gapxsumpw

Add a constant to each strided array element and compute the sum using pairwise summation.

Usage

var gapxsumpw = require( '@stdlib/blas/ext/base/gapxsumpw' );

gapxsumpw( N, alpha, x, stride )

Adds a constant to each strided array element and computes the sum using pairwise summation.

var x = [ 1.0, -2.0, 2.0 ];
var N = x.length;

var v = gapxsumpw( N, 5.0, x, 1 );
// returns 16.0

The function has the following parameters:

N: number of indexed elements.
x: input Array or typed array.
stride: index increment for x.

The N and stride parameters determine which elements in x are accessed at runtime. For example, to access every other element in x,

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 N = floor( x.length / 2 );

var v = gapxsumpw( N, 5.0, x, 2 );
// returns 25.0

Note that indexing is relative to the first index. To introduce an offset, use typed array views.

var Float64Array = require( '@stdlib/array/float64' );
var floor = require( '@stdlib/math/base/special/floor' );

var x0 = new Float64Array( [ 2.0, 1.0, 2.0, -2.0, -2.0, 2.0, 3.0, 4.0 ] );
var x1 = new Float64Array( x0.buffer, x0.BYTES_PER_ELEMENT*1 ); // start at 2nd element

var N = floor( x0.length / 2 );

var v = gapxsumpw( N, 5.0, x1, 2 );
// returns 25.0

gapxsumpw.ndarray( N, alpha, x, stride, offset )

Adds a constant to each strided array element and computes the sum using pairwise summation and alternative indexing semantics.

var x = [ 1.0, -2.0, 2.0 ];
var N = x.length;

var v = gapxsumpw.ndarray( N, 5.0, x, 1, 0 );
// returns 16.0

The function has the following additional parameters:

offset: starting index for x.

While typed array views mandate a view offset based on the underlying buffer, the offset parameter supports indexing semantics based on a starting index. For example, to access every other value in x starting from the second value

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 N = floor( x.length / 2 );

var v = gapxsumpw.ndarray( N, 5.0, x, 2, 1 );
// returns 25.0

Notes

If N <= 0, both functions return 0.0.
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 (dapxsumpw, sapxsumpw, etc.) are likely to be significantly more performant.

Examples

var randu = require( '@stdlib/random/base/randu' );
var round = require( '@stdlib/math/base/special/round' );
var Float64Array = require( '@stdlib/array/float64' );
var gapxsumpw = require( '@stdlib/blas/ext/base/gapxsumpw' );

var x;
var i;

x = new Float64Array( 10 );
for ( i = 0; i < x.length; i++ ) {
    x[ i ] = round( randu()*100.0 );
}
console.log( x );

var v = gapxsumpw( x.length, 5.0, x, 1 );
console.log( v );

References

Higham, Nicholas J. 1993. "The Accuracy of Floating Point Summation." SIAM Journal on Scientific Computing 14 (4): 783–99. doi:10.1137/0914050.