1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
|
Use("testers.mpi");
DoNext(_string) <--
[
NextTest("<font color=0000ff>" : string : "</font>");
NewLine();
];
Echo({"<HTML><BODY BGCOLOR=\"ffffff\"><PRE><font size=4>"});
Echo({"An assorted selection of the tests found in the Wester benchmark"});
StartTests();
/*
*/
DoNext("Compute 50!");
BenchShow(50!);
DoNext("Compute the prime decomposition of 6!.");
BenchShow(ans:=Factors(6!));
Echo({"This list contains lists of two elements. This list should
be interpreted as "});
BenchShow(PrettyForm(FW(ans)));
DoNext("Compute 1/2 + ... + 1/10.");
BenchShow(Sum(i,2,10,1/i));
DoNext("Compute a numerical approximation of e^(Pi*sqrt(163)) to 50 digits.");
BenchCall(BuiltinPrecisionSet(50));
BenchShow(N(Exp(Pi*Sqrt(163))));
DoNext("Compute an infinite decimal representation of 1/7");
BenchShow(Decimal(1/7));
DoNext("Compute the first terms of the continued fraction of Pi.");
BenchShow(PrettyForm(ContFrac(Internal'Pi())));
DoNext("Simplify sqrt(2*sqrt(3)+4).");
BenchShow(RadSimp(Sqrt(2*Sqrt(3)+4)));
DoNext("Simplify sqrt(14+3*sqrt(3+2*sqrt(5-12*sqrt(3-2*sqrt(2))))).");
BenchShow(RadSimp(Sqrt(14+3*Sqrt(3+2*Sqrt(5-12*Sqrt(3-2*Sqrt(2)))))));
DoNext("Simplify 2*infinity-3.");
BenchShow(2*Infinity-3);
Echo({"Infinity is also defined for comparisons like a < Infinity"});
DoNext("Compute the normal form of (x^2-4)/(x^2+4x+4).");
BenchShow(PrettyForm(GcdReduce((x^2-4)/(x^2+4*x+4),x)));
DoNext("Expand (x+1)^5, then differentiate and factorize.");
BenchCall(ans:=Factors(D(x)Expand((x+1)^5)));
BenchShow(PrettyForm(FW(ans)));
DoNext("Simplify sqrt(997) - (997^3)^(1/6).");
BenchShow(RadSimp(Sqrt(997) - (997^3)^(1/6)));
DoNext("Simplify sqrt(999983) - (999983^3)^(1/6).");
BenchShow(RadSimp(Sqrt(999983) - (999983^3)^(1/6)));
DoNext("Recognize that (2^(1/3)+4^(1/3))^3-6*(2^(1/3)+4^(1/3)) - 6 is 0.");
BenchShow(RadSimp( (2^(1/3)+4^(1/3))^3-6*(2^(1/3)+4^(1/3)) - 6));
DoNext("Simplify log e^z into z only for -Pi < Im(z) <= Pi.");
BenchShow(Simplify(Ln(Exp(z))));
DoNext("Invert the 2x2 matrix [[a,b],[1,ab]]. ");
BenchCall(A:={{a,b},{1,a*b}});
BenchShow(ans:=Inverse(A));
BenchShow(TableForm(Simplify(ans)));
DoNext("Find the eigenvalues of the matrix [[5, -3, -7],[-2, 1, 2],[ 2, -3, -4]].");
BenchCall(A:={{5,-3,-7},{-2,1,2},{2,-3,-4}});
BenchShow(EigenValues(A));
DoNext("Compute the limit of (1-cos x)/x^2 when x goes to zero. ");
BenchShow(Limit(x,0)(1-Cos(x))/(x^2));
DoNext("Compute the derivative of |x|.");
BenchShow(D(x)Abs(x));
DoNext("Compute an antiderivative of |x|.");
BenchShow(AntiDeriv(x,Abs(x)));
DoNext("Compute the derivative of |x| (piecewise defined).");
BenchShow(D(x)(if (x<0) (-x) else x));
DoNext("Compute the antiderivative of |x| (piecewise defined).");
BenchShow( AntiDeriv(x,if (x<0) (-x) else x));
DoNext("Compute the first terms of the Taylor expansion of 1/sqrt(1-v^2/c^2) at v=0. ");
BenchCall(ans:=Taylor(v,0,4)Sqrt(1/(1-v^2/c^2)));
BenchCall(ans:=Simplify(ans));
BenchShow(PrettyForm(ans));
DoNext("Compute the inverse of the square of the above expansion. ");
BenchCall(ans:=Taylor(v,0,4)(1/ans)^2);
BenchShow(PrettyForm(Simplify(ans)));
DoNext("Compute the Taylor expansion of tan(x) at x=0 by dividing the expansion of sin(x) by that of cos(x). ");
BenchCall(ans1:=Taylor(x,0,5)(Sin(x)/Cos(x)));
BenchCall(PrettyForm(ans1));
BenchCall(ans2:=Taylor(x,0,5)Tan(x));
BenchCall(PrettyForm(ans2));
BenchShow(ans1-ans2);
DoNext("Compute the Legendre polynomials directly.");
BenchCall(10 # Legendre(0,_x) <-- 1);
BenchCall(20 # Legendre(n_IsInteger,_x) <--
[
Local(result);
result:=[
Local(x);
Expand((1/(2^n*(n!))) * Deriv(x,n)Expand((x^2-1)^n,x));
];
Eval(result);
]);
BenchCall(ForEach(item,Table(Legendre(i,x),i,0,4,1))PrettyForm(item));
DoNext("Compute the Legendre polynomials recursively, using their recurrence of order 2. ");
BenchCall(10 # LegendreRecursive(0,_x) <-- 1);
BenchCall(20 # LegendreRecursive(1,_x) <-- x);
BenchCall(30 # LegendreRecursive(n_IsPositiveInteger,_x) <--
Expand(((2*n - 1)*x*LegendreRecursive(n-1,x)-(n - 1)*LegendreRecursive(n-2,x))/n));
BenchCall(ForEach(item,Table(LegendreRecursive(i,x),i,0,4,1))PrettyForm(item));
DoNext("Evaluate the fourth Legendre polynomial at 1. ");
BenchShow(Legendre(4,1));
DoNext("Define the polynomial p = sum( i=1..5, ai*x^i ). ");
BenchCall(ans:=Sum(MakeVector(a,5)*(FillList(x,5)^(1 .. 5))));
BenchCall(PrettyForm(ans));
DoNext("Apply Horner\'s rule to the above polynomial.");
BenchCall(ans:=Sum(MakeVector(a,5)*(FillList(x,5)^(1 .. 5))));
BenchCall(PrettyForm(Horner(ans,x)));
DoNext("Compute the first terms of the continued fraction of Pi. ");
BenchCall(pi:=N(Pi,20));
BenchCall(a:=ContFrac(pi,6));
BenchCall(PrettyForm(a));
DoNext("Compute an infinite decimal representation of 1/7. ");
BenchShow(Decimal(1/7));
Echo({"This result means that the decimal expansion of 1/7 is
0.142857142857142...."});
DoNext("Evaluate TRUE and FALSE. ");
BenchShow(True And False);
DoNext("Solve the equation tan(x) = 1. ");
BenchShow(Solve(Tan(x)==1,x));
DoNext("Revert the Taylor expansion of sin(y) + cos(y) at y=0. ");
BenchCall(t:=InverseTaylor(y,0,6)Sin(y)+Cos(y));
BenchCall(PrettyForm(t));
Echo({"And check that it is in fact the inverse up to degree 5:"});
BenchCall(s:=Taylor(y,0,6)Sin(y)+Cos(y));
BenchShow(BigOh(Subst(y,s)t,y,6));
DoNext("Solve the linear (dependent) system x+y+z=6,2x+y+2z=10,x+3y+z=10.");
BenchShow(OldSolve({x+y+z==6,2*x+y+2*z==10,x+3*y+z==10},{x,y,z}));
DoNext("Evaluate True And False");
BenchShow(True And False);
BenchShow(CanProve(True And False));
DoNext("Simplify x or (not x)");
BenchShow(CanProve(x Or Not x));
DoNext("Simplify x or y or (x and y)");
BenchShow(CanProve(x Or y Or (x And y)));
Echo({curline," examples done"});
Echo({"</FONT></PRE></BODY></HTML>"});
|
