# gcusumpw
> Calculate the cumulative sum of strided array elements using pairwise summation.
## Usage
```javascript
var gcusumpw = require( '@stdlib/blas/ext/base/gcusumpw' );
```
#### gcusumpw( N, sum, x, strideX, y, strideY )
Computes the cumulative sum of strided array elements using pairwise summation.
```javascript
var x = [ 1.0, -2.0, 2.0 ];
var y = [ 0.0, 0.0, 0.0 ];
gcusumpw( 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 ];
gcusumpw( 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 = gcusumpw( 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 );
gcusumpw( N, 0.0, x1, -2, y1, 1 );
// y0 => [ 0.0, 0.0, 0.0, 4.0, 6.0, 4.0, 5.0, 0.0 ]
```
#### gcusumpw.ndarray( N, sum, x, strideX, offsetX, y, strideY, offsetY )
Computes the cumulative sum of strided array elements using pairwise summation and alternative indexing semantics.
```javascript
var x = [ 1.0, -2.0, 2.0 ];
var y = [ 0.0, 0.0, 0.0 ];
gcusumpw.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 );
gcusumpw.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.
- In general, pairwise summation is more numerically stable than ordinary recursive summation (i.e., "simple" summation), with slightly worse performance. While not the most numerically stable summation technique (e.g., compensated summation techniques such as the Kahan–Babuška-Neumaier algorithm are generally more numerically stable), pairwise summation strikes a reasonable balance between numerical stability and performance. If either numerical stability or performance is more desirable for your use case, consider alternative summation techniques.
- Depending on the environment, the typed versions ([`dcusumpw`][@stdlib/blas/ext/base/dcusumpw], [`scusumpw`][@stdlib/blas/ext/base/scusumpw], 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 gcusumpw = require( '@stdlib/blas/ext/base/gcusumpw' );
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 );
gcusumpw( x.length, 0.0, x, 1, y, -1 );
console.log( y );
```
* * *
## References
- Higham, Nicholas J. 1993. "The Accuracy of Floating Point Summation." _SIAM Journal on Scientific Computing_ 14 (4): 783–99. doi:[10.1137/0914050][@higham:1993a].
[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/dcusumpw]: https://www.npmjs.com/package/@stdlib/blas/tree/main/ext/base/dcusumpw
[@stdlib/blas/ext/base/scusumpw]: https://www.npmjs.com/package/@stdlib/blas/tree/main/ext/base/scusumpw
[@higham:1993a]: https://doi.org/10.1137/0914050