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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
>If a_i = 1
for all i
, the above expression reduces to a simple continued fraction.
where the values b_i
are called the coefficients or terms of the continued fraction and the rationals
are called convergents.
Installation
npm install @stdlib/mathitersequencescontinuedfraction
Usage
var iterContinuedFractionSeq = require( '@stdlib/mathitersequencescontinuedfraction' );
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 protocolcompliant object has the following properties:

next: function which returns an iterator protocolcompliant object containing the next iterated value (if one exists) assigned to a
value
property and adone
property having aboolean
value indicating whether the iterator is finished.  return: function which closes an iterator and returns a single (optional) argument in an iterator protocolcompliant 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: floatingpoint 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 twoelement array:
[ <term>, <convergent> ]
.
Default:
'terms'
.
By default, in theory, the function returns an infinite iterator; however, in practice, due to limited precision, every floatingpoint 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 floatingpoint epsilon (~2.22e16
). 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': 1.0e7
};
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 forx
(i.e., if the terms forx
are[b0; b1, b2, ..., bn]
, the returned iterator returns[b0; b1, b2, ..., bn]
). While other continued fraction representations are possible, floatingpoint rounding error can introduce asymmetries when evaluating terms to recover the original values forx
andx < 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/constantsfloat64pi' );
var iterContinuedFractionSeq = require( '@stdlib/mathitersequencescontinuedfraction' );
function evaluate( terms ) {
var sum;
var N;
var i;
N = terms.length;
sum = 0.0;
if ( N > 1 ) {
sum = 1.0 / terms[ N1 ];
for ( i = N2; 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 ) );
See Also

@stdlib/mathiter/utils/continuedfraction
: evaluate the terms of a continued fraction.
Notice
This package is part of stdlib, a standard library for JavaScript and Node.js, with an emphasis on numerical and scientific computing. The library provides a collection of robust, high performance libraries for mathematics, statistics, streams, utilities, and more.
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
License
See LICENSE.
Copyright
Copyright © 20162024. The Stdlib Authors.