BUILT-IN MATHEMATICA SYMBOL

FindIntegerNullVector

finds a list of integers

such that

.

FindIntegerNullVector

finds a list of integers

with

such that

.

MORE INFORMATION

Not all are zero. The numbers can be real or complex. For complex numbers the numbers are Gaussian integers.

In FindIntegerNullVector no integer null vector may exist with the given norm bound. The input is then returned unevaluated.

The following options can be given:

	WorkingPrecision	Automatic	precision to use in internal computation
	ZeroTest	Automatic	method to test whether a number is zero

The setting ZeroTest->Automatic automatically determines the method to test the integer relation .

For inexact numbers , the relation found holds up to the precision of the input. For exact numbers , the relation found is validated using PossibleZeroQ.

For inexact numbers and WorkingPrecision->Automatic the precision is taken to be the precision of the input.

For exact numbers and WorkingPrecision->Automatic the precision is taken to start with MachinePrecision and use up to $MaxExtraPrecision extra precision when searching for an integer null vector when no norm bound d is specified. In the case of a norm bound d, enough precision is used to either find a null vector or prove that none exist.

EXAMPLES

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Basic Examples (3)

Exact input:

Inexact input:

Exact input with norm bound:

There is no null vector with norm less than or equal to 2:

Exact input:

In[1]:=

Out[1]=

In[2]:=

Out[2]=

Inexact input:

In[1]:=

Out[1]=

Exact input with norm bound:

In[1]:=

Out[1]=

There is no null vector with norm less than or equal to 2:

In[2]:=

Out[2]=

Scope (6)

This finds an integer null vector for a vector of exact real numbers:

This proves that there is no null vector with norm less than or equal to 8:

For a bound close to the norm of a null vector you may not get proof that no null vector exists:

The returned null vector does not satisfy the norm bound:

FindIntegerNullVector cannot prove that numbers are linearly independent over the integers:

It can prove that there is no integer null vector with norm less than or equal to a given bound:

For inexact input, the relation is true up to the precision of the input:

No null vector exists for the given norm bound:

Here no null vector is found, but nonexistence of a null vector is proven only for a smaller norm bound:

This finds a null vector for a 20-digit approximation of

:

The result is not a null vector for the exact vector

:

A null vector found for a higher-precision approximation of

is also a null vector for

:

This gives a Gaussian integer null vector for a vector of exact complex numbers:

This finds a Gaussian integer null vector for a vector of approximate complex numbers:

Options (2)

By default, precision of at most $MachinePrecision+$MaxExtraPrecision digits is used:

Using a higher WorkingPrecision allows you to find a null vector:

By default, PossibleZeroQ with Method

is used to prove relations:

This uses exact methods to prove the relation:

This uses a high-precision numeric test instead:

Applications (2)

Find coefficients of the minimal polynomial of an algebraic number:

This finds the minimal polynomial using symbolic methods:

Find a relation between transcendental numbers:

Properties & Relations (3)

FindIntegerNullVector returns an integer null vector for the given vector:

An integer null vector is a nontrivial integer solution of a homogeneous linear equation:

Use FindInstance to find solutions of equations:

Find coefficients of the minimal polynomial of an algebraic number using its approximation:

Use RootApproximant to find an algebraic number using its approximation:

Possible Issues (2)

The precision of an approximation may not be sufficient to find a relation between numbers:

Using a higher-precision approximation, you get a true relation:

Numeric zero testing used for nonalgebraic numbers may allow results that are not null vectors:

This gives the precision used in zero testing when no symbolic zero-testing method applies:

With a higher zero-testing precision, FindIntegerNullVector correctly rejects the vector:

Reset the system option to the original value: