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When x is a floating point number or bigfloat,
guess_exact_value
tries to find an exact expression
(in terms of radicals, logarithms, exponentials, and the constant %pi
)
which is nearly equal to the given number.
If guess_exact_value
cannot find such an expression,
x is returned unchanged.
When x is rational number or other mapatom (other than a float or bigfloat), x is returned unchanged.
Otherwise, x is a nonatomic expression,
and guess_exact_value
is applied to each of the arguments of x.
Example:
(%i1) load ("pslq.mac"); (%o1) pslq.mac (%i2) root: float (sin (%pi/12)); (%o2) 0.2588190451025207 (%i3) guess_exact_value (root); sqrt(2 - sqrt(3)) (%o3) ----------------- 2 (%i4) L: makelist (root^i, i, 0, 4); (%o4) [1.0, 0.2588190451025207, 0.06698729810778066, 0.01733758853025369, 0.004487298107780675] (%i5) m: pslq_integer_relation(%); (%o5) [- 1, 0, 16, 0, - 16] (%i6) makelist (x^i, i, 0, 4) . m; 4 2 (%o6) (- 16 x ) + 16 x - 1 (%i7) solve(%); sqrt(sqrt(3) + 2) sqrt(sqrt(3) + 2) (%o7) [x = - -----------------, x = -----------------, 2 2 sqrt(2 - sqrt(3)) sqrt(2 - sqrt(3)) x = - -----------------, x = -----------------] 2 2
Implements the PSLQ algorithm [1] to find integer relations between bigfloat numbers.
For a given list L of floating point numbers,
pslq_integer_relation
returns a list of integers m
such that m . L = 0
(with absolute residual error less than pslq_threshold
).
[1] D.H.Bailey: Integer Relation Detection and Lattice Reduction.
Example:
(%i1) load ("pslq.mac"); (%o1) pslq.mac (%i2) root: float (sin (%pi/12)); (%o2) 0.2588190451025207 (%i3) L: makelist (root^i, i, 0, 4); (%o3) [1.0, 0.2588190451025207, 0.06698729810778066, 0.01733758853025369, 0.004487298107780675] (%i4) m: pslq_integer_relation(%); (%o4) [- 1, 0, 16, 0, - 16] (%i5) m . L; (%o5) - 2.359223927328458E-16 (%i6) float (10^(2 - fpprec)); (%o6) 1.0E-14 (%i7) is (abs (m . L) < 10^(2 - fpprec)); (%o7) true
Default value: 10^(fpprec - 2)
Maximum magnitude of some intermediate results in pslq_integer_relation
.
The search fails if one of the intermediate results has elements
larger than pslq_precision
.
Default value: 10^(2 - fpprec)
Threshold for absolute residual error of integer relation found by pslq_integer_relation
.
Default value: 20 * n
Number of iterations of the PSLQ algorithm.
The default value is 20 times n,
where n is the length of the list of numbers supplied to pslq_integer_relation
.
Indicates success or failure for an integer relation search by pslq_integer_relation
.
When pslq_status
is 1, it indicates an integer relation was found,
and the absolute residual error is less than pslq_threshold
.
When pslq_status
is 2, it indicates an integer relation was not found
because some intermediate results are larger than pslq_precision
.
When pslq_status
is 3, it indicates an integer relation was not found
because the number of iterations pslq_depth
was reached.
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