## 86 Package pslq ¶

### 86.1 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.

### 86.2 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.