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IntegerRelations

  

PSLQ

  

find an integer dependence (relation)

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

PSLQ(v)

Parameters

v

-

list or Vector of (complex) floating-point numbers

Description

• 

Given a list (or a Vector) v of  real numbers, the PSLQ(v) command outputs a list (or a Vector) u of  integers such that  is minimized.  Thus the PSLQ function finds an integer relation between a vector of linearly dependent real numbers if the input has enough precision.

• 

Given a list (or a Vector) v of  complex numbers, the PSLQ(v) command outputs a list (or a vector) u of  complex integers (Gaussian integers) such that the norm of  is minimized.

• 

This is an implementation of Bailey and Ferguson's PSLQ algorithm. You can also use the Lenstra-Lenstra-Lovasz (LLL) lattice reduction algorithm to find a linear relation.  For more information, see IntegerRelations[LLL]. Generally speaking, PSLQ is faster.

• 

One application of PSLQ is to find the minimal polynomial of an algebraic number given a decimal approximation of the algebraic number.  The examples below illustrate this.  Generally speaking, if the height of the minimal polynomial is  and its degree is , then you need more than  correct decimal digits for the algebraic number.  If the input has  digits and this is insufficient, the output of PSLQ will typically be a list of  integers each with approximately / digits long.

• 

The internal working precision of the PSLQ command corresponds to the value of Digits. For best results, the same value of Digits should be used with which the input approximation was obtained.

Examples

(1)

(2)

Check that the following linear combination is small.

(3)

Finding a integer relation between non-algebraic constants.

(4)

(5)

(6)

Using PSLQ to find the minimal polynomial for .

(7)

(8)

Approximate with  digits and round to  digits.

(9)

(10)

(11)

(12)

The minimal polynomial for

(13)

Check that  is a root of

(14)

The next example involves complex numbers. First define a tenth root of unity.

(15)

(16)

(17)

(18)

(19)

(20)

In the next example, a Gaussian integer relation is found. We subsequently find an integer relation from the Gaussian integer relation by eliminating I.

(21)

(22)

(23)

(24)

(25)

(26)

(27)

(28)

The last example is of much larger degree requiring more than the default  digits of precision.  In the example, we are using PSLQ to test if the algebraic number  is of degree  or less.

(29)

(30)

Compute to  digits and round to  digits.

(31)

(32)

(33)

(34)

Check.

(35)

(36)

Thus, the minimal polynomial for  must be the degree 9 factor.

Here is what happens if we mistakenly assume that algebraic number  is of degree  or less. The output of PSLQ looks like random  digit integers, which indicates that it has not found anything interesting.

(37)

(38)

See Also

identify

IntegerRelations

IntegerRelations[LinearDependency]

IntegerRelations[LLL]

 


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