Hide keyboard shortcuts

Hot-keys on this page

r m x p   toggle line displays

j k   next/prev highlighted chunk

0   (zero) top of page

1   (one) first highlighted chunk

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

198

199

200

201

202

203

204

205

206

207

208

209

210

211

212

213

214

215

216

217

218

219

220

## -*- encoding: utf-8 -*- 

""" 

This file (./linsolve_doctest.sage) was *autogenerated* from ./linsolve.tex, 

with sagetex.sty version 2011/05/27 v2.3.1. 

It contains the contents of all the sageexample environments from this file. 

You should be able to doctest this file with: 

sage -t ./linsolve_doctest.sage 

It is always safe to delete this file; it is not used in typesetting your 

document. 

 

Sage example in ./linsolve.tex, line 235:: 

 

sage: def cond_hilbert(n): 

....: A = matrix(QQ, [[1/(i+j-1) for j in [1..n]] for i in [1..n]]) 

....: return A.norm(Infinity) * (A^-1).norm(Infinity) 

 

Sage example in ./linsolve.tex, line 269:: 

 

sage: n = 8 

sage: x = vector(QQ,[1 for i in range(0,n)]) 

sage: A = matrix(QQ, [[1/(i+j-1) for j in [1..n]] for i in [1..n]]) 

sage: y = A*x 

sage: A[n-1,n-1] = (1/(2*n-1))*(1+1/(10^5)) # perturbe la matrice 

sage: sol = A\y 

sage: diff = max(float(sol[i]-x[i]) for i in range(0,n)) 

 

Sage example in ./linsolve.tex, line 313:: 

 

sage: n = 8 

sage: A = matrix(RR, [[1/(i+j-1) for j in [1..n]] for i in [1..n]]) 

sage: x = vector(RR, [1 for i in range(0,n)]) 

sage: y = A*x 

sage: s = A.solve_right(y) 

sage: diff = [float(s[i]-x[i]) for i in range(0,n)] 

 

Sage example in ./linsolve.tex, line 422:: 

 

sage: n = 20; cout = (n+1)*factorial(n); cout 

51090942171709440000 

 

Sage example in ./linsolve.tex, line 433:: 

 

sage: v = 3*10^9 

sage: print("%3.3f" % float(cout/v/3600/24/365)) 

540.028 

 

Sage example in ./linsolve.tex, line 502:: 

 

sage: A = matrix(RDF, [[-1,2],[3,4]]) 

sage: b = vector(RDF, [2,3]) 

sage: x = A\b; x # rel tol 3e-15 

(-0.20000000000000018, 0.9000000000000001) 

 

Sage example in ./linsolve.tex, line 512:: 

 

sage: x = A.solve_right(b) 

 

Sage example in ./linsolve.tex, line 520:: 

 

sage: A = matrix(RDF, [[-1,2],[3,4]]) 

sage: P, L, U = A.LU() 

 

Sage example in ./linsolve.tex, line 561:: 

 

sage: A = random_matrix(RDF, 1000) 

sage: b = vector(RDF, range(1000)) 

 

Sage example in ./linsolve.tex, line 600:: 

 

sage: m = random_matrix(RDF, 10) 

sage: A = transpose(m)*m 

sage: C = A.cholesky() 

 

Sage example in ./linsolve.tex, line 655:: 

 

sage: A = random_matrix(RDF,6,5) 

sage: Q, R = A.QR() 

 

Sage example in ./linsolve.tex, line 786:: 

 

sage: A = matrix(RDF, [[1,3,2],[1,4,2],[0,5,2],[1,3,2]]) 

sage: b = vector(RDF, [1,2,3,4]) 

sage: Z = transpose(A)*A 

sage: C = Z.cholesky() 

sage: R = transpose(A)*b 

sage: Z.solve_right(R) # rel tol 1e-13 

(-1.5000000000000044, -0.5000000000000009, 2.750000000000003) 

 

Sage example in ./linsolve.tex, line 822:: 

 

sage: A = matrix(RDF, [[1,3,2],[1,4,2],[0,5,2],[1,3,2]]) 

sage: b = vector(RDF, [1,2,3,4]) 

sage: Q, R = A.QR() 

sage: R1 = R[0:3,0:3] 

sage: b1 = transpose(Q)*b 

sage: c = b1[0:3] 

sage: R1.solve_right(c) # rel tol 2e-14 

(-1.499999999999999, -0.49999999999999867, 2.7499999999999973) 

 

Sage example in ./linsolve.tex, line 834:: 

 

sage: Z = A.transpose()*A 

sage: Z.norm(Infinity)*(Z^-1).norm(Infinity) # rel tol 1e-14 

1992.3750000000084 

 

Sage example in ./linsolve.tex, line 876:: 

 

sage: A = matrix(RDF, [[1,3,2],[1,3,2],[0,5,2],[1,3,2]]) 

sage: b = vector(RDF, [1,2,3,4]) 

sage: U, Sig, V = A.SVD() 

sage: m = A.ncols() 

sage: x = vector(RDF, [0]*m) 

sage: lamb = vector(RDF, [0]*m) 

sage: for i in range(0,m): 

....: s = Sig[i,i] 

....: if s < 1e-12: 

....: break 

....: lamb[i] = U.column(i)*b / s 

sage: x = V*lamb; x # rel tol 1e-14 

(0.2370370370370367, 0.4518518518518521, 0.3703703703703702) 

 

Sage example in ./linsolve.tex, line 968:: 

 

sage: A = matrix(RDF, [[1,2],[3,4],[5,6],[7,8]]) 

 

Sage example in ./linsolve.tex, line 974:: 

 

sage: th = 0.7 

sage: R = matrix(RDF, [[cos(th),sin(th)],[-sin(th),cos(th)]]) 

sage: B = (A + 0.1*random_matrix(RDF,4,2)) * transpose(R) 

 

Sage example in ./linsolve.tex, line 1189:: 

 

sage: A = matrix(RDF, [[1,3,2],[1,2,3],[0,5,2]]) 

 

Sage example in ./linsolve.tex, line 1382:: 

 

sage: def pol2companion(p): 

....: n = len(p) 

....: m = matrix(RDF,n) 

....: for i in range(1,n): 

....: m[i,i-1]=1 

....: for i in range(0,n): 

....: m[i,n-1]=-p[i] 

....: return m 

 

Sage example in ./linsolve.tex, line 1392:: 

 

sage: q = [1,-1,2,3,5,-1,10,11] 

sage: comp = pol2companion(q); comp 

[ 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -1.0] 

[ 1.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0] 

[ 0.0 1.0 0.0 0.0 0.0 0.0 0.0 -2.0] 

[ 0.0 0.0 1.0 0.0 0.0 0.0 0.0 -3.0] 

[ 0.0 0.0 0.0 1.0 0.0 0.0 0.0 -5.0] 

[ 0.0 0.0 0.0 0.0 1.0 0.0 0.0 1.0] 

[ 0.0 0.0 0.0 0.0 0.0 1.0 0.0 -10.0] 

[ 0.0 0.0 0.0 0.0 0.0 0.0 1.0 -11.0] 

sage: racines = comp.eigenvalues(); racines # abs tol 1e-10 

[0.347521510119 + 0.566550553398*I, 

0.347521510119 - 0.566550553398*I, 

0.345023776962 + 0.439908702386*I, 

0.345023776962 - 0.439908702386*I, 

-0.517257614325 + 0.512958206789*I, 

-0.517257614325 - 0.512958206789*I, 

-1.36699716455, 

-9.98357818097] 

 

Sage example in ./linsolve.tex, line 1515:: 

 

sage: def eval(P,x): 

....: if len(P) == 0: 

....: return 0 

....: else: 

....: return P[0]+x*eval(P[1:],x) 

 

Sage example in ./linsolve.tex, line 1523:: 

 

sage: def pscal(P,Q,lx): 

....: return float(sum(eval(P,s)*eval(Q,s) for s in lx)) 

 

Sage example in ./linsolve.tex, line 1528:: 

 

sage: def padd(P,a,Q): 

....: for i in range(0,len(Q)): 

....: P[i] += a*Q[i] 

 

Sage example in ./linsolve.tex, line 1536:: 

 

sage: class BadParamsforOrthop(Exception): 

....: def __init__(self, degreplusun, npoints): 

....: self.deg = degreplusun 

....: self.np = npoints 

....: def __str__(self): 

....: return "degre: " + str(self.deg) + \ 

....: " nb. points: " + repr(self.np) 

 

Sage example in ./linsolve.tex, line 1546:: 

 

sage: def orthopoly(n,x): 

....: if n > len(x): 

....: raise BadParamsforOrthop(n-1, len(x)) 

....: orth = [[1./sqrt(float(len(x)))]] 

....: for p in range(1,n): 

....: nextp = copy(orth[p-1]) 

....: nextp.insert(0,0) 

....: s = [] 

....: for i in range(p-1,max(p-3,-1),-1): 

....: s.append(pscal(nextp, orth[i], x)) 

....: j = 0 

....: for i in range(p-1,max(p-3,-1),-1): 

....: padd(nextp, -s[j], orth[i]) 

....: j += 1 

....: norm = sqrt(pscal(nextp, nextp, x)) 

....: nextpn = [nextp[i]/norm for i in range(len(nextp))] 

....: orth.append(nextpn) 

....: return orth 

 

""" 

# This file was *autogenerated* from the file linsolve_doctest.sage.