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# gsyr

> Perform the symmetric rank 1 operation `A = α*x*x^T + A`.

<section class="usage">

## Usage

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

#### gsyr( order, uplo, N, α, x, sx, A, LDA )

Performs the symmetric rank 1 operation `A = α*x*x^T + A` where `α` is a scalar, `x` is an `N` element vector, and `A` is an `N` by `N` symmetric matrix.

```javascript
var A = [ 1.0, 2.0, 3.0, 2.0, 1.0, 2.0, 3.0, 2.0, 1.0 ];
var x = [ 1.0, 2.0, 3.0 ];

gsyr( 'row-major', 'upper', 3, 1.0, x, 1, A, 3 );
// A => [ 2.0, 4.0, 6.0, 2.0, 5.0, 8.0, 3.0, 2.0, 10.0 ]
```

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**: input array.
-   **sx**: stride length for `x`.
-   **A**: input matrix stored in linear memory.
-   **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 the elements of `x` in reverse order,

```javascript
var A = [ 1.0, 2.0, 3.0, 2.0, 1.0, 2.0, 3.0, 2.0, 1.0 ];
var x = [ 3.0, 2.0, 1.0 ];

gsyr( 'row-major', 'upper', 3, 1.0, x, -1, A, 3 );
// A => [ 2.0, 4.0, 6.0, 2.0, 5.0, 8.0, 3.0, 2.0, 10.0 ]
```

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' );

// Initial arrays...
var x0 = new Float64Array( [ 0.0, 3.0, 2.0, 1.0 ] );
var A = new Float64Array( [ 1.0, 2.0, 3.0, 2.0, 1.0, 2.0, 3.0, 2.0, 1.0 ] );

// Create offset views...
var x1 = new Float64Array( x0.buffer, x0.BYTES_PER_ELEMENT*1 ); // start at 2nd element

gsyr( 'row-major', 'upper', 3, 1.0, x1, -1, A, 3 );
// A => <Float64Array>[ 2.0, 4.0, 6.0, 2.0, 5.0, 8.0, 3.0, 2.0, 10.0 ]
```

#### gsyr.ndarray( uplo, N, α, x, sx, ox, A, sa1, sa2, oa )

Performs the symmetric rank 1 operation `A = α*x*x^T + A`, using alternative indexing semantics and where `α` is a scalar, `x` is an `N` element vector, and `A` is an `N` by `N` symmetric matrix.

```javascript
var A = [ 1.0, 2.0, 3.0, 2.0, 1.0, 2.0, 3.0, 2.0, 1.0 ];
var x = [ 1.0, 2.0, 3.0 ];

gsyr.ndarray( 'upper', 3, 1.0, x, 1, 0, A, 3, 1, 0 );
// A => [ 2.0, 4.0, 6.0, 2.0, 5.0, 8.0, 3.0, 2.0, 10.0 ]
```

The function has the following additional parameters:

-   **ox**: starting index for `x`.
-   **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 A = [ 1.0, 2.0, 3.0, 2.0, 1.0, 2.0, 3.0, 2.0, 1.0 ];
var x = [ 1.0, 2.0, 3.0, 4.0, 5.0 ];

gsyr.ndarray( 'upper', 3, 1.0, x, -2, 4, A, 3, 1, 0 );
// A => [ 26.0, 17.0, 8.0, 2.0, 10.0, 5.0, 3.0, 2.0, 2.0 ]
```

</section>

<!-- /.usage -->

<section class="notes">

## Notes

-   `gsyr()` corresponds to the [BLAS][blas] level 2 function [`dsyr`][dsyr] with the exception that this implementation works with any array type, not just Float64Arrays. Depending on the environment, the typed versions ([`dsyr`][@stdlib/blas/base/dsyr], [`ssyr`][@stdlib/blas/base/ssyr], etc.) are likely to be significantly more performant.
-   Both functions support array-like objects having getter and setter accessors for array element access (e.g., [`@stdlib/array/base/accessor`][@stdlib/array/base/accessor]).

</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 gsyr = require( '@stdlib/blas/base/gsyr' );

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

var N = 3;

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

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

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

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

</section>

<!-- /.examples -->

<!-- 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

[dsyr]: https://www.netlib.org/lapack/explore-html/dc/d82/group__her_ga07f0e3f8592107877f12a554a41c7413.html#ga07f0e3f8592107877f12a554a41c7413

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

[@stdlib/blas/base/dsyr]: https://github.com/stdlib-js/blas/tree/main/base/dsyr

[@stdlib/blas/base/ssyr]: https://github.com/stdlib-js/blas/tree/main/base/ssyr

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

</section>

<!-- /.links -->
