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# drsskbn

> Calculate the [residual sum of squares][wikipedia-residual-sum-of-squares] of two double-precision floating-point strided arrays using an improved Kahan–Babuška algorithm.

<section class="intro">

The [residual sum of squares][wikipedia-residual-sum-of-squares] (also referred to as the **sum of squared residuals** (SSR) and the **sum of squared errors** (SSE)) is defined as

<!-- <equation class="equation" label="eq:residual_sum_of_squares" align="center" raw="d = \sum_{i=0}^{N-1} (y_i - x_i)^2" alt="Equation for residual sum of squares."> -->

<div class="equation" align="center" data-raw-text="d = \sum_{i=0}^{N-1} (y_i - x_i)^2" data-equation="eq:residual_sum_of_squares">
    <img src="" alt="residual sum of squares.">
    <br>
</div>

<!-- </equation> -->

</section>

<!-- /.intro -->

<section class="usage">

## Usage

```javascript
var drsskbn = require( '@stdlib/blas/ext/base/drsskbn' );
```

#### drsskbn( N, x, strideX, y, strideY )

Computes the [residual sum of squares][wikipedia-residual-sum-of-squares] of two double-precision floating-point strided arrays using an improved Kahan–Babuška algorithm.

```javascript
var Float64Array = require( '@stdlib/array/float64' );

var x = new Float64Array( [ 1.0, -2.0, 2.0 ] );
var y = new Float64Array( [ 1.0, 1.0, -4.0 ] );

var z = drsskbn( x.length, x, 1, y, 1 );
// returns 45.0
```

The function has the following parameters:

-   **N**: number of indexed elements.
-   **x**: first input [`Float64Array`][@stdlib/array/float64].
-   **strideX**: stride length for `x`.
-   **y**: second input [`Float64Array`][@stdlib/array/float64].
-   **strideY**: stride length for `y`.

The `N` and stride parameters determine which elements in strided arrays are accessed at runtime. For example, to compute the [residual sum of squares][wikipedia-residual-sum-of-squares] of every other element in `x` and `y`

```javascript
var Float64Array = require( '@stdlib/array/float64' );

var x = new Float64Array( [ 1.0, 2.0, 2.0, -7.0, -2.0, 3.0, 4.0, 2.0 ] );
var y = new Float64Array( [ 2.0, 1.0, 2.0, 1.0, -2.0, 2.0, 3.0, 4.0 ] );

var z = drsskbn( x.length, x, 1, y, 1 );
// returns 72.0
```

Note that indexing is relative to the first index. To introduce an offset, use [`typed array`][mdn-typed-array] views.

<!-- eslint-disable stdlib/capitalized-comments -->

```javascript
var Float64Array = require( '@stdlib/array/float64' );

var x0 = new Float64Array( [ 2.0, 1.0, 2.0, -2.0, -2.0, 2.0, 3.0, 4.0 ] );
var y0 = new Float64Array( [ 8.0, -2.0, 3.0, -2.0, 7.0, -2.0, 0.0, -1.0 ] );

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*1 ); // start at 2nd element

var z = drsskbn( 4, x1, 2, y1, 2 );
// returns 50.0
```

#### drsskbn.ndarray( N, x, strideX, offsetX,  y, strideY, offsetY )

Computes the [residual sum of squares][wikipedia-residual-sum-of-squares] of two double-precision floating-point strided arrays using an improved Kahan–Babuška algorithm and alternative indexing semantics.

```javascript
var Float64Array = require( '@stdlib/array/float64' );

var x = new Float64Array( [ 1.0, -2.0, 2.0 ] );
var y = new Float64Array( [ 1.0, 1.0, -4.0 ] );

var z = drsskbn.ndarray( x.length, x, 1, 0, y, 1, 0 );
// returns 45.0
```

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, the offset parameters support indexing semantics based on starting indices. For example, to calculate the [residual sum of squares][wikipedia-residual-sum-of-squares] for every other element in `x` and `y` starting from the second element

```javascript
var Float64Array = require( '@stdlib/array/float64' );

var x = new Float64Array( [ 2.0, 1.0, 2.0, -2.0, -2.0, 2.0, 3.0, 4.0, 6.0 ] );
var y = new Float64Array( [ 8.0, -2.0, 3.0, -2.0, 7.0, -2.0, 0.0, -1.0, 4.0 ] );

var z = drsskbn.ndarray( 4, x, 2, 1, y, 2, 1 );
// returns 50.0
```

</section>

<!-- /.usage -->

<section class="notes">

## Notes

-   If `N <= 0`, both functions return `0.0`.

</section>

<!-- /.notes -->

<section class="examples">

## Examples

<!-- eslint no-undef: "error" -->

```javascript
var discreteUniform = require( '@stdlib/random/array/discrete-uniform' );
var drsskbn = require( '@stdlib/blas/ext/base/drsskbn' );

var opts = {
    'dtype': 'float64'
};
var x = discreteUniform( 10, -50, 50, opts );
console.log( x );

var y = discreteUniform( 10, -50, 50, opts );
console.log( y );

var d = drsskbn( x.length, x, 1, y, 1 );
console.log( d );
```

</section>

<!-- /.examples -->

<!-- C interface documentation. -->

* * *

<section class="c">

## C APIs

<!-- Section to include introductory text. Make sure to keep an empty line after the intro `section` element and another before the `/section` close. -->

<section class="intro">

</section>

<!-- /.intro -->

<!-- C usage documentation. -->

<section class="usage">

### Usage

```c
#include "stdlib/blas/ext/base/drsskbn.h"
```

#### stdlib_strided_drsskbn( N, \*X, strideX, \*Y, strideY )

Computes the [residual sum of squares][wikipedia-residual-sum-of-squares] of two double-precision floating-point strided arrays using an improved Kahan–Babuška algorithm.

```c
const double x[] = { 1.0, -2.0, 2.0 };
const double y[] = { 1.0, 1.0, -4.0 };

double z = stdlib_strided_drsskbn( 3, x, 1, y, 1 );
// returns 45.0
```

The function accepts the following arguments:

-   **N**: `[in] CBLAS_INT` number of indexed elements.
-   **X**: `[in] double*` first input array.
-   **strideX**: `[in] CBLAS_INT` stride length for `X`.
-   **Y**: `[in] double*` second input array.
-   **strideY**: `[in] CBLAS_INT` stride length for `Y`.

```c
double stdlib_strided_drsskbn( const CBLAS_INT N, const double *X, const CBLAS_INT strideX, const double *Y, const CBLAS_INT strideY );
```

<!--lint ignore maximum-heading-length-->

#### stdlib_strided_drsskbn_ndarray( N, \*X, strideX, offsetX, \*Y, strideY, offsetY )

<!--lint enable maximum-heading-length-->

Computes the [residual sum of squares][wikipedia-residual-sum-of-squares] of two double-precision floating-point strided arrays using an improved Kahan–Babuška algorithm and alternative indexing semantics.

```c
const double x[] = { 1.0, -2.0, 2.0 };
const double y[] = { 1.0, 1.0, -4.0 };

double v = stdlib_strided_drsskbn_ndarray( 3, x, 1, 0, y, 1, 0 );
// returns 45.0
```

The function accepts the following arguments:

-   **N**: `[in] CBLAS_INT` number of indexed elements.
-   **X**: `[in] double*` first input array.
-   **strideX**: `[in] CBLAS_INT` stride length for `X`.
-   **offsetX**: `[in] CBLAS_INT` starting index for `X`.
-   **Y**: `[in] double*` second input array.
-   **strideY**: `[in] CBLAS_INT` stride length for `Y`.
-   **offsetY**: `[in] CBLAS_INT` starting index for `Y`.

```c
double stdlib_strided_drsskbn_ndarray( const CBLAS_INT N, const double *X, const CBLAS_INT strideX, const CBLAS_INT offsetX, const double *Y, const CBLAS_INT strideY, const CBLAS_INT offsetY );
```

</section>

<!-- /.usage -->

<!-- C API usage notes. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->

<section class="notes">

</section>

<!-- /.notes -->

<!-- C API usage examples. -->

<section class="examples">

### Examples

```c
#include "stdlib/blas/ext/base/drsskbn.h"
#include <stdio.h>

int main( void ) {
    // Create two strided arrays:
    const double x[] = { 1.0, -2.0, -4.0, 5.0, 0.0, 3.0 };
    const double y[] = { 5.0, 12.0, -8.0, 15.0, 9.0, 0.0 };

    // Specify the number of elements:
    const int N = 5;

    // Specify the stride lengths:
    const int strideX = 1;
    const int strideY = 1;

    // Compute the residual sum of squares of `x` and `y`:
    double d = stdlib_strided_drsskbn( N, x, strideX, y, strideY );

    // Print the result:
    printf( "rss: %lf\n", d );

    // Specify index offsets:
    const int offsetX = 1;
    const int offsetY = 1;

    // Compute the residual sum of squares of `x` and `y` with offsets:
    d = stdlib_strided_drsskbn_ndarray( N, x, strideX, offsetX, y, strideY, offsetY );

    // Print the result:
    printf( "rss: %lf\n", d );
}
```

</section>

<!-- /.examples -->

</section>

<!-- /.c -->

* * *

<section class="references">

## 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].

</section>

<!-- /.references -->

<!-- Section for related `stdlib` packages. Do not manually edit this section, as it is automatically populated. -->

<section class="related">

</section>

<!-- /.related -->

<!-- Section for all links. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->

<section class="links">

[wikipedia-residual-sum-of-squares]: https://en.wikipedia.org/wiki/Residual_sum_of_squares

[@stdlib/array/float64]: https://www.npmjs.com/package/@stdlib/array-float64

[mdn-typed-array]: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/TypedArray

[@neumaier:1974a]: https://doi.org/10.1002/zamm.19740540106

</section>

<!-- /.links -->
