m_i(0,0) != pow(a,-1)) {
126 clog << "inversion of 1x1 matrix " << m
127 << " erroneously returned " << m_i << endl;
134 static unsigned matrix_invert2()
138 symbol a("a"), b("b"), c("c"), d("d");
139 m.set(0,0,a).set(0,1,b);
140 m.set(1,0,c).set(1,1,d);
141 matrix m_i = m.inverse();
142 ex det = m.determinant();
144 if ((normal(m_i(0,0)*det) != d) ||
145 (normal(m_i(0,1)*det) != -b) ||
146 (normal(m_i(1,0)*det) != -c) ||
147 (normal(m_i(1,1)*det) != a)) {
148 clog << "inversion of 2x2 matrix " << m
149 << " erroneously returned " << m_i << endl;
156 static unsigned matrix_invert3()
160 symbol a("a"), b("b"), c("c");
161 symbol d("d"), e("e"), f("f");
162 symbol g("g"), h("h"), i("i");
163 m.set(0,0,a).set(0,1,b).set(0,2,c);
164 m.set(1,0,d).set(1,1,e).set(1,2,f);
165 m.set(2,0,g).set(2,1,h).set(2,2,i);
166 matrix m_i = m.inverse();
167 ex det = m.determinant();
169 if ((normal(m_i(0,0)*det) != (e*i-f*h)) ||
170 (normal(m_i(0,1)*det) != (c*h-b*i)) ||
171 (normal(m_i(0,2)*det) != (b*f-c*e)) ||
172 (normal(m_i(1,0)*det) != (f*g-d*i)) ||
173 (normal(m_i(1,1)*det) != (a*i-c*g)) ||
174 (normal(m_i(1,2)*det) != (c*d-a*f)) ||
175 (normal(m_i(2,0)*det) != (d*h-e*g)) ||
176 (normal(m_i(2,1)*det) != (b*g-a*h)) ||
177 (normal(m_i(2,2)*det) != (a*e-b*d))) {
178 clog << "inversion of 3x3 matrix " << m
179 << " erroneously returned " << m_i << endl;
186 static unsigned matrix_solve2()
188 // check the solution of the multiple system A*X = B:
189 // [ 1 2 -1 ] [ x0 y0 ] [ 4 0 ]
190 // [ 1 4 -2 ]*[ x1 y1 ] = [ 7 0 ]
191 // [ a -2 2 ] [ x2 y2 ] [ a 4 ]
194 symbol x0("x0"), x1("x1"), x2("x2");
195 symbol y0("y0"), y1("y1"), y2("y2");
197 A.set(0,0,1).set(0,1,2).set(0,2,-1);
198 A.set(1,0,1).set(1,1,4).set(1,2,-2);
199 A.set(2,0,a).set(2,1,-2).set(2,2,2);
201 B.set(0,0,4).set(1,0,7).set(2,0,a);
202 B.set(0,1,0).set(1,1,0).set(2,1,4);
204 X.set(0,0,x0).set(1,0,x1).set(2,0,x2);
205 X.set(0,1,y0).set(1,1,y1).set(2,1,y2);
207 cmp.set(0,0,1).set(1,0,3).set(2,0,3);
208 cmp.set(0,1,0).set(1,1,2).set(2,1,4);
209 matrix sol(A.solve(X, B));
210 for (unsigned ro=0; ro<3; ++ro)
211 for (unsigned co=0; co<2; ++co)
212 if (cmp(ro,co) != sol(ro,co))
215 clog << "Solving " << A << " * " << X << " == " << B << endl
216 << "erroneously returned " << sol << endl;
222 static unsigned matrix_evalm()
237 ex e = ((S + T) * (S + 2*T));
239 if (!f.is_equal(R)) {
240 clog << "Evaluating " << e << " erroneously returned " << f << " instead of " << R << endl;
247 static unsigned matrix_rank()
250 symbol x("x"), y("y");
253 // the zero matrix always has rank 0
255 clog << "The rank of " << m << " was not computed correctly." << endl;
259 // a trivial rank one example
264 clog << "The rank of " << m << " was not computed correctly." << endl;
268 // an example from Maple's help with rank two
273 clog << "The rank of " << m << " was not computed correctly." << endl;
277 // the 3x3 unit matrix has rank 3
278 m = ex_to<matrix>(unit_matrix(3,3));
280 clog << "The rank of " << m << " was not computed correctly." << endl;
287 static unsigned matrix_misc()
291 symbol a("a"), b("b"), c("c"), d("d"), e("e"), f("f");
292 m1.set(0,0,a).set(0,1,b);
293 m1.set(1,0,c).set(1,1,d);
296 // check a simple trace
297 if (tr.compare(a+d)) {
298 clog << "trace of 2x2 matrix " << m1
299 << " erroneously returned " << tr << endl;
303 // and two simple transpositions
304 matrix m2 = transpose(m1);
305 if (m2(0,0) != a || m2(0,1) != c || m2(1,0) != b || m2(1,1) != d) {
306 clog << "transpose of 2x2 matrix " << m1
307 << " erroneously returned " << m2 << endl;
311 m3.set(0,0,a).set(0,1,b);
312 m3.set(1,0,c).set(1,1,d);
313 m3.set(2,0,e).set(2,1,f);
314 if (transpose(transpose(m3)) != m3) {
315 clog << "transposing 3x2 matrix " << m3 << " twice"
316 << " erroneously returned " << transpose(transpose(m3)) << endl;
320 // produce a runtime-error by inverting a singular matrix and catch it
326 } catch (std::runtime_error err) {
330 cerr << "singular 2x2 matrix " << m4
331 << " erroneously inverted to " << m5 << endl;
338 unsigned exam_matrices()
342 cout << "examining symbolic matrix manipulations" << flush;
344 result += matrix_determinants(); cout << '.' << flush;
345 result += matrix_invert1(); cout << '.' << flush;
346 result += matrix_invert2(); cout << '.' << flush;
347 result += matrix_invert3(); cout << '.' << flush;
348 result += matrix_solve2(); cout << '.' << flush;
349 result += matrix_evalm(); cout << "." << flush;
350 result += matrix_rank(); cout << "." << flush;
351 result += matrix_misc(); cout << '.' << flush;
356 int main(int argc, char** argv)
358 return exam_matrices();