GAP (FLOAT) - Chapter 3: Polynomials

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3 Polynomials
3.1 The Floats pseudo-field
3.2 Roots of polynomials
3.3 Finding integer relations
3.3-1 PSLQ
3.4 LLL lattice reduction

### 3 Polynomials

#### 3.1 The Floats pseudo-field

Polynomials with floating-point coefficients may be manipulated in GAP; though they behave, in subtle ways, quite differently than polynomials over rings.

The "pseudo-field" of floating-point numbers is an object in GAP, called `FLOAT_PSEUDOFIELD`. (It is not really a field, e.g. because addition of floating-point numbers in not associative). It may be used to create indeterminates, for example as

    gap> x := Indeterminate(FLOAT_PSEUDOFIELD,"x");
    x
    gap> 2*x^2+3;
    2.0*x^2+3.0
    gap> Value(last,10);
    203.0

#### 3.2 Roots of polynomials

The Jenkins-Traub algorithm has been implemented, in arbitrary precision for MPFR and MPC.

Furthermore, CXSC can provide complex enclosures for the roots of a complex polynomial.

#### 3.3 Finding integer relations

The PSLQ algorithm has been implemented by Steve A. Linton, as an external contribution to Float. This algorithm receives as input a vector of floats x and a required precision ϵ, and seeks an integer vector v such that |x⋅ v|<ϵ. The implementation follows quite closely the original article [BB01].

##### 3.3-1 PSLQ

`‣ PSLQ`( x, epsilon[, gamma] ) | ( function )

`‣ PSLQ_MP`( x, epsilon[, gamma[, beta]] ) | ( function )

Returns: An integer vector v with |x⋅ v|<ϵ.

The PSLQ algorithm by Bailey and Broadhurst (see [BB01]) searches for an integer relation between the entries in x.

β and γ are algorithm tuning parameters, and default to 4/10 and 2/sqrt(3) respectively.

The second form implements the "Multi-pair" variant of the algorithm, which is better suited to parallelization.

    gap> PSLQ([1.0,(1+Sqrt(5.0))/2],1.e-2);

    [ 55, -34 ] # Fibonacci numbers
    gap> RootsFloat([1,-4,2]*1.0);

    [ 0.292893, 1.70711 ] # roots of 2x^2-4x+1
    gap> PSLQ(List([0..2],i->last[1]^i),1.e-7);

    [ 1, -4, 2 ] # a degree-2 polynomial fitting well

#### 3.4 LLL lattice reduction

A faster implementation of the LLL lattice reduction algorithm has also been implemented. It is accessible via the commands `FPLLLReducedBasis(m)` and `FPLLLShortestVector(m)`.

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