# gcusumkbn
> Calculate the cumulative sum of strided array elements using an improved Kahan–Babuška algorithm.
## Usage
```javascript
var gcusumkbn = require( '@stdlib/blas/ext/base/gcusumkbn' );
```
#### gcusumkbn( N, sum, x, strideX, y, strideY )
Computes the cumulative sum of strided array elements using an improved Kahan–Babuška algorithm.
```javascript
var x = [ 1.0, -2.0, 2.0 ];
var y = [ 0.0, 0.0, 0.0 ];
gcusumkbn( x.length, 0.0, x, 1, y, 1 );
// y => [ 1.0, -1.0, 1.0 ]
x = [ 1.0, -2.0, 2.0 ];
y = [ 0.0, 0.0, 0.0 ];
gcusumkbn( x.length, 10.0, x, 1, y, 1 );
// y => [ 11.0, 9.0, 11.0 ]
```
The function has the following parameters:
- **N**: number of indexed elements.
- **sum**: initial sum.
- **x**: input [`Array`][mdn-array] or [`typed array`][mdn-typed-array].
- **strideX**: index increment for `x`.
- **y**: output [`Array`][mdn-array] or [`typed array`][mdn-typed-array].
- **strideY**: index increment for `y`.
The `N` and `stride` parameters determine which elements in `x` and `y` are accessed at runtime. For example, to compute the cumulative sum of every other element in `x`,
```javascript
var floor = require( '@stdlib/math/base/special/floor' );
var x = [ 1.0, 2.0, 2.0, -7.0, -2.0, 3.0, 4.0, 2.0 ];
var y = [ 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0 ];
var N = floor( x.length / 2 );
var v = gcusumkbn( N, 0.0, x, 2, y, 1 );
// y => [ 1.0, 3.0, 1.0, 5.0, 0.0, 0.0, 0.0, 0.0 ]
```
Note that indexing is relative to the first index. To introduce an offset, use [`typed array`][mdn-typed-array] views.
```javascript
var Float64Array = require( '@stdlib/array/float64' );
var floor = require( '@stdlib/math/base/special/floor' );
// Initial arrays...
var x0 = new Float64Array( [ 2.0, 1.0, 2.0, -2.0, -2.0, 2.0, 3.0, 4.0 ] );
var y0 = new Float64Array( x0.length );
// Create offset views...
var x1 = new Float64Array( x0.buffer, x0.BYTES_PER_ELEMENT*1 ); // start at 2nd element
var y1 = new Float64Array( y0.buffer, y0.BYTES_PER_ELEMENT*3 ); // start at 4th element
var N = floor( x0.length / 2 );
gcusumkbn( N, 0.0, x1, -2, y1, 1 );
// y0 => [ 0.0, 0.0, 0.0, 4.0, 6.0, 4.0, 5.0, 0.0 ]
```
#### gcusumkbn.ndarray( N, sum, x, strideX, offsetX, y, strideY, offsetY )
Computes the cumulative sum of strided array elements using an improved Kahan–Babuška algorithm and alternative indexing semantics.
```javascript
var x = [ 1.0, -2.0, 2.0 ];
var y = [ 0.0, 0.0, 0.0 ];
gcusumkbn.ndarray( x.length, 0.0, x, 1, 0, y, 1, 0 );
// y => [ 1.0, -1.0, 1.0 ]
```
The function has the following additional parameters:
- **offsetX**: starting index for `x`.
- **offsetY**: starting index for `y`.
While [`typed array`][mdn-typed-array] views mandate a view offset based on the underlying `buffer`, `offsetX` and `offsetY` parameters support indexing semantics based on a starting indices. For example, to calculate the cumulative sum of every other value in `x` starting from the second value and to store in the last `N` elements of `y` starting from the last element
```javascript
var floor = require( '@stdlib/math/base/special/floor' );
var x = [ 2.0, 1.0, 2.0, -2.0, -2.0, 2.0, 3.0, 4.0 ];
var y = [ 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0 ];
var N = floor( x.length / 2 );
gcusumkbn.ndarray( N, 0.0, x, 2, 1, y, -1, y.length-1 );
// y => [ 0.0, 0.0, 0.0, 0.0, 5.0, 1.0, -1.0, 1.0 ]
```
## Notes
- If `N <= 0`, both functions return `y` unchanged.
- Depending on the environment, the typed versions ([`dcusum`][@stdlib/blas/ext/base/dcusum], [`scusum`][@stdlib/blas/ext/base/scusum], etc.) are likely to be significantly more performant.
## Examples
```javascript
var randu = require( '@stdlib/random/base/randu' );
var round = require( '@stdlib/math/base/special/round' );
var Float64Array = require( '@stdlib/array/float64' );
var gcusumkbn = require( '@stdlib/blas/ext/base/gcusumkbn' );
var y;
var x;
var i;
x = new Float64Array( 10 );
y = new Float64Array( x.length );
for ( i = 0; i < x.length; i++ ) {
x[ i ] = round( randu()*100.0 );
}
console.log( x );
console.log( y );
gcusumkbn( x.length, 0.0, x, 1, y, -1 );
console.log( y );
```
* * *
## References
- Neumaier, Arnold. 1974. "Rounding Error Analysis of Some Methods for Summing Finite Sums." _Zeitschrift Für Angewandte Mathematik Und Mechanik_ 54 (1): 39–51. doi:[10.1002/zamm.19740540106][@neumaier:1974a].
[mdn-array]: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Array
[mdn-typed-array]: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/TypedArray
[@stdlib/blas/ext/base/dcusum]: https://www.npmjs.com/package/@stdlib/blas/tree/main/ext/base/dcusum
[@stdlib/blas/ext/base/scusum]: https://www.npmjs.com/package/@stdlib/blas/tree/main/ext/base/scusum
[@neumaier:1974a]: https://doi.org/10.1002/zamm.19740540106