The diamond operator for real algebraic expressions Susanne Schmitt - - PowerPoint PPT Presentation

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I N F O R M A T I K

The diamond operator for real algebraic expressions

Susanne Schmitt

sschmitt@mpi-sb.mpg.de

Max Planck Institut f¨ ur Informatik Saarbr¨ ucken

The diamond operator for real algebraic expressions – p.1/5

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I N F O R M A T I K

The diamond operator

Definition real algebraic expressions Sign computation separation bound sign computation Implementation Examples

The diamond operator for real algebraic expressions – p.2/5

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Definition

Real algebraic numbers are all real numbers which are roots of polynomials with integral coefficients.

The diamond operator for real algebraic expressions – p.3/5

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I N F O R M A T I K

Definition

Real algebraic expressions are arithmetic expressions which are recursively built up from the integers, using the

• perations of

, and

, or the diamond operator

.

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Definition

Real algebraic expressions are arithmetic expressions which are recursively built up from the integers, using the

• perations of

, and

, or the diamond operator

. The value val

is the real algebraic number defined by the expression. val

is the

• th smallest real root of

the polynomial val

• val

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• val

The diamond operator for real algebraic expressions – p.3/5

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I N F O R M A T I K

Sign computation

A separation bound for a real algebraic expression

is a positive real number sep

such that val

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val

sep

The diamond operator for real algebraic expressions – p.4/5

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I N F O R M A T I K

Sign computation

A separation bound for a real algebraic expression

is a positive real number sep

such that val

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val

sep

(Burnikel, Funke, Mehlhorn, Schirra, S.: A Separation Bound for Real Algebraic Expressions, ESA’01)

The diamond operator for real algebraic expressions – p.4/5

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I N F O R M A T I K

Sign computation

A separation bound for a real algebraic expression

is a positive real number sep

such that val

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val

sep

Computation of sign

val

: Initialize an error bound

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Compute

with

val

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If

• , the sign of val

is equal to the sign of

. Otherwise,

• and hence

val

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If

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sep

, then val

. Else halve

• and repeat.

The diamond operator for real algebraic expressions – p.4/5

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Implementation

leda_real: implement real algebraic root expressions. If a leda real is not a double, the expression is stored and an interval which contains the real algebraic number is computed. In most cases the sign of the real algebraic number can be determined using this interval approximation. If that is not possible, one repeats computing bigfloat approximations with gradually higher accuracy. If the separation bound is reached, the real algebraic number is

.

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Implementation

Interval approximation: Uspensky’s algorithm Bigfloat approximation: Newton method

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