1 /** @file exam_matrices.cpp
3 * Here we examine manipulations on GiNaC's symbolic matrices. */
6 * GiNaC Copyright (C) 1999-2016 Johannes Gutenberg University Mainz, Germany
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24 using namespace GiNaC;
30 static unsigned matrix_determinants()
34 matrix m1(1,1), m2(2,2), m3(3,3), m4(4,4);
35 symbol a("a"), b("b"), c("c");
36 symbol d("d"), e("e"), f("f");
37 symbol g("g"), h("h"), i("i");
39 // check symbolic trivial matrix determinant
41 det = m1.determinant();
43 clog << "determinant of 1x1 matrix " << m1
44 << " erroneously returned " << det << endl;
48 // check generic dense symbolic 2x2 matrix determinant
49 m2.set(0,0,a).set(0,1,b);
50 m2.set(1,0,c).set(1,1,d);
51 det = m2.determinant();
52 if (det != (a*d-b*c)) {
53 clog << "determinant of 2x2 matrix " << m2
54 << " erroneously returned " << det << endl;
58 // check generic dense symbolic 3x3 matrix determinant
59 m3.set(0,0,a).set(0,1,b).set(0,2,c);
60 m3.set(1,0,d).set(1,1,e).set(1,2,f);
61 m3.set(2,0,g).set(2,1,h).set(2,2,i);
62 det = m3.determinant();
63 if (det != (a*e*i - a*f*h - d*b*i + d*c*h + g*b*f - g*c*e)) {
64 clog << "determinant of 3x3 matrix " << m3
65 << " erroneously returned " << det << endl;
69 // check dense numeric 3x3 matrix determinant
70 m3.set(0,0,numeric(0)).set(0,1,numeric(-1)).set(0,2,numeric(3));
71 m3.set(1,0,numeric(3)).set(1,1,numeric(-2)).set(1,2,numeric(2));
72 m3.set(2,0,numeric(3)).set(2,1,numeric(4)).set(2,2,numeric(-2));
73 det = m3.determinant();
75 clog << "determinant of 3x3 matrix " << m3
76 << " erroneously returned " << det << endl;
80 // check dense symbolic 2x2 matrix determinant
81 m2.set(0,0,a/(a-b)).set(0,1,1);
82 m2.set(1,0,b/(a-b)).set(1,1,1);
83 det = m2.determinant();
85 if (det.normal() == 1) // only half wrong
86 clog << "determinant of 2x2 matrix " << m2
87 << " was returned unnormalized as " << det << endl;
89 clog << "determinant of 2x2 matrix " << m2
90 << " erroneously returned " << det << endl;
94 // check sparse symbolic 4x4 matrix determinant
95 m4.set(0,1,a).set(1,0,b).set(3,2,c).set(2,3,d);
96 det = m4.determinant();
98 clog << "determinant of 4x4 matrix " << m4
99 << " erroneously returned " << det << endl;
103 // check characteristic polynomial
104 m3.set(0,0,a).set(0,1,-2).set(0,2,2);
105 m3.set(1,0,3).set(1,1,a-1).set(1,2,2);
106 m3.set(2,0,3).set(2,1,4).set(2,2,a-3);
107 ex p = m3.charpoly(a);
109 clog << "charpoly of 3x3 matrix " << m3
110 << " erroneously returned " << p << endl;
117 static unsigned matrix_invert1()
124 matrix m_i = m.inverse();
126 if (m_i(0,0) != pow(a,-1)) {
127 clog << "inversion of 1x1 matrix " << m
128 << " erroneously returned " << m_i << endl;
135 static unsigned matrix_invert2()
139 symbol a("a"), b("b"), c("c"), d("d");
140 m.set(0,0,a).set(0,1,b);
141 m.set(1,0,c).set(1,1,d);
142 matrix m_i = m.inverse();
143 ex det = m.determinant();
145 if ((normal(m_i(0,0)*det) != d) ||
146 (normal(m_i(0,1)*det) != -b) ||
147 (normal(m_i(1,0)*det) != -c) ||
148 (normal(m_i(1,1)*det) != a)) {
149 clog << "inversion of 2x2 matrix " << m
150 << " erroneously returned " << m_i << endl;
157 static unsigned matrix_invert3()
161 symbol a("a"), b("b"), c("c");
162 symbol d("d"), e("e"), f("f");
163 symbol g("g"), h("h"), i("i");
164 m.set(0,0,a).set(0,1,b).set(0,2,c);
165 m.set(1,0,d).set(1,1,e).set(1,2,f);
166 m.set(2,0,g).set(2,1,h).set(2,2,i);
167 matrix m_i = m.inverse();
168 ex det = m.determinant();
170 if ((normal(m_i(0,0)*det) != (e*i-f*h)) ||
171 (normal(m_i(0,1)*det) != (c*h-b*i)) ||
172 (normal(m_i(0,2)*det) != (b*f-c*e)) ||
173 (normal(m_i(1,0)*det) != (f*g-d*i)) ||
174 (normal(m_i(1,1)*det) != (a*i-c*g)) ||
175 (normal(m_i(1,2)*det) != (c*d-a*f)) ||
176 (normal(m_i(2,0)*det) != (d*h-e*g)) ||
177 (normal(m_i(2,1)*det) != (b*g-a*h)) ||
178 (normal(m_i(2,2)*det) != (a*e-b*d))) {
179 clog << "inversion of 3x3 matrix " << m
180 << " erroneously returned " << m_i << endl;
187 static unsigned matrix_solve2()
189 // check the solution of the multiple system A*X = B:
190 // [ 1 2 -1 ] [ x0 y0 ] [ 4 0 ]
191 // [ 1 4 -2 ]*[ x1 y1 ] = [ 7 0 ]
192 // [ a -2 2 ] [ x2 y2 ] [ a 4 ]
195 symbol x0("x0"), x1("x1"), x2("x2");
196 symbol y0("y0"), y1("y1"), y2("y2");
198 A.set(0,0,1).set(0,1,2).set(0,2,-1);
199 A.set(1,0,1).set(1,1,4).set(1,2,-2);
200 A.set(2,0,a).set(2,1,-2).set(2,2,2);
202 B.set(0,0,4).set(1,0,7).set(2,0,a);
203 B.set(0,1,0).set(1,1,0).set(2,1,4);
205 X.set(0,0,x0).set(1,0,x1).set(2,0,x2);
206 X.set(0,1,y0).set(1,1,y1).set(2,1,y2);
208 cmp.set(0,0,1).set(1,0,3).set(2,0,3);
209 cmp.set(0,1,0).set(1,1,2).set(2,1,4);
210 matrix sol(A.solve(X, B));
211 for (unsigned ro=0; ro<3; ++ro)
212 for (unsigned co=0; co<2; ++co)
213 if (cmp(ro,co) != sol(ro,co))
216 clog << "Solving " << A << " * " << X << " == " << B << endl
217 << "erroneously returned " << sol << endl;
223 static unsigned matrix_evalm()
238 ex e = ((S + T) * (S + 2*T));
240 if (!f.is_equal(R)) {
241 clog << "Evaluating " << e << " erroneously returned " << f << " instead of " << R << endl;
248 static unsigned matrix_rank()
251 symbol x("x"), y("y");
254 // the zero matrix always has rank 0
256 clog << "The rank of " << m << " was not computed correctly." << endl;
260 // a trivial rank one example
265 clog << "The rank of " << m << " was not computed correctly." << endl;
269 // an example from Maple's help with rank two
274 clog << "The rank of " << m << " was not computed correctly." << endl;
278 // the 3x3 unit matrix has rank 3
279 m = ex_to<matrix>(unit_matrix(3,3));
281 clog << "The rank of " << m << " was not computed correctly." << endl;
288 static unsigned matrix_misc()
292 symbol a("a"), b("b"), c("c"), d("d"), e("e"), f("f");
293 m1.set(0,0,a).set(0,1,b);
294 m1.set(1,0,c).set(1,1,d);
297 // check a simple trace
298 if (tr.compare(a+d)) {
299 clog << "trace of 2x2 matrix " << m1
300 << " erroneously returned " << tr << endl;
304 // and two simple transpositions
305 matrix m2 = transpose(m1);
306 if (m2(0,0) != a || m2(0,1) != c || m2(1,0) != b || m2(1,1) != d) {
307 clog << "transpose of 2x2 matrix " << m1
308 << " erroneously returned " << m2 << endl;
312 m3.set(0,0,a).set(0,1,b);
313 m3.set(1,0,c).set(1,1,d);
314 m3.set(2,0,e).set(2,1,f);
315 if (transpose(transpose(m3)) != m3) {
316 clog << "transposing 3x2 matrix " << m3 << " twice"
317 << " erroneously returned " << transpose(transpose(m3)) << endl;
321 // produce a runtime-error by inverting a singular matrix and catch it
327 } catch (std::runtime_error err) {
331 cerr << "singular 2x2 matrix " << m4
332 << " erroneously inverted to " << m5 << endl;
339 unsigned exam_matrices()
343 cout << "examining symbolic matrix manipulations" << flush;
345 result += matrix_determinants(); cout << '.' << flush;
346 result += matrix_invert1(); cout << '.' << flush;
347 result += matrix_invert2(); cout << '.' << flush;
348 result += matrix_invert3(); cout << '.' << flush;
349 result += matrix_solve2(); cout << '.' << flush;
350 result += matrix_evalm(); cout << "." << flush;
351 result += matrix_rank(); cout << "." << flush;
352 result += matrix_misc(); cout << '.' << flush;
357 int main(int argc, char** argv)
359 return exam_matrices();