evalrational

Evaluate a rational function.

A rational function f(x) is defined as

f left-parenthesis x right-parenthesis equals StartFraction upper P left-parenthesis x right-parenthesis Over upper Q left-parenthesis x right-parenthesis EndFraction

where both P(x) and Q(x) are polynomials in x. A polynomial in x can be expressed

c Subscript n Baseline x Superscript n Baseline plus c Subscript n minus 1 Baseline x Superscript n minus 1 Baseline plus ellipsis plus c 1 x Superscript 1 Baseline plus c 0 equals sigma-summation Underscript i equals 0 Overscript n Endscripts c Subscript i Baseline x Superscript i

where c_n, c_{n-1}, ..., c_0 are constants.

Usage

var evalrational = require( '@stdlib/math/base/tools/evalrational' );

evalrational( P, Q, x )

Evaluates a rational function at a value x. The coefficients P and Q are expected to be arrays of the same length.

var P = [ -6.0, -5.0 ];
var Q = [ 3.0, 0.5 ];

var v = evalrational( P, Q, 6.0 ); //  => ( -6*6^0 - 5*6^1 ) / ( 3*6^0 + 0.5*6^1 ) = (-6-30)/(3+3)
// returns -6.0

For polynomials of different degree, the coefficient array for the lower degree polynomial should be padded with zeros.

// 2x^3 + 4x^2 - 5x^1 - 6x^0 => degree 4
var P = [ -6.0, -5.0, 4.0, 2.0 ];

// 0.5x^1 + 3x^0 => degree 2
var Q = [ 3.0, 0.5, 0.0, 0.0 ]; // zero-padded

var v = evalrational( P, Q, 6.0 ); // => ( -6*6^0 - 5*6^1 + 4*6^2 + 2*6^3 ) / ( 3*6^0 + 0.5*6^1 + 0*6^2 + 0*6^3 ) = (-6-30+144+432)/(3+3)
// returns 90.0

Coefficients should be ordered in ascending degree, thus matching summation notation.

evalrational.factory( P, Q )

Uses code generation to in-line coefficients and return a function for evaluating a rational function.

var P = [ 20.0, 8.0, 3.0 ];
var Q = [ 10.0, 9.0, 1.0 ];

var rational = evalrational.factory( P, Q );

var v = rational( 10.0 ); // => (20*10^0 + 8*10^1 + 3*10^2) / (10*10^0 + 9*10^1 + 1*10^2) = (20+80+300)/(10+90+100)
// returns 2.0

v = rational( 2.0 ); // => (20*2^0 + 8*2^1 + 3*2^2) / (10*2^0 + 9*2^1 + 1*2^2) = (20+16+12)/(10+18+4)
// returns 1.5

Notes

  • For hot code paths in which coefficients are invariant, a compiled function will be more performant than evalrational().
  • While code generation can boost performance, its use may be problematic in browser contexts enforcing a strict content security policy (CSP). If running in or targeting an environment with a CSP, avoid using code generation.
## Examples
var randu = require( '@stdlib/random/base/randu' );
var round = require( '@stdlib/math/base/special/round' );
var Float64Array = require( '@stdlib/array/float64' );
var evalrational = require( '@stdlib/math/base/tools/evalrational' );

var rational;
var sign;
var len;
var P;
var Q;
var v;
var i;

// Create two arrays of random coefficients...
len = 10;
P = new Float64Array( len );
Q = new Float64Array( len );
for ( i = 0; i < len; i++ ) {
    if ( randu() < 0.5 ) {
        sign = -1.0;
    } else {
        sign = 1.0;
    }
    P[ i ] = sign * round( randu()*100 );
    Q[ i ] = sign * round( randu()*100 );
}

// Evaluate the rational function at random values...
for ( i = 0; i < 100; i++ ) {
    v = randu() * 100.0;
    console.log( 'f(%d) = %d', v, evalrational( P, Q, v ) );
}

// Generate an `evalrational` function...
rational = evalrational.factory( P, Q );
for ( i = 0; i < 100; i++ ) {
    v = (randu()*100.0) - 50.0;
    console.log( 'f(%d) = %d', v, rational( v ) );
}