82.1 Introduction to pslq

Package pslq contains two functions.

(1) guess_exact_value tries to find an exact equivalent for an inexact number (float or bigfloat).

(2) pslq_integer_relation tries to find integer coefficients such that a linear combination of inexact numbers is approximately zero.

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82.2 Functions and Variables for pslq

Function: guess_exact_value (x) ¶

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

Function: pslq_integer_relation (L) ¶

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

Variable: pslq_precision ¶

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.

Variable: pslq_threshold ¶

Default value: 10^(2 - fpprec)

Threshold for absolute residual error of integer relation found by pslq_integer_relation.

Variable: pslq_depth ¶

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.

Variable: pslq_status ¶

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.