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## -*- 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. |