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# ssyr2

> Perform the symmetric rank 2 operation `A = α*x*y^T + α*y*x^T + A`.

<section class="usage">

## Usage

```javascript
var ssyr2 = require( '@stdlib/blas/base/ssyr2' );
```

#### ssyr2( order, uplo, N, α, x, sx, y, sy, A, LDA )

Performs the symmetric rank 2 operation `A = α*x*y^T + α*y*x^T + A`, where `α` is a scalar, `x` and `y` are `N` element vectors, and `A` is an `N` by `N` symmetric matrix.

```javascript
var Float32Array = require( '@stdlib/array/float32' );

var A = new Float32Array( [ 1.0, 2.0, 3.0, 2.0, 1.0, 2.0, 3.0, 2.0, 1.0 ] );
var x = new Float32Array( [ 1.0, 2.0, 3.0 ] );
var y = new Float32Array( [ 1.0, 2.0, 3.0 ] );

ssyr2( 'row-major', 'upper', 3, 1.0, x, 1, y, 1, A, 3 );
// A => <Float32Array>[ 3.0, 6.0, 9.0, 2.0, 9.0, 14.0, 3.0, 2.0, 19.0 ]
```

The function has the following parameters:

-   **order**: storage layout.
-   **uplo**: specifies whether the upper or lower triangular part of the symmetric matrix `A` should be referenced.
-   **N**: number of elements along each dimension of `A`.
-   **α**: scalar constant.
-   **x**: first input [`Float32Array`][mdn-float32array].
-   **sx**: stride length for `x`.
-   **y**: second input [`Float32Array`][mdn-float32array].
-   **sy**: stride length for `y`.
-   **A**: input matrix stored in linear memory as a [`Float32Array`][mdn-float32array].
-   **LDA**: stride of the first dimension of `A` (a.k.a., leading dimension of the matrix `A`).

The stride parameters determine how elements in the input arrays are accessed at runtime. For example, to iterate over every other element of `x`,

```javascript
var Float32Array = require( '@stdlib/array/float32' );

var A = new Float32Array( [ 1.0, 2.0, 3.0, 2.0, 1.0, 2.0, 3.0, 2.0, 1.0 ] );
var x = new Float32Array( [ 1.0, 2.0, 3.0, 4.0, 5.0 ] );
var y = new Float32Array( [ 1.0, 2.0, 3.0 ] );

ssyr2( 'row-major', 'upper', 3, 1.0, x, 2, y, 1, A, 3 );
// A => <Float32Array>[ 3.0, 7.0, 11.0, 2.0, 13.0, 21.0, 3.0, 2.0, 31.0 ]
```

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 Float32Array = require( '@stdlib/array/float32' );

// Initial arrays...
var x0 = new Float32Array( [ 0.0, 1.0, 1.0, 1.0 ] );
var y0 = new Float32Array( [ 0.0, 1.0, 2.0, 3.0 ] );
var A = new Float32Array( [ 1.0, 2.0, 3.0, 2.0, 1.0, 2.0, 3.0, 2.0, 1.0 ] );

// Create offset views...
var x1 = new Float32Array( x0.buffer, x0.BYTES_PER_ELEMENT*1 ); // start at 2nd element
var y1 = new Float32Array( y0.buffer, y0.BYTES_PER_ELEMENT*1 ); // start at 2nd element

ssyr2( 'row-major', 'upper', 3, 1.0, x1, 1, y1, 1, A, 3 );
// A => <Float32Array>[ 3.0, 5.0, 7.0, 2.0, 5.0, 7.0, 3.0, 2.0, 7.0 ]
```

#### ssyr2.ndarray( uplo, N, α, x, sx, ox, y, sy, oy, A, sa1, sa2, oa )

Performs the symmetric rank 2 operation `A = α*x*y^T + α*y*x^T + A`, using alternative indexing semantics and where `α` is a scalar, `x` and `y` are `N` element vectors, and `A` is an `N` by `N` symmetric matrix.

```javascript
var Float32Array = require( '@stdlib/array/float32' );

var A = new Float32Array( [ 1.0, 2.0, 3.0, 2.0, 1.0, 2.0, 3.0, 2.0, 1.0 ] );
var x = new Float32Array( [ 1.0, 2.0, 3.0 ] );
var y = new Float32Array( [ 1.0, 2.0, 3.0 ] );

ssyr2.ndarray( 'upper', 3, 1.0, x, 1, 0, y, 1, 0, A, 3, 1, 0 );
// A => <Float32Array>[ 3.0, 6.0, 9.0, 2.0, 9.0, 14.0, 3.0, 2.0, 19.0 ]
```

The function has the following additional parameters:

-   **ox**: starting index for `x`.
-   **oy**: starting index for `y`.
-   **sa1**: stride of the first dimension of `A`.
-   **sa2**: stride of the second dimension of `A`.
-   **oa**: starting index for `A`.

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,

```javascript
var Float32Array = require( '@stdlib/array/float32' );

var A = new Float32Array( [ 1.0, 2.0, 3.0, 2.0, 1.0, 2.0, 3.0, 2.0, 1.0 ] );
var x = new Float32Array( [ 1.0, 2.0, 3.0, 4.0, 5.0 ] );
var y = new Float32Array( [ 1.0, 2.0, 3.0 ] );

ssyr2.ndarray( 'upper', 3, 1.0, x, -2, 4, y, 1, 0, A, 3, 1, 0 );
// A => <Float32Array>[ 11.0, 15.0, 19.0, 2.0, 13.0, 13.0, 3.0, 2.0, 7.0 ]
```

</section>

<!-- /.usage -->

<section class="notes">

## Notes

-   `ssyr2()` corresponds to the [BLAS][blas] level 2 function [`ssyr2`][blas-ssyr2].

</section>

<!-- /.notes -->

<section class="examples">

## Examples

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

```javascript
var discreteUniform = require( '@stdlib/random/array/discrete-uniform' );
var ones = require( '@stdlib/array/ones' );
var ssyr2 = require( '@stdlib/blas/base/ssyr2' );

var opts = {
    'dtype': 'float32'
};

var N = 3;

// Create N-by-N symmetric matrices:
var A1 = ones( N*N, opts.dtype );
var A2 = ones( N*N, opts.dtype );

// Create random vectors:
var x = discreteUniform( N, -10.0, 10.0, opts );
var y = discreteUniform( N, -10.0, 10.0, opts );

ssyr2( 'row-major', 'upper', 3, 1.0, x, 1, y, 1, A1, 3 );
console.log( A1 );

ssyr2.ndarray( 'upper', 3, 1.0, x, 1, 0, y, 1, 0, A2, 3, 1, 0 );
console.log( A2 );
```

</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/base/ssyr2.h"
```

#### c_ssyr2( order, uplo, N, alpha, \*X, sx, \*Y, sy, \*A, LDA )

Performs the symmetric rank 2 operation `A = α*x*y^T + α*y*x^T + A` where `α` is a scalar, `x` and `y` are `N` element vectors, and `A` is an `N` by `N` symmetric matrix.

```c
#include "stdlib/blas/base/shared.h"

float A[] = { 1.0f, 2.0f, 3.0f, 2.0f, 1.0f, 2.0f, 3.0f, 2.0f, 1.0f };
const float x[] = { 1.0f, 2.0f, 3.0f };
const float y[] = { 1.0f, 2.0f, 3.0f };

c_ssyr2( CblasColMajor, CblasUpper, 3, 1.0f, x, 1, y, 1, A, 3 );
```

The function accepts the following arguments:

-   **order**: `[in] CBLAS_LAYOUT` storage layout.
-   **uplo**: `[in] CBLAS_UPLO` specifies whether the upper or lower triangular part of the symmetric matrix `A` should be referenced.
-   **N**: `[in] CBLAS_INT` number of elements along each dimension of `A`.
-   **alpha**: `[in] float` scalar constant.
-   **X**: `[in] float*` first input array.
-   **strideX**: `[in] CBLAS_INT` stride length for `X`.
-   **Y**: `[in] float*` second input array.
-   **strideY**: `[in] CBLAS_INT` stride length for `Y`.
-   **A**: `[inout] float*` input matrix.
-   **LDA**: `[in] CBLAS_INT` stride of the first dimension of `A` (a.k.a., leading dimension of the matrix `A`).

```c
void c_ssyr2( const CBLAS_LAYOUT order, const CBLAS_UPLO uplo, const CBLAS_INT N, const float alpha, const float *X, const CBLAS_INT strideX, const float *Y, const CBLAS_INT strideY, float *A, const CBLAS_INT LDA )
```

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

#### c_ssyr2_ndarray( uplo, N, alpha, \*X, sx, ox, \*Y, sy, oy, \*A, sa1, sa2, oa )

Performs the symmetric rank 2 operation `A = α*x*y^T + α*y*x^T + A`, using alternative indexing semantics and where `α` is a scalar, `x` and `y` are `N` element vectors, and `A` is an `N` by `N` symmetric matrix.

```c
#include "stdlib/blas/base/shared.h"

float A[] = { 1.0f, 2.0f, 3.0f, 2.0f, 1.0f, 2.0f, 3.0f, 2.0f, 1.0f };
const float x[] = { 1.0f, 2.0f, 3.0f };
const float y[] = { 1.0f, 2.0f, 3.0f };

c_ssyr2_ndarray( CblasUpper, 3, 1.0f, x, 1, 0, y, 1, 0, A, 3, 1, 0 );
```

The function accepts the following arguments:

-   **uplo**: `[in] CBLAS_UPLO` specifies whether the upper or lower triangular part of the symmetric matrix `A` should be referenced.
-   **N**: `[in] CBLAS_INT` number of elements along each dimension of `A`.
-   **alpha**: `[in] float` scalar constant.
-   **X**: `[in] float*` first input array.
-   **sx**: `[in] CBLAS_INT` stride length for `X`.
-   **ox**: `[in] CBLAS_INT` starting index for `X`.
-   **Y**: `[in] float` second input array.
-   **sy**: `[in] CBLAS_INT` stride length for `Y`.
-   **oy**: `[in] CBLAS_INT` starting index for `Y`.
-   **A**: `[inout] float*` input matrix.
-   **sa1**: `[in] CBLAS_INT` stride of the first dimension of `A`.
-   **sa2**: `[in] CBLAS_INT` stride of the second dimension of `A`.
-   **oa**: `[in] CBLAS_INT` starting index for `A`.

```c
void c_ssyr2_ndarray( const CBLAS_UPLO uplo, const CBLAS_INT N, const float alpha, const float *X, const CBLAS_INT strideX, const CBLAS_INT offsetX, const float *Y, CBLAS_INT strideY, const CBLAS_INT offsetY, float *A, const CBLAS_INT strideA1, const CBLAS_INT strideA2, const CBLAS_INT offsetA )
```

</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/base/ssyr2.h"
#include "stdlib/blas/base/shared.h"
#include <stdio.h>

int main( void ) {
    // Define 3x3 symmetric matrices stored in row-major layout:
    float A1[ 3*3 ] = {
        1.0f, 2.0f, 3.0f,
        2.0f, 1.0f, 2.0f,
        3.0f, 2.0f, 1.0f
    };

    float A2[ 3*3 ] = {
        1.0f, 2.0f, 3.0f,
        2.0f, 1.0f, 2.0f,
        3.0f, 2.0f, 1.0f
    };

    // Define `x` and `y` vectors:
    const float x[ 3 ] = { 1.0f, 2.0f, 3.0f };
    const float y[ 3 ] = { 1.0f, 2.0f, 3.0f };

    // Specify the number of elements along each dimension of `A1` and `A2`:
    const int N = 3;

    // Perform the symmetric rank 2 operation `A = α*x*y^T + α*y*x^T + A`:
    c_ssyr2( CblasColMajor, CblasUpper, N, 1.0f, x, 1, y, 1, A1, N );

    // Print the result:
    for ( int i = 0; i < N*N; i++ ) {
        printf( "A1[ %i ] = %f\n", i, A1[ i ] );
    }

    // Perform the symmetric rank 2 operation `A = α*x*y^T + α*y*x^T + A` using alternative indexing semantics:
    c_ssyr2_ndarray( CblasUpper, N, 1.0f, x, 1, 0, y, 1, 0, A2, N, 1, 0 );

    // Print the result:
    for ( int i = 0; i < N*N; i++ ) {
        printf( "A2[ %i ] = %f\n", i, A2[ i ] );
    }
}
```

</section>

<!-- /.examples -->

</section>

<!-- /.c -->

<!-- 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">

[blas]: http://www.netlib.org/blas

[blas-ssyr2]: https://netlib.org/lapack/explore-html/dd/de5/group__her2_ga6741f2ac8fe025042fd994ccc6625b45.html

[mdn-float32array]: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Float32Array

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

</section>

<!-- /.links -->
