def test_levin_3(): mp.dps = 17 z = mp.mpf(2) eps = mp.mpf(mp.eps) with mp.extraprec( 7 * mp.prec ): # we need copious amount of precision to sum this highly divergent series L = mp.levin(method="levin", variant="t") n, s = 0, 0 while 1: s += (-z)**n * mp.fac(4 * n) / (mp.fac(n) * mp.fac(2 * n) * (4**n)) n += 1 v, e = L.step_psum(s) if e < eps: break if n > 1000: raise RuntimeError("iteration limit exceeded") eps = mp.exp(0.8 * mp.log(eps)) exact = mp.quad(lambda x: mp.exp(-x * x / 2 - z * x**4), [0, mp.inf]) * 2 / mp.sqrt(2 * mp.pi) # there is also a symbolic expression for the integral: # exact = mp.exp(mp.one / (32 * z)) * mp.besselk(mp.one / 4, mp.one / (32 * z)) / (4 * mp.sqrt(z * mp.pi)) err = abs(v - exact) assert err < eps w = mp.nsum(lambda n: (-z)**n * mp.fac(4 * n) / (mp.fac(n) * mp.fac(2 * n) * (4**n)), [0, mp.inf], method="levin", levin_variant="t", workprec=8 * mp.prec, steps=[2] + [1 for x in xrange(1000)]) err = abs(v - w) assert err < eps
def test_gauss_quadrature_dynamic(verbose = False): n = 5 A = mp.randmatrix(2 * n, 1) def F(x): r = 0 for i in xrange(len(A) - 1, -1, -1): r = r * x + A[i] return r def run(qtype, FW, R, alpha = 0, beta = 0): X, W = mp.gauss_quadrature(n, qtype, alpha = alpha, beta = beta) a = 0 for i in xrange(len(X)): a += W[i] * F(X[i]) b = mp.quad(lambda x: FW(x) * F(x), R) c = mp.fabs(a - b) if verbose: print(qtype, c, a, b) assert c < 1e-5 run("legendre", lambda x: 1, [-1, 1]) run("legendre01", lambda x: 1, [0, 1]) run("hermite", lambda x: mp.exp(-x*x), [-mp.inf, mp.inf]) run("laguerre", lambda x: mp.exp(-x), [0, mp.inf]) run("glaguerre", lambda x: mp.sqrt(x)*mp.exp(-x), [0, mp.inf], alpha = 1 / mp.mpf(2)) run("chebyshev1", lambda x: 1/mp.sqrt(1-x*x), [-1, 1]) run("chebyshev2", lambda x: mp.sqrt(1-x*x), [-1, 1]) run("jacobi", lambda x: (1-x)**(1/mp.mpf(3)) * (1+x)**(1/mp.mpf(5)), [-1, 1], alpha = 1 / mp.mpf(3), beta = 1 / mp.mpf(5) )
def test_levin_2(): # [2] A. Sidi - "Pratical Extrapolation Methods" p.373 mp.dps = 17 z = mp.mpf(10) eps = mp.mpf(mp.eps) with mp.extraprec(2 * mp.prec): L = mp.levin(method="sidi", variant="t") n = 0 while 1: s = (-1)**n * mp.fac(n) * z**(-n) v, e = L.step(s) n += 1 if e < eps: break if n > 1000: raise RuntimeError("iteration limit exceeded") eps = mp.exp(0.9 * mp.log(eps)) exact = mp.quad(lambda x: mp.exp(-x) / (1 + x / z), [0, mp.inf]) # there is also a symbolic expression for the integral: # exact = z * mp.exp(z) * mp.expint(1,z) err = abs(v - exact) assert err < eps w = mp.nsum(lambda n: (-1)**n * mp.fac(n) * z**(-n), [0, mp.inf], method="sidi", levin_variant="t") assert err < eps
def test_svd_test_case(): # a test case from Golub and Reinsch # (see wilkinson/reinsch: handbook for auto. comp., vol ii-linear algebra, 134-151(1971).) eps = mp.exp(0.8 * mp.log(mp.eps)) a = [[22, 10, 2, 3, 7], [14, 7, 10, 0, 8], [-1, 13, -1, -11, 3], [-3, -2, 13, -2, 4], [ 9, 8, 1, -2, 4], [ 9, 1, -7, 5, -1], [ 2, -6, 6, 5, 1], [ 4, 5, 0, -2, 2]] a = mp.matrix(a) b = mp.matrix([mp.sqrt(1248), 20, mp.sqrt(384), 0, 0]) S = mp.svd_r(a, compute_uv = False) S -= b assert mp.mnorm(S) < eps S = mp.svd_c(a, compute_uv = False) S -= b assert mp.mnorm(S) < eps
def run_eigsy(A, verbose=False): if verbose: print("original matrix:\n", str(A)) D, Q = mp.eigsy(A) B = Q * mp.diag(D) * Q.transpose() C = A - B E = Q * Q.transpose() - mp.eye(A.rows) if verbose: print("eigenvalues:\n", D) print("eigenvectors:\n", Q) NC = mp.mnorm(C) NE = mp.mnorm(E) if verbose: print("difference:", NC, "\n", C, "\n") print("difference:", NE, "\n", E, "\n") eps = mp.exp(0.8 * mp.log(mp.eps)) assert NC < eps assert NE < eps return NC
def run_eigsy(A, verbose = False): if verbose: print("original matrix:\n", str(A)) D, Q = mp.eigsy(A) B = Q * mp.diag(D) * Q.transpose() C = A - B E = Q * Q.transpose() - mp.eye(A.rows) if verbose: print("eigenvalues:\n", D) print("eigenvectors:\n", Q) NC = mp.mnorm(C) NE = mp.mnorm(E) if verbose: print("difference:", NC, "\n", C, "\n") print("difference:", NE, "\n", E, "\n") eps = mp.exp( 0.8 * mp.log(mp.eps)) assert NC < eps assert NE < eps return NC
def run_eig(A, verbose=0): if verbose > 1: print("original matrix (eig):\n", A) n = A.rows E, EL, ER = mp.eig(A, left=True, right=True) if verbose > 1: print("E:\n", E) print("EL:\n", EL) print("ER:\n", ER) eps = mp.exp(0.8 * mp.log(mp.eps)) err0 = 0 for i in xrange(n): B = A * ER[:, i] - E[i] * ER[:, i] err0 = max(err0, mp.mnorm(B)) B = EL[i, :] * A - EL[i, :] * E[i] err0 = max(err0, mp.mnorm(B)) err0 /= n * n if verbose > 0: print("difference (E):", err0) assert err0 < eps
def run_hessenberg(A, verbose=0): if verbose > 1: print("original matrix (hessenberg):\n", A) n = A.rows Q, H = mp.hessenberg(A) if verbose > 1: print("Q:\n", Q) print("H:\n", H) B = Q * H * Q.transpose_conj() eps = mp.exp(0.8 * mp.log(mp.eps)) err0 = 0 for x in xrange(n): for y in xrange(n): err0 += abs(A[y, x] - B[y, x]) err0 /= n * n err1 = 0 for x in xrange(n): for y in xrange(x + 2, n): err1 += abs(H[y, x]) if verbose > 0: print("difference (H):", err0, err1) if verbose > 1: print("B:\n", B) assert err0 < eps assert err1 == 0
def test_gauss_quadrature_dynamic(verbose=False): n = 5 A = mp.randmatrix(2 * n, 1) def F(x): r = 0 for i in xrange(len(A) - 1, -1, -1): r = r * x + A[i] return r def run(qtype, FW, R, alpha=0, beta=0): X, W = mp.gauss_quadrature(n, qtype, alpha=alpha, beta=beta) a = 0 for i in xrange(len(X)): a += W[i] * F(X[i]) b = mp.quad(lambda x: FW(x) * F(x), R) c = mp.fabs(a - b) if verbose: print(qtype, c, a, b) assert c < 1e-5 run("legendre", lambda x: 1, [-1, 1]) run("legendre01", lambda x: 1, [0, 1]) run("hermite", lambda x: mp.exp(-x * x), [-mp.inf, mp.inf]) run("laguerre", lambda x: mp.exp(-x), [0, mp.inf]) run("glaguerre", lambda x: mp.sqrt(x) * mp.exp(-x), [0, mp.inf], alpha=1 / mp.mpf(2)) run("chebyshev1", lambda x: 1 / mp.sqrt(1 - x * x), [-1, 1]) run("chebyshev2", lambda x: mp.sqrt(1 - x * x), [-1, 1]) run("jacobi", lambda x: (1 - x)**(1 / mp.mpf(3)) * (1 + x)**(1 / mp.mpf(5)), [-1, 1], alpha=1 / mp.mpf(3), beta=1 / mp.mpf(5))
def test_levin_3(): mp.dps = 17 z=mp.mpf(2) eps = mp.mpf(mp.eps) with mp.extraprec(7*mp.prec): # we need copious amount of precision to sum this highly divergent series L = mp.levin(method = "levin", variant = "t") n, s = 0, 0 while 1: s += (-z)**n * mp.fac(4 * n) / (mp.fac(n) * mp.fac(2 * n) * (4 ** n)) n += 1 v, e = L.step_psum(s) if e < eps: break if n > 1000: raise RuntimeError("iteration limit exceeded") eps = mp.exp(0.8 * mp.log(eps)) exact = mp.quad(lambda x: mp.exp( -x * x / 2 - z * x ** 4), [0,mp.inf]) * 2 / mp.sqrt(2 * mp.pi) # there is also a symbolic expression for the integral: # exact = mp.exp(mp.one / (32 * z)) * mp.besselk(mp.one / 4, mp.one / (32 * z)) / (4 * mp.sqrt(z * mp.pi)) err = abs(v - exact) assert err < eps w = mp.nsum(lambda n: (-z)**n * mp.fac(4 * n) / (mp.fac(n) * mp.fac(2 * n) * (4 ** n)), [0, mp.inf], method = "levin", levin_variant = "t", workprec = 8*mp.prec, steps = [2] + [1 for x in xrange(1000)]) err = abs(v - w) assert err < eps
def test_levin_2(): # [2] A. Sidi - "Pratical Extrapolation Methods" p.373 mp.dps = 17 z=mp.mpf(10) eps = mp.mpf(mp.eps) with mp.extraprec(2 * mp.prec): L = mp.levin(method = "sidi", variant = "t") n = 0 while 1: s = (-1)**n * mp.fac(n) * z ** (-n) v, e = L.step(s) n += 1 if e < eps: break if n > 1000: raise RuntimeError("iteration limit exceeded") eps = mp.exp(0.9 * mp.log(eps)) exact = mp.quad(lambda x: mp.exp(-x)/(1+x/z),[0,mp.inf]) # there is also a symbolic expression for the integral: # exact = z * mp.exp(z) * mp.expint(1,z) err = abs(v - exact) assert err < eps w = mp.nsum(lambda n: (-1) ** n * mp.fac(n) * z ** (-n), [0, mp.inf], method = "sidi", levin_variant = "t") assert err < eps
def test_cohen_alt_0(): mp.dps = 17 AC = mp.cohen_alt() S, s, n = [], 0, 1 while 1: s += -((-1)**n) * mp.one / (n * n) n += 1 S.append(s) v, e = AC.update_psum(S) if e < mp.eps: break if n > 1000: raise RuntimeError("iteration limit exceeded") eps = mp.exp(0.9 * mp.log(mp.eps)) err = abs(v - mp.pi**2 / 12) assert err < eps
def test_cohen_alt_0(): mp.dps = 17 AC = mp.cohen_alt() S, s, n = [], 0, 1 while 1: s += -((-1) ** n) * mp.one / (n * n) n += 1 S.append(s) v, e = AC.update_psum(S) if e < mp.eps: break if n > 1000: raise RuntimeError("iteration limit exceeded") eps = mp.exp(0.9 * mp.log(mp.eps)) err = abs(v - mp.pi ** 2 / 12) assert err < eps
def test_cohen_alt_1(): mp.dps = 17 A = [] AC = mp.cohen_alt() n = 1 while 1: A.append(mp.loggamma(1 + mp.one / (2 * n - 1))) A.append(-mp.loggamma(1 + mp.one / (2 * n))) n += 1 v, e = AC.update(A) if e < mp.eps: break if n > 1000: raise RuntimeError("iteration limit exceeded") v = mp.exp(v) err = abs(v - 1.06215090557106) assert err < 1e-12
def test_cohen_alt_1(): mp.dps = 17 A = [] AC = mp.cohen_alt() n = 1 while 1: A.append( mp.loggamma(1 + mp.one / (2 * n - 1))) A.append(-mp.loggamma(1 + mp.one / (2 * n))) n += 1 v, e = AC.update(A) if e < mp.eps: break if n > 1000: raise RuntimeError("iteration limit exceeded") v = mp.exp(v) err = abs(v - 1.06215090557106) assert err < 1e-12
def run_schur(A, verbose=0): if verbose > 1: print("original matrix (schur):\n", A) n = A.rows Q, R = mp.schur(A) if verbose > 1: print("Q:\n", Q) print("R:\n", R) B = Q * R * Q.transpose_conj() C = Q * Q.transpose_conj() eps = mp.exp(0.8 * mp.log(mp.eps)) err0 = 0 for x in xrange(n): for y in xrange(n): err0 += abs(A[y, x] - B[y, x]) err0 /= n * n err1 = 0 for x in xrange(n): for y in xrange(n): if x == y: C[y, x] -= 1 err1 += abs(C[y, x]) err1 /= n * n err2 = 0 for x in xrange(n): for y in xrange(x + 1, n): err2 += abs(R[y, x]) if verbose > 0: print("difference (S):", err0, err1, err2) if verbose > 1: print("B:\n", B) assert err0 < eps assert err1 < eps assert err2 == 0
def test_levin_nsum(): mp.dps = 17 with mp.extraprec(mp.prec): z = mp.mpf(10) ** (-10) a = mp.nsum(lambda n: n**(-(1+z)), [1, mp.inf], method = "l") - 1 / z assert abs(a - mp.euler) < 1e-10 eps = mp.exp(0.8 * mp.log(mp.eps)) a = mp.nsum(lambda n: (-1)**(n-1) / n, [1, mp.inf], method = "sidi") assert abs(a - mp.log(2)) < eps z = 2 + 1j f = lambda n: mp.rf(2 / mp.mpf(3), n) * mp.rf(4 / mp.mpf(3), n) * z**n / (mp.rf(1 / mp.mpf(3), n) * mp.fac(n)) v = mp.nsum(f, [0, mp.inf], method = "levin", steps = [10 for x in xrange(1000)]) exact = mp.hyp2f1(2 / mp.mpf(3), 4 / mp.mpf(3), 1 / mp.mpf(3), z) assert abs(exact - v) < eps
def test_levin_0(): mp.dps = 17 eps = mp.mpf(mp.eps) with mp.extraprec(2 * mp.prec): L = mp.levin(method = "levin", variant = "u") S, s, n = [], 0, 1 while 1: s += mp.one / (n * n) n += 1 S.append(s) v, e = L.update_psum(S) if e < eps: break if n > 1000: raise RuntimeError("iteration limit exceeded") eps = mp.exp(0.9 * mp.log(eps)) err = abs(v - mp.pi ** 2 / 6) assert err < eps w = mp.nsum(lambda n: 1/(n * n), [1, mp.inf], method = "levin", levin_variant = "u") err = abs(v - w) assert err < eps
def test_svd_test_case(): # a test case from Golub and Reinsch # (see wilkinson/reinsch: handbook for auto. comp., vol ii-linear algebra, 134-151(1971).) eps = mp.exp(0.8 * mp.log(mp.eps)) a = [[22, 10, 2, 3, 7], [14, 7, 10, 0, 8], [-1, 13, -1, -11, 3], [-3, -2, 13, -2, 4], [9, 8, 1, -2, 4], [9, 1, -7, 5, -1], [2, -6, 6, 5, 1], [4, 5, 0, -2, 2]] a = mp.matrix(a) b = mp.matrix([mp.sqrt(1248), 20, mp.sqrt(384), 0, 0]) S = mp.svd_r(a, compute_uv=False) S -= b assert mp.mnorm(S) < eps S = mp.svd_c(a, compute_uv=False) S -= b assert mp.mnorm(S) < eps
def test_levin_1(): mp.dps = 17 eps = mp.mpf(mp.eps) with mp.extraprec(2 * mp.prec): L = mp.levin(method = "levin", variant = "v") A, n = [], 1 while 1: s = mp.mpf(n) ** (2 + 3j) n += 1 A.append(s) v, e = L.update(A) if e < eps: break if n > 1000: raise RuntimeError("iteration limit exceeded") eps = mp.exp(0.9 * mp.log(eps)) err = abs(v - mp.zeta(-2-3j)) assert err < eps w = mp.nsum(lambda n: n ** (2 + 3j), [1, mp.inf], method = "levin", levin_variant = "v") err = abs(v - w) assert err < eps
def test_levin_nsum(): mp.dps = 17 with mp.extraprec(mp.prec): z = mp.mpf(10)**(-10) a = mp.nsum(lambda n: n**(-(1 + z)), [1, mp.inf], method="l") - 1 / z assert abs(a - mp.euler) < 1e-10 eps = mp.exp(0.8 * mp.log(mp.eps)) a = mp.nsum(lambda n: (-1)**(n - 1) / n, [1, mp.inf], method="sidi") assert abs(a - mp.log(2)) < eps z = 2 + 1j f = lambda n: mp.rf(2 / mp.mpf(3), n) * mp.rf(4 / mp.mpf(3), n) * z**n / ( mp.rf(1 / mp.mpf(3), n) * mp.fac(n)) v = mp.nsum(f, [0, mp.inf], method="levin", steps=[10 for x in xrange(1000)]) exact = mp.hyp2f1(2 / mp.mpf(3), 4 / mp.mpf(3), 1 / mp.mpf(3), z) assert abs(exact - v) < eps
def test_levin_0(): mp.dps = 17 eps = mp.mpf(mp.eps) with mp.extraprec(2 * mp.prec): L = mp.levin(method="levin", variant="u") S, s, n = [], 0, 1 while 1: s += mp.one / (n * n) n += 1 S.append(s) v, e = L.update_psum(S) if e < eps: break if n > 1000: raise RuntimeError("iteration limit exceeded") eps = mp.exp(0.9 * mp.log(eps)) err = abs(v - mp.pi**2 / 6) assert err < eps w = mp.nsum(lambda n: 1 / (n * n), [1, mp.inf], method="levin", levin_variant="u") err = abs(v - w) assert err < eps
def test_levin_1(): mp.dps = 17 eps = mp.mpf(mp.eps) with mp.extraprec(2 * mp.prec): L = mp.levin(method="levin", variant="v") A, n = [], 1 while 1: s = mp.mpf(n)**(2 + 3j) n += 1 A.append(s) v, e = L.update(A) if e < eps: break if n > 1000: raise RuntimeError("iteration limit exceeded") eps = mp.exp(0.9 * mp.log(eps)) err = abs(v - mp.zeta(-2 - 3j)) assert err < eps w = mp.nsum(lambda n: n**(2 + 3j), [1, mp.inf], method="levin", levin_variant="v") err = abs(v - w) assert err < eps
def run_svd_r(A, full_matrices=False, verbose=True): m, n = A.rows, A.cols eps = mp.exp(0.8 * mp.log(mp.eps)) if verbose: print("original matrix:\n", str(A)) print("full", full_matrices) U, S0, V = mp.svd_r(A, full_matrices=full_matrices) S = mp.zeros(U.cols, V.rows) for j in xrange(min(m, n)): S[j, j] = S0[j] if verbose: print("U:\n", str(U)) print("S:\n", str(S0)) print("V:\n", str(V)) C = U * S * V - A err = mp.mnorm(C) if verbose: print("C\n", str(C), "\n", err) assert err < eps D = V * V.transpose() - mp.eye(V.rows) err = mp.mnorm(D) if verbose: print("D:\n", str(D), "\n", err) assert err < eps E = U.transpose() * U - mp.eye(U.cols) err = mp.mnorm(E) if verbose: print("E:\n", str(E), "\n", err) assert err < eps
def run_svd_r(A, full_matrices = False, verbose = True): m, n = A.rows, A.cols eps = mp.exp(0.8 * mp.log(mp.eps)) if verbose: print("original matrix:\n", str(A)) print("full", full_matrices) U, S0, V = mp.svd_r(A, full_matrices = full_matrices) S = mp.zeros(U.cols, V.rows) for j in xrange(min(m, n)): S[j,j] = S0[j] if verbose: print("U:\n", str(U)) print("S:\n", str(S0)) print("V:\n", str(V)) C = U * S * V - A err = mp.mnorm(C) if verbose: print("C\n", str(C), "\n", err) assert err < eps D = V * V.transpose() - mp.eye(V.rows) err = mp.mnorm(D) if verbose: print("D:\n", str(D), "\n", err) assert err < eps E = U.transpose() * U - mp.eye(U.cols) err = mp.mnorm(E) if verbose: print("E:\n", str(E), "\n", err) assert err < eps
def pdf_gauss_mp(x, sigma, mean): return mp.mpf(1.) / mp.sqrt(mp.mpf("2.") * sigma**2 * mp.pi) * mp.exp( -(x - mean)**2 / (mp.mpf("2.") * sigma**2))
def pdf_gauss_mp(x, sigma, mean): return mp.mpf(1.) / mp.sqrt(mp.mpf("2.") * sigma ** 2 * mp.pi) * mp.exp( - (x - mean) ** 2 / (mp.mpf("2.") * sigma ** 2))
def exp(self): """ Exponencial de un intervalo: 'self.exp()' """ return Intervalo(mp.exp(self.lo), mp.exp(self.hi))
def exp(a): try: return a.exp() except: return mp.exp(a)
def exp(self): """ Exponencial de un intervalo: 'self.exp()' """ return Intervalo( mp.exp(self.lo), mp.exp(self.hi) )