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iterContinuedFractionSeq

Create an iterator which generates a list of all continued fraction terms which can be obtained given the precision of a provided number.

A generalized continued fraction has the form

$$x = b_0 + \frac{a_1}{b_1+\frac{a_2}{b_2+\frac{a_3}{b_3+\frac{a_4}{b_4+\ldots}}}}$$

If a_i = 1 for all i, the above expression reduces to a simple continued fraction.

$$x = b_0 + \frac{1}{b_1+\frac{1}{b_2+\frac{1}{b_3+\frac{1}{b_4+\ldots}}}}$$

where the values b_i are called the coefficients or terms of the continued fraction and the rationals

$$b_0, b_0 + \frac{1}{b_1},b_0 + \frac{1}{b_1 + \frac{1}{b2}},b_0 + \frac{1}{b_1 + \frac{1}{b2+\frac{1}{b3}}},\ldots$$

are called convergents.

Installation

npm install @stdlib/math-iter-sequences-continued-fraction

Alternatively,

• To load the package in a website via a script tag without installation and bundlers, use the ES Module available on the esm branch (see README).
• If you are using Deno, visit the deno branch (see README for usage intructions).
• For use in Observable, or in browser/node environments, use the Universal Module Definition (UMD) build available on the umd branch (see README).

The branches.md file summarizes the available branches and displays a diagram illustrating their relationships.

To view installation and usage instructions specific to each branch build, be sure to explicitly navigate to the respective README files on each branch, as linked to above.

Usage

var iterContinuedFractionSeq = require( '@stdlib/math-iter-sequences-continued-fraction' );

iterContinuedFractionSeq( x, [options] )

Returns an iterator which generates a list of all continued fraction terms (b_i) which can be obtained given the precision of x.

var it = iterContinuedFractionSeq( 3.245 );
// returns <Object>

var v = it.next().value;
// returns 3

v = it.next().value;
// returns 4

v = it.next().value;
// returns 12

v = it.next().value;
// returns 4

var bool = it.next().done;
// returns true

The returned iterator protocol-compliant object has the following properties:

• next: function which returns an iterator protocol-compliant object containing the next iterated value (if one exists) assigned to a value property and a done property having a boolean value indicating whether the iterator is finished.
• return: function which closes an iterator and returns a single (optional) argument in an iterator protocol-compliant object.

The function supports the following options:

• iter: maximum number of iterations. Default: 1e308.

• tol: tolerance at which to terminate further evaluation of the continued fraction. Default: floating-point epsilon.

• returns: specifies the type of result to return. Must be one of the following options:

  • terms: return continued fraction terms.
  • convergents: return continued fraction convergents.
  • *: return both continued fraction terms and their associated convergents as a two-element array: [ <term>, <convergent> ].

  Default: 'terms'.

By default, in theory, the function returns an infinite iterator; however, in practice, due to limited precision, every floating-point number is a rational number, and, thus, every returned iterator will end in a finite number of iterations. To explicitly cap the maximum number of iterations, set the iter option.

var opts = {
    'iter': 2
};
var it = iterContinuedFractionSeq( 3.245, opts );
// returns <Object>

var v = it.next().value;
// returns 3

v = it.next().value;
// returns 4

var bool = it.next().done;
// returns true

The returned iterator terminates once the difference between the input value x and a continued fraction approximation is sufficiently small. The default tolerance is floating-point epsilon (~2.22e-16). Once an update to a continued fraction approximation is less than or equal to this tolerance, the iterator terminates. To adjust the tolerance (e.g., to return a rough approximation of an input value x), set the tol option.

var opts = {
    'tol': 1.0e-7
};
var it = iterContinuedFractionSeq( 3.141592653589793, opts );
// returns <Object>

var v = it.next().value;
// returns 3

v = it.next().value;
// returns 7

v = it.next().value;
// returns 16

var bool = it.next().done;
// returns true

// The returned terms [3; 7, 16] evaluate to 3.1415929203539825

By default, the returned iterator returns continued fraction terms. To return convergents, set the returns option to 'convergents'.

var it = iterContinuedFractionSeq( 3.245, {
    'returns': 'convergents'
});
// returns <Object>

var v = it.next().value;
// returns 3.0

v = it.next().value;
// returns 3.25

v = it.next().value;
// returns ~3.2449

v = it.next().value;
// returns 3.245

var bool = it.next().done;
// returns true

To return both continued fraction terms and their associated convergents, set the returns option to *.

var it = iterContinuedFractionSeq( 3.245, {
    'returns': '*'
});
// returns <Object>

var v = it.next().value;
// returns [ 3, 3.0 ]

v = it.next().value;
// returns [ 4, 3.25 ]

v = it.next().value;
// returns [ 12, ~3.2449 ]

v = it.next().value;
// returns [ 4, 3.245 ]

var bool = it.next().done;
// returns true

Notes

• The returned iterator returns the terms for a simple continued fraction.
• For x < 0, the returned iterator returns negated terms for |x| (i.e., if the terms for |x| are [b0; b1, b2, ..., bn], the returned iterator returns [-b0; -b1, -b2, ..., -bn]). While other continued fraction representations are possible, floating-point rounding error can introduce asymmetries when evaluating terms to recover the original values for |x| and x < 0. Accordingly, alternative continued fraction representations for negative input values are not supported.
• If an environment supports Symbol.iterator, the returned iterator is iterable.

Examples

var PI = require( '@stdlib/constants-float64-pi' );
var iterContinuedFractionSeq = require( '@stdlib/math-iter-sequences-continued-fraction' );

function evaluate( terms ) {
    var sum;
    var N;
    var i;

    N = terms.length;
    sum = 0.0;
    if ( N > 1 ) {
        sum = 1.0 / terms[ N-1 ];
        for ( i = N-2; i > 0; i-- ) {
            sum = 1.0 / ( terms[ i ] + sum );
        }
    }
    sum += terms[ 0 ];
    return sum;
}

// Create an iterator:
var opts = {
    'iter': 20
};
var it = iterContinuedFractionSeq( PI, opts );

// Perform manual iteration...
var terms = [];
var v;
while ( true ) {
    v = it.next();
    if ( v.done ) {
        break;
    }
    terms.push( v.value );
}
console.log( 'original: %d', PI );
console.log( terms );
console.log( 'computed: %d', evaluate( terms ) );

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See Also

• @stdlib/math-iter/utils/continued-fraction: evaluate the terms of a continued fraction.

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Notice

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For more information on the project, filing bug reports and feature requests, and guidance on how to develop stdlib, see the main project repository.

Community

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License

See LICENSE.

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