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845 lines
29 KiB
845 lines
29 KiB
"""
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Implements the PSLQ algorithm for integer relation detection,
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and derivative algorithms for constant recognition.
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"""
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from .libmp.backend import xrange
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from .libmp import int_types, sqrt_fixed
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# round to nearest integer (can be done more elegantly...)
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def round_fixed(x, prec):
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return ((x + (1<<(prec-1))) >> prec) << prec
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class IdentificationMethods(object):
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pass
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def pslq(ctx, x, tol=None, maxcoeff=1000, maxsteps=100, verbose=False):
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r"""
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Given a vector of real numbers `x = [x_0, x_1, ..., x_n]`, ``pslq(x)``
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uses the PSLQ algorithm to find a list of integers
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`[c_0, c_1, ..., c_n]` such that
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.. math ::
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|c_1 x_1 + c_2 x_2 + ... + c_n x_n| < \mathrm{tol}
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and such that `\max |c_k| < \mathrm{maxcoeff}`. If no such vector
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exists, :func:`~mpmath.pslq` returns ``None``. The tolerance defaults to
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3/4 of the working precision.
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**Examples**
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Find rational approximations for `\pi`::
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>>> from mpmath import *
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>>> mp.dps = 15; mp.pretty = True
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>>> pslq([-1, pi], tol=0.01)
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[22, 7]
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>>> pslq([-1, pi], tol=0.001)
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[355, 113]
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>>> mpf(22)/7; mpf(355)/113; +pi
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3.14285714285714
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3.14159292035398
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3.14159265358979
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Pi is not a rational number with denominator less than 1000::
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>>> pslq([-1, pi])
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>>>
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To within the standard precision, it can however be approximated
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by at least one rational number with denominator less than `10^{12}`::
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>>> p, q = pslq([-1, pi], maxcoeff=10**12)
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>>> print(p); print(q)
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238410049439
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75888275702
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>>> mpf(p)/q
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3.14159265358979
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The PSLQ algorithm can be applied to long vectors. For example,
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we can investigate the rational (in)dependence of integer square
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roots::
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>>> mp.dps = 30
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>>> pslq([sqrt(n) for n in range(2, 5+1)])
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>>>
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>>> pslq([sqrt(n) for n in range(2, 6+1)])
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>>>
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>>> pslq([sqrt(n) for n in range(2, 8+1)])
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[2, 0, 0, 0, 0, 0, -1]
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**Machin formulas**
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A famous formula for `\pi` is Machin's,
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.. math ::
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\frac{\pi}{4} = 4 \operatorname{acot} 5 - \operatorname{acot} 239
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There are actually infinitely many formulas of this type. Two
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others are
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.. math ::
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\frac{\pi}{4} = \operatorname{acot} 1
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\frac{\pi}{4} = 12 \operatorname{acot} 49 + 32 \operatorname{acot} 57
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+ 5 \operatorname{acot} 239 + 12 \operatorname{acot} 110443
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We can easily verify the formulas using the PSLQ algorithm::
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>>> mp.dps = 30
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>>> pslq([pi/4, acot(1)])
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[1, -1]
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>>> pslq([pi/4, acot(5), acot(239)])
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[1, -4, 1]
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>>> pslq([pi/4, acot(49), acot(57), acot(239), acot(110443)])
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[1, -12, -32, 5, -12]
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We could try to generate a custom Machin-like formula by running
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the PSLQ algorithm with a few inverse cotangent values, for example
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acot(2), acot(3) ... acot(10). Unfortunately, there is a linear
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dependence among these values, resulting in only that dependence
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being detected, with a zero coefficient for `\pi`::
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>>> pslq([pi] + [acot(n) for n in range(2,11)])
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[0, 1, -1, 0, 0, 0, -1, 0, 0, 0]
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We get better luck by removing linearly dependent terms::
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>>> pslq([pi] + [acot(n) for n in range(2,11) if n not in (3, 5)])
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[1, -8, 0, 0, 4, 0, 0, 0]
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In other words, we found the following formula::
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>>> 8*acot(2) - 4*acot(7)
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3.14159265358979323846264338328
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>>> +pi
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3.14159265358979323846264338328
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**Algorithm**
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This is a fairly direct translation to Python of the pseudocode given by
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David Bailey, "The PSLQ Integer Relation Algorithm":
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http://www.cecm.sfu.ca/organics/papers/bailey/paper/html/node3.html
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The present implementation uses fixed-point instead of floating-point
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arithmetic, since this is significantly (about 7x) faster.
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"""
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n = len(x)
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if n < 2:
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raise ValueError("n cannot be less than 2")
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# At too low precision, the algorithm becomes meaningless
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prec = ctx.prec
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if prec < 53:
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raise ValueError("prec cannot be less than 53")
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if verbose and prec // max(2,n) < 5:
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print("Warning: precision for PSLQ may be too low")
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target = int(prec * 0.75)
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if tol is None:
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tol = ctx.mpf(2)**(-target)
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else:
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tol = ctx.convert(tol)
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extra = 60
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prec += extra
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if verbose:
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print("PSLQ using prec %i and tol %s" % (prec, ctx.nstr(tol)))
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tol = ctx.to_fixed(tol, prec)
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assert tol
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# Convert to fixed-point numbers. The dummy None is added so we can
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# use 1-based indexing. (This just allows us to be consistent with
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# Bailey's indexing. The algorithm is 100 lines long, so debugging
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# a single wrong index can be painful.)
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x = [None] + [ctx.to_fixed(ctx.mpf(xk), prec) for xk in x]
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# Sanity check on magnitudes
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minx = min(abs(xx) for xx in x[1:])
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if not minx:
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raise ValueError("PSLQ requires a vector of nonzero numbers")
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if minx < tol//100:
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if verbose:
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print("STOPPING: (one number is too small)")
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return None
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g = sqrt_fixed((4<<prec)//3, prec)
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A = {}
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B = {}
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H = {}
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# Initialization
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# step 1
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for i in xrange(1, n+1):
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for j in xrange(1, n+1):
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A[i,j] = B[i,j] = (i==j) << prec
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H[i,j] = 0
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# step 2
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s = [None] + [0] * n
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for k in xrange(1, n+1):
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t = 0
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for j in xrange(k, n+1):
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t += (x[j]**2 >> prec)
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s[k] = sqrt_fixed(t, prec)
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t = s[1]
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y = x[:]
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for k in xrange(1, n+1):
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y[k] = (x[k] << prec) // t
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s[k] = (s[k] << prec) // t
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# step 3
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for i in xrange(1, n+1):
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for j in xrange(i+1, n):
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H[i,j] = 0
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if i <= n-1:
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if s[i]:
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H[i,i] = (s[i+1] << prec) // s[i]
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else:
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H[i,i] = 0
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for j in range(1, i):
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sjj1 = s[j]*s[j+1]
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if sjj1:
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H[i,j] = ((-y[i]*y[j])<<prec)//sjj1
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else:
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H[i,j] = 0
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# step 4
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for i in xrange(2, n+1):
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for j in xrange(i-1, 0, -1):
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#t = floor(H[i,j]/H[j,j] + 0.5)
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if H[j,j]:
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t = round_fixed((H[i,j] << prec)//H[j,j], prec)
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else:
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#t = 0
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continue
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y[j] = y[j] + (t*y[i] >> prec)
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for k in xrange(1, j+1):
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H[i,k] = H[i,k] - (t*H[j,k] >> prec)
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for k in xrange(1, n+1):
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A[i,k] = A[i,k] - (t*A[j,k] >> prec)
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B[k,j] = B[k,j] + (t*B[k,i] >> prec)
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# Main algorithm
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for REP in range(maxsteps):
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# Step 1
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m = -1
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szmax = -1
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for i in range(1, n):
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h = H[i,i]
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sz = (g**i * abs(h)) >> (prec*(i-1))
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if sz > szmax:
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m = i
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szmax = sz
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# Step 2
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y[m], y[m+1] = y[m+1], y[m]
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for i in xrange(1,n+1): H[m,i], H[m+1,i] = H[m+1,i], H[m,i]
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for i in xrange(1,n+1): A[m,i], A[m+1,i] = A[m+1,i], A[m,i]
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for i in xrange(1,n+1): B[i,m], B[i,m+1] = B[i,m+1], B[i,m]
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# Step 3
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if m <= n - 2:
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t0 = sqrt_fixed((H[m,m]**2 + H[m,m+1]**2)>>prec, prec)
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# A zero element probably indicates that the precision has
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# been exhausted. XXX: this could be spurious, due to
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# using fixed-point arithmetic
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if not t0:
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break
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t1 = (H[m,m] << prec) // t0
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t2 = (H[m,m+1] << prec) // t0
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for i in xrange(m, n+1):
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t3 = H[i,m]
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t4 = H[i,m+1]
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H[i,m] = (t1*t3+t2*t4) >> prec
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H[i,m+1] = (-t2*t3+t1*t4) >> prec
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# Step 4
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for i in xrange(m+1, n+1):
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for j in xrange(min(i-1, m+1), 0, -1):
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try:
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t = round_fixed((H[i,j] << prec)//H[j,j], prec)
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# Precision probably exhausted
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except ZeroDivisionError:
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break
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y[j] = y[j] + ((t*y[i]) >> prec)
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for k in xrange(1, j+1):
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H[i,k] = H[i,k] - (t*H[j,k] >> prec)
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for k in xrange(1, n+1):
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A[i,k] = A[i,k] - (t*A[j,k] >> prec)
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B[k,j] = B[k,j] + (t*B[k,i] >> prec)
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# Until a relation is found, the error typically decreases
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# slowly (e.g. a factor 1-10) with each step TODO: we could
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# compare err from two successive iterations. If there is a
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# large drop (several orders of magnitude), that indicates a
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# "high quality" relation was detected. Reporting this to
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# the user somehow might be useful.
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best_err = maxcoeff<<prec
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for i in xrange(1, n+1):
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err = abs(y[i])
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# Maybe we are done?
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if err < tol:
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# We are done if the coefficients are acceptable
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vec = [int(round_fixed(B[j,i], prec) >> prec) for j in \
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range(1,n+1)]
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if max(abs(v) for v in vec) < maxcoeff:
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if verbose:
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print("FOUND relation at iter %i/%i, error: %s" % \
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(REP, maxsteps, ctx.nstr(err / ctx.mpf(2)**prec, 1)))
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return vec
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best_err = min(err, best_err)
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# Calculate a lower bound for the norm. We could do this
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# more exactly (using the Euclidean norm) but there is probably
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# no practical benefit.
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recnorm = max(abs(h) for h in H.values())
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if recnorm:
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norm = ((1 << (2*prec)) // recnorm) >> prec
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norm //= 100
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else:
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norm = ctx.inf
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if verbose:
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print("%i/%i: Error: %8s Norm: %s" % \
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(REP, maxsteps, ctx.nstr(best_err / ctx.mpf(2)**prec, 1), norm))
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if norm >= maxcoeff:
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break
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if verbose:
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print("CANCELLING after step %i/%i." % (REP, maxsteps))
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print("Could not find an integer relation. Norm bound: %s" % norm)
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return None
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def findpoly(ctx, x, n=1, **kwargs):
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r"""
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``findpoly(x, n)`` returns the coefficients of an integer
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polynomial `P` of degree at most `n` such that `P(x) \approx 0`.
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If no polynomial having `x` as a root can be found,
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:func:`~mpmath.findpoly` returns ``None``.
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:func:`~mpmath.findpoly` works by successively calling :func:`~mpmath.pslq` with
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the vectors `[1, x]`, `[1, x, x^2]`, `[1, x, x^2, x^3]`, ...,
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`[1, x, x^2, .., x^n]` as input. Keyword arguments given to
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:func:`~mpmath.findpoly` are forwarded verbatim to :func:`~mpmath.pslq`. In
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particular, you can specify a tolerance for `P(x)` with ``tol``
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and a maximum permitted coefficient size with ``maxcoeff``.
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For large values of `n`, it is recommended to run :func:`~mpmath.findpoly`
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at high precision; preferably 50 digits or more.
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**Examples**
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By default (degree `n = 1`), :func:`~mpmath.findpoly` simply finds a linear
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polynomial with a rational root::
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>>> from mpmath import *
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>>> mp.dps = 15; mp.pretty = True
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>>> findpoly(0.7)
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[-10, 7]
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The generated coefficient list is valid input to ``polyval`` and
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``polyroots``::
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>>> nprint(polyval(findpoly(phi, 2), phi), 1)
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-2.0e-16
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>>> for r in polyroots(findpoly(phi, 2)):
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... print(r)
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...
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-0.618033988749895
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1.61803398874989
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Numbers of the form `m + n \sqrt p` for integers `(m, n, p)` are
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solutions to quadratic equations. As we find here, `1+\sqrt 2`
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is a root of the polynomial `x^2 - 2x - 1`::
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>>> findpoly(1+sqrt(2), 2)
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[1, -2, -1]
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>>> findroot(lambda x: x**2 - 2*x - 1, 1)
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2.4142135623731
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Despite only containing square roots, the following number results
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in a polynomial of degree 4::
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>>> findpoly(sqrt(2)+sqrt(3), 4)
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[1, 0, -10, 0, 1]
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In fact, `x^4 - 10x^2 + 1` is the *minimal polynomial* of
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`r = \sqrt 2 + \sqrt 3`, meaning that a rational polynomial of
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lower degree having `r` as a root does not exist. Given sufficient
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precision, :func:`~mpmath.findpoly` will usually find the correct
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minimal polynomial of a given algebraic number.
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**Non-algebraic numbers**
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If :func:`~mpmath.findpoly` fails to find a polynomial with given
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coefficient size and tolerance constraints, that means no such
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polynomial exists.
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We can verify that `\pi` is not an algebraic number of degree 3 with
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coefficients less than 1000::
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>>> mp.dps = 15
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>>> findpoly(pi, 3)
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>>>
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It is always possible to find an algebraic approximation of a number
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using one (or several) of the following methods:
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1. Increasing the permitted degree
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2. Allowing larger coefficients
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3. Reducing the tolerance
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One example of each method is shown below::
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>>> mp.dps = 15
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>>> findpoly(pi, 4)
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[95, -545, 863, -183, -298]
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>>> findpoly(pi, 3, maxcoeff=10000)
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[836, -1734, -2658, -457]
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>>> findpoly(pi, 3, tol=1e-7)
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[-4, 22, -29, -2]
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It is unknown whether Euler's constant is transcendental (or even
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irrational). We can use :func:`~mpmath.findpoly` to check that if is
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an algebraic number, its minimal polynomial must have degree
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at least 7 and a coefficient of magnitude at least 1000000::
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>>> mp.dps = 200
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>>> findpoly(euler, 6, maxcoeff=10**6, tol=1e-100, maxsteps=1000)
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>>>
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Note that the high precision and strict tolerance is necessary
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for such high-degree runs, since otherwise unwanted low-accuracy
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approximations will be detected. It may also be necessary to set
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maxsteps high to prevent a premature exit (before the coefficient
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bound has been reached). Running with ``verbose=True`` to get an
|
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idea what is happening can be useful.
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"""
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x = ctx.mpf(x)
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if n < 1:
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raise ValueError("n cannot be less than 1")
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if x == 0:
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return [1, 0]
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xs = [ctx.mpf(1)]
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for i in range(1,n+1):
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xs.append(x**i)
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a = ctx.pslq(xs, **kwargs)
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if a is not None:
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return a[::-1]
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|
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def fracgcd(p, q):
|
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x, y = p, q
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|
while y:
|
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x, y = y, x % y
|
|
if x != 1:
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p //= x
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q //= x
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|
if q == 1:
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return p
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return p, q
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|
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def pslqstring(r, constants):
|
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q = r[0]
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r = r[1:]
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s = []
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for i in range(len(r)):
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p = r[i]
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if p:
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z = fracgcd(-p,q)
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cs = constants[i][1]
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if cs == '1':
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cs = ''
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|
else:
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cs = '*' + cs
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if isinstance(z, int_types):
|
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if z > 0: term = str(z) + cs
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|
else: term = ("(%s)" % z) + cs
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|
else:
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|
term = ("(%s/%s)" % z) + cs
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s.append(term)
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|
s = ' + '.join(s)
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|
if '+' in s or '*' in s:
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s = '(' + s + ')'
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return s or '0'
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|
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def prodstring(r, constants):
|
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q = r[0]
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r = r[1:]
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num = []
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den = []
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for i in range(len(r)):
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p = r[i]
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if p:
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z = fracgcd(-p,q)
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cs = constants[i][1]
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|
if isinstance(z, int_types):
|
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if abs(z) == 1: t = cs
|
|
else: t = '%s**%s' % (cs, abs(z))
|
|
([num,den][z<0]).append(t)
|
|
else:
|
|
t = '%s**(%s/%s)' % (cs, abs(z[0]), z[1])
|
|
([num,den][z[0]<0]).append(t)
|
|
num = '*'.join(num)
|
|
den = '*'.join(den)
|
|
if num and den: return "(%s)/(%s)" % (num, den)
|
|
if num: return num
|
|
if den: return "1/(%s)" % den
|
|
|
|
def quadraticstring(ctx,t,a,b,c):
|
|
if c < 0:
|
|
a,b,c = -a,-b,-c
|
|
u1 = (-b+ctx.sqrt(b**2-4*a*c))/(2*c)
|
|
u2 = (-b-ctx.sqrt(b**2-4*a*c))/(2*c)
|
|
if abs(u1-t) < abs(u2-t):
|
|
if b: s = '((%s+sqrt(%s))/%s)' % (-b,b**2-4*a*c,2*c)
|
|
else: s = '(sqrt(%s)/%s)' % (-4*a*c,2*c)
|
|
else:
|
|
if b: s = '((%s-sqrt(%s))/%s)' % (-b,b**2-4*a*c,2*c)
|
|
else: s = '(-sqrt(%s)/%s)' % (-4*a*c,2*c)
|
|
return s
|
|
|
|
# Transformation y = f(x,c), with inverse function x = f(y,c)
|
|
# The third entry indicates whether the transformation is
|
|
# redundant when c = 1
|
|
transforms = [
|
|
(lambda ctx,x,c: x*c, '$y/$c', 0),
|
|
(lambda ctx,x,c: x/c, '$c*$y', 1),
|
|
(lambda ctx,x,c: c/x, '$c/$y', 0),
|
|
(lambda ctx,x,c: (x*c)**2, 'sqrt($y)/$c', 0),
|
|
(lambda ctx,x,c: (x/c)**2, '$c*sqrt($y)', 1),
|
|
(lambda ctx,x,c: (c/x)**2, '$c/sqrt($y)', 0),
|
|
(lambda ctx,x,c: c*x**2, 'sqrt($y)/sqrt($c)', 1),
|
|
(lambda ctx,x,c: x**2/c, 'sqrt($c)*sqrt($y)', 1),
|
|
(lambda ctx,x,c: c/x**2, 'sqrt($c)/sqrt($y)', 1),
|
|
(lambda ctx,x,c: ctx.sqrt(x*c), '$y**2/$c', 0),
|
|
(lambda ctx,x,c: ctx.sqrt(x/c), '$c*$y**2', 1),
|
|
(lambda ctx,x,c: ctx.sqrt(c/x), '$c/$y**2', 0),
|
|
(lambda ctx,x,c: c*ctx.sqrt(x), '$y**2/$c**2', 1),
|
|
(lambda ctx,x,c: ctx.sqrt(x)/c, '$c**2*$y**2', 1),
|
|
(lambda ctx,x,c: c/ctx.sqrt(x), '$c**2/$y**2', 1),
|
|
(lambda ctx,x,c: ctx.exp(x*c), 'log($y)/$c', 0),
|
|
(lambda ctx,x,c: ctx.exp(x/c), '$c*log($y)', 1),
|
|
(lambda ctx,x,c: ctx.exp(c/x), '$c/log($y)', 0),
|
|
(lambda ctx,x,c: c*ctx.exp(x), 'log($y/$c)', 1),
|
|
(lambda ctx,x,c: ctx.exp(x)/c, 'log($c*$y)', 1),
|
|
(lambda ctx,x,c: c/ctx.exp(x), 'log($c/$y)', 0),
|
|
(lambda ctx,x,c: ctx.ln(x*c), 'exp($y)/$c', 0),
|
|
(lambda ctx,x,c: ctx.ln(x/c), '$c*exp($y)', 1),
|
|
(lambda ctx,x,c: ctx.ln(c/x), '$c/exp($y)', 0),
|
|
(lambda ctx,x,c: c*ctx.ln(x), 'exp($y/$c)', 1),
|
|
(lambda ctx,x,c: ctx.ln(x)/c, 'exp($c*$y)', 1),
|
|
(lambda ctx,x,c: c/ctx.ln(x), 'exp($c/$y)', 0),
|
|
]
|
|
|
|
def identify(ctx, x, constants=[], tol=None, maxcoeff=1000, full=False,
|
|
verbose=False):
|
|
r"""
|
|
Given a real number `x`, ``identify(x)`` attempts to find an exact
|
|
formula for `x`. This formula is returned as a string. If no match
|
|
is found, ``None`` is returned. With ``full=True``, a list of
|
|
matching formulas is returned.
|
|
|
|
As a simple example, :func:`~mpmath.identify` will find an algebraic
|
|
formula for the golden ratio::
|
|
|
|
>>> from mpmath import *
|
|
>>> mp.dps = 15; mp.pretty = True
|
|
>>> identify(phi)
|
|
'((1+sqrt(5))/2)'
|
|
|
|
:func:`~mpmath.identify` can identify simple algebraic numbers and simple
|
|
combinations of given base constants, as well as certain basic
|
|
transformations thereof. More specifically, :func:`~mpmath.identify`
|
|
looks for the following:
|
|
|
|
1. Fractions
|
|
2. Quadratic algebraic numbers
|
|
3. Rational linear combinations of the base constants
|
|
4. Any of the above after first transforming `x` into `f(x)` where
|
|
`f(x)` is `1/x`, `\sqrt x`, `x^2`, `\log x` or `\exp x`, either
|
|
directly or with `x` or `f(x)` multiplied or divided by one of
|
|
the base constants
|
|
5. Products of fractional powers of the base constants and
|
|
small integers
|
|
|
|
Base constants can be given as a list of strings representing mpmath
|
|
expressions (:func:`~mpmath.identify` will ``eval`` the strings to numerical
|
|
values and use the original strings for the output), or as a dict of
|
|
formula:value pairs.
|
|
|
|
In order not to produce spurious results, :func:`~mpmath.identify` should
|
|
be used with high precision; preferably 50 digits or more.
|
|
|
|
**Examples**
|
|
|
|
Simple identifications can be performed safely at standard
|
|
precision. Here the default recognition of rational, algebraic,
|
|
and exp/log of algebraic numbers is demonstrated::
|
|
|
|
>>> mp.dps = 15
|
|
>>> identify(0.22222222222222222)
|
|
'(2/9)'
|
|
>>> identify(1.9662210973805663)
|
|
'sqrt(((24+sqrt(48))/8))'
|
|
>>> identify(4.1132503787829275)
|
|
'exp((sqrt(8)/2))'
|
|
>>> identify(0.881373587019543)
|
|
'log(((2+sqrt(8))/2))'
|
|
|
|
By default, :func:`~mpmath.identify` does not recognize `\pi`. At standard
|
|
precision it finds a not too useful approximation. At slightly
|
|
increased precision, this approximation is no longer accurate
|
|
enough and :func:`~mpmath.identify` more correctly returns ``None``::
|
|
|
|
>>> identify(pi)
|
|
'(2**(176/117)*3**(20/117)*5**(35/39))/(7**(92/117))'
|
|
>>> mp.dps = 30
|
|
>>> identify(pi)
|
|
>>>
|
|
|
|
Numbers such as `\pi`, and simple combinations of user-defined
|
|
constants, can be identified if they are provided explicitly::
|
|
|
|
>>> identify(3*pi-2*e, ['pi', 'e'])
|
|
'(3*pi + (-2)*e)'
|
|
|
|
Here is an example using a dict of constants. Note that the
|
|
constants need not be "atomic"; :func:`~mpmath.identify` can just
|
|
as well express the given number in terms of expressions
|
|
given by formulas::
|
|
|
|
>>> identify(pi+e, {'a':pi+2, 'b':2*e})
|
|
'((-2) + 1*a + (1/2)*b)'
|
|
|
|
Next, we attempt some identifications with a set of base constants.
|
|
It is necessary to increase the precision a bit.
|
|
|
|
>>> mp.dps = 50
|
|
>>> base = ['sqrt(2)','pi','log(2)']
|
|
>>> identify(0.25, base)
|
|
'(1/4)'
|
|
>>> identify(3*pi + 2*sqrt(2) + 5*log(2)/7, base)
|
|
'(2*sqrt(2) + 3*pi + (5/7)*log(2))'
|
|
>>> identify(exp(pi+2), base)
|
|
'exp((2 + 1*pi))'
|
|
>>> identify(1/(3+sqrt(2)), base)
|
|
'((3/7) + (-1/7)*sqrt(2))'
|
|
>>> identify(sqrt(2)/(3*pi+4), base)
|
|
'sqrt(2)/(4 + 3*pi)'
|
|
>>> identify(5**(mpf(1)/3)*pi*log(2)**2, base)
|
|
'5**(1/3)*pi*log(2)**2'
|
|
|
|
An example of an erroneous solution being found when too low
|
|
precision is used::
|
|
|
|
>>> mp.dps = 15
|
|
>>> identify(1/(3*pi-4*e+sqrt(8)), ['pi', 'e', 'sqrt(2)'])
|
|
'((11/25) + (-158/75)*pi + (76/75)*e + (44/15)*sqrt(2))'
|
|
>>> mp.dps = 50
|
|
>>> identify(1/(3*pi-4*e+sqrt(8)), ['pi', 'e', 'sqrt(2)'])
|
|
'1/(3*pi + (-4)*e + 2*sqrt(2))'
|
|
|
|
**Finding approximate solutions**
|
|
|
|
The tolerance ``tol`` defaults to 3/4 of the working precision.
|
|
Lowering the tolerance is useful for finding approximate matches.
|
|
We can for example try to generate approximations for pi::
|
|
|
|
>>> mp.dps = 15
|
|
>>> identify(pi, tol=1e-2)
|
|
'(22/7)'
|
|
>>> identify(pi, tol=1e-3)
|
|
'(355/113)'
|
|
>>> identify(pi, tol=1e-10)
|
|
'(5**(339/269))/(2**(64/269)*3**(13/269)*7**(92/269))'
|
|
|
|
With ``full=True``, and by supplying a few base constants,
|
|
``identify`` can generate almost endless lists of approximations
|
|
for any number (the output below has been truncated to show only
|
|
the first few)::
|
|
|
|
>>> for p in identify(pi, ['e', 'catalan'], tol=1e-5, full=True):
|
|
... print(p)
|
|
... # doctest: +ELLIPSIS
|
|
e/log((6 + (-4/3)*e))
|
|
(3**3*5*e*catalan**2)/(2*7**2)
|
|
sqrt(((-13) + 1*e + 22*catalan))
|
|
log(((-6) + 24*e + 4*catalan)/e)
|
|
exp(catalan*((-1/5) + (8/15)*e))
|
|
catalan*(6 + (-6)*e + 15*catalan)
|
|
sqrt((5 + 26*e + (-3)*catalan))/e
|
|
e*sqrt(((-27) + 2*e + 25*catalan))
|
|
log(((-1) + (-11)*e + 59*catalan))
|
|
((3/20) + (21/20)*e + (3/20)*catalan)
|
|
...
|
|
|
|
The numerical values are roughly as close to `\pi` as permitted by the
|
|
specified tolerance:
|
|
|
|
>>> e/log(6-4*e/3)
|
|
3.14157719846001
|
|
>>> 135*e*catalan**2/98
|
|
3.14166950419369
|
|
>>> sqrt(e-13+22*catalan)
|
|
3.14158000062992
|
|
>>> log(24*e-6+4*catalan)-1
|
|
3.14158791577159
|
|
|
|
**Symbolic processing**
|
|
|
|
The output formula can be evaluated as a Python expression.
|
|
Note however that if fractions (like '2/3') are present in
|
|
the formula, Python's :func:`~mpmath.eval()` may erroneously perform
|
|
integer division. Note also that the output is not necessarily
|
|
in the algebraically simplest form::
|
|
|
|
>>> identify(sqrt(2))
|
|
'(sqrt(8)/2)'
|
|
|
|
As a solution to both problems, consider using SymPy's
|
|
:func:`~mpmath.sympify` to convert the formula into a symbolic expression.
|
|
SymPy can be used to pretty-print or further simplify the formula
|
|
symbolically::
|
|
|
|
>>> from sympy import sympify # doctest: +SKIP
|
|
>>> sympify(identify(sqrt(2))) # doctest: +SKIP
|
|
2**(1/2)
|
|
|
|
Sometimes :func:`~mpmath.identify` can simplify an expression further than
|
|
a symbolic algorithm::
|
|
|
|
>>> from sympy import simplify # doctest: +SKIP
|
|
>>> x = sympify('-1/(-3/2+(1/2)*5**(1/2))*(3/2-1/2*5**(1/2))**(1/2)') # doctest: +SKIP
|
|
>>> x # doctest: +SKIP
|
|
(3/2 - 5**(1/2)/2)**(-1/2)
|
|
>>> x = simplify(x) # doctest: +SKIP
|
|
>>> x # doctest: +SKIP
|
|
2/(6 - 2*5**(1/2))**(1/2)
|
|
>>> mp.dps = 30 # doctest: +SKIP
|
|
>>> x = sympify(identify(x.evalf(30))) # doctest: +SKIP
|
|
>>> x # doctest: +SKIP
|
|
1/2 + 5**(1/2)/2
|
|
|
|
(In fact, this functionality is available directly in SymPy as the
|
|
function :func:`~mpmath.nsimplify`, which is essentially a wrapper for
|
|
:func:`~mpmath.identify`.)
|
|
|
|
**Miscellaneous issues and limitations**
|
|
|
|
The input `x` must be a real number. All base constants must be
|
|
positive real numbers and must not be rationals or rational linear
|
|
combinations of each other.
|
|
|
|
The worst-case computation time grows quickly with the number of
|
|
base constants. Already with 3 or 4 base constants,
|
|
:func:`~mpmath.identify` may require several seconds to finish. To search
|
|
for relations among a large number of constants, you should
|
|
consider using :func:`~mpmath.pslq` directly.
|
|
|
|
The extended transformations are applied to x, not the constants
|
|
separately. As a result, ``identify`` will for example be able to
|
|
recognize ``exp(2*pi+3)`` with ``pi`` given as a base constant, but
|
|
not ``2*exp(pi)+3``. It will be able to recognize the latter if
|
|
``exp(pi)`` is given explicitly as a base constant.
|
|
|
|
"""
|
|
|
|
solutions = []
|
|
|
|
def addsolution(s):
|
|
if verbose: print("Found: ", s)
|
|
solutions.append(s)
|
|
|
|
x = ctx.mpf(x)
|
|
|
|
# Further along, x will be assumed positive
|
|
if x == 0:
|
|
if full: return ['0']
|
|
else: return '0'
|
|
if x < 0:
|
|
sol = ctx.identify(-x, constants, tol, maxcoeff, full, verbose)
|
|
if sol is None:
|
|
return sol
|
|
if full:
|
|
return ["-(%s)"%s for s in sol]
|
|
else:
|
|
return "-(%s)" % sol
|
|
|
|
if tol:
|
|
tol = ctx.mpf(tol)
|
|
else:
|
|
tol = ctx.eps**0.7
|
|
M = maxcoeff
|
|
|
|
if constants:
|
|
if isinstance(constants, dict):
|
|
constants = [(ctx.mpf(v), name) for (name, v) in sorted(constants.items())]
|
|
else:
|
|
namespace = dict((name, getattr(ctx,name)) for name in dir(ctx))
|
|
constants = [(eval(p, namespace), p) for p in constants]
|
|
else:
|
|
constants = []
|
|
|
|
# We always want to find at least rational terms
|
|
if 1 not in [value for (name, value) in constants]:
|
|
constants = [(ctx.mpf(1), '1')] + constants
|
|
|
|
# PSLQ with simple algebraic and functional transformations
|
|
for ft, ftn, red in transforms:
|
|
for c, cn in constants:
|
|
if red and cn == '1':
|
|
continue
|
|
t = ft(ctx,x,c)
|
|
# Prevent exponential transforms from wreaking havoc
|
|
if abs(t) > M**2 or abs(t) < tol:
|
|
continue
|
|
# Linear combination of base constants
|
|
r = ctx.pslq([t] + [a[0] for a in constants], tol, M)
|
|
s = None
|
|
if r is not None and max(abs(uw) for uw in r) <= M and r[0]:
|
|
s = pslqstring(r, constants)
|
|
# Quadratic algebraic numbers
|
|
else:
|
|
q = ctx.pslq([ctx.one, t, t**2], tol, M)
|
|
if q is not None and len(q) == 3 and q[2]:
|
|
aa, bb, cc = q
|
|
if max(abs(aa),abs(bb),abs(cc)) <= M:
|
|
s = quadraticstring(ctx,t,aa,bb,cc)
|
|
if s:
|
|
if cn == '1' and ('/$c' in ftn):
|
|
s = ftn.replace('$y', s).replace('/$c', '')
|
|
else:
|
|
s = ftn.replace('$y', s).replace('$c', cn)
|
|
addsolution(s)
|
|
if not full: return solutions[0]
|
|
|
|
if verbose:
|
|
print(".")
|
|
|
|
# Check for a direct multiplicative formula
|
|
if x != 1:
|
|
# Allow fractional powers of fractions
|
|
ilogs = [2,3,5,7]
|
|
# Watch out for existing fractional powers of fractions
|
|
logs = []
|
|
for a, s in constants:
|
|
if not sum(bool(ctx.findpoly(ctx.ln(a)/ctx.ln(i),1)) for i in ilogs):
|
|
logs.append((ctx.ln(a), s))
|
|
logs = [(ctx.ln(i),str(i)) for i in ilogs] + logs
|
|
r = ctx.pslq([ctx.ln(x)] + [a[0] for a in logs], tol, M)
|
|
if r is not None and max(abs(uw) for uw in r) <= M and r[0]:
|
|
addsolution(prodstring(r, logs))
|
|
if not full: return solutions[0]
|
|
|
|
if full:
|
|
return sorted(solutions, key=len)
|
|
else:
|
|
return None
|
|
|
|
IdentificationMethods.pslq = pslq
|
|
IdentificationMethods.findpoly = findpoly
|
|
IdentificationMethods.identify = identify
|
|
|
|
|
|
if __name__ == '__main__':
|
|
import doctest
|
|
doctest.testmod()
|