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# dspmv

> Perform the matrix-vector operation `y = α*A*x + β*y` where `α` and `β` are scalars, `x` and `y` are `N` element vectors and, `A` is an `N` by `N` symmetric matrix supplied in packed form.

<section class="usage">

## Usage

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

#### dspmv( order, uplo, N, α, AP, x, sx, β, y, sy )

Performs the matrix-vector operation `y = α*A*x + β*y` where `α` and `β` are scalars, `x` and `y` are `N` element vectors, and `A` is an `N` by `N` symmetric matrix supplied in packed form `AP`.

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

var AP = new Float64Array( [ 1.0, 2.0, 3.0, 4.0, 5.0, 6.0 ] );
var x = new Float64Array( [ 1.0, 1.0, 1.0 ] );
var y = new Float64Array( [ 1.0, 1.0, 1.0 ] );

dspmv( 'column-major', 'lower', 3, 1.0, AP, x, 1, 1.0, y, 1 );
// y => <Float64Array>[ 7.0, 12.0, 15.0 ]
```

The function has the following parameters:

-   **order**: storage layout.
-   **uplo**: specifies whether the upper or lower triangular part of the symmetric matrix `A` is supplied.
-   **N**: specifies the order of the matrix `A`.
-   **α**: scalar constant.
-   **AP**: packed form of a symmetric matrix `A` stored in linear memory as a [`Float64Array`][mdn-float64array].
-   **x**: input [`Float64Array`][mdn-float64array].
-   **sx**: index increment for `x`.
-   **β**: scalar constant.
-   **y**: output [`Float64Array`][mdn-float64array].
-   **sy**: index increment for `y`.

The stride parameters determine how elements in the input arrays are accessed at runtime. For example, to iterate over the elements of `y` in reverse order,

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

var AP = new Float64Array( [ 1.0, 2.0, 3.0, 4.0, 5.0, 6.0 ] );
var x = new Float64Array( [ 1.0, 1.0, 1.0 ] );
var y = new Float64Array( [ 1.0, 1.0, 1.0 ] );

dspmv( 'column-major', 'lower', 3, 1.0, AP, x, 1, 1.0, y, -1 );
// y => <Float64Array>[ 15.0, 12.0, 7.0 ]
```

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, 1.0, 1.0 ] );
var y0 = new Float64Array( [ 0.0, 1.0, 1.0 ] );
var AP = new Float64Array( [ 1.0, 2.0, 3.0 ] );

// 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*1 ); // start at 2nd element

dspmv( 'row-major', 'upper', 2, 1.0, AP, x1, -1, 1.0, y1, -1 );
// y0 => <Float64Array>[ 0.0, 6.0, 4.0 ]
```

#### dspmv.ndarray( order, uplo, N, α, AP, oa, x, sx, ox, β, y, sy, oy )

Performs the matrix-vector operation `y = α*A*x + β*y` using alternative indexing semantics and where `α` and `β` are scalars, `x` and `y` are `N` element vectors, and `A` is an `N` by `N` symmetric matrix supplied in packed form `AP`.

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

var AP = new Float64Array( [ 1.0, 2.0, 3.0, 4.0, 5.0, 6.0 ] );
var x = new Float64Array( [ 1.0, 1.0, 1.0 ] );
var y = new Float64Array( [ 1.0, 1.0, 1.0 ] );

dspmv.ndarray( 'column-major', 'lower', 3, 1.0, AP, 0, x, 1, 0, 1.0, y, 1, 0 );
// y => <Float64Array>[ 7.0, 12.0, 15.0 ]
```

The function has the following additional parameters:

-   **oa**: starting index for `AP`.
-   **ox**: starting index for `x`.
-   **oy**: 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,

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

var AP = new Float64Array( [ 0.0, 0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0 ] );
var x = new Float64Array( [ 1.0, 1.0, 1.0 ] );
var y = new Float64Array( [ 1.0, 1.0, 1.0 ] );

dspmv.ndarray( 'column-major', 'lower', 3, 1.0, AP, 2, x, 1, 0, 1.0, y, -1, 2 );
// y => <Float64Array>[ 15.0, 12.0, 7.0 ]
```

</section>

<!-- /.usage -->

<section class="notes">

## Notes

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

</section>

<!-- /.notes -->

<section class="examples">

## Examples

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

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

var opts = {
    'dtype': 'float64'
};

var N = 3;
var AP = discreteUniform( N * ( N + 1 ) / 2, -10, 10, opts );

var x = discreteUniform( N, -10, 10, opts );
var y = discreteUniform( N, -10, 10, opts );

dspmv.ndarray( 'row-major', 'upper', N, 1.0, AP, 0, x, 1, 0, 1.0, y, 1, 0 );
console.log( y );
```

</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
TODO
```

#### TODO

TODO.

```c
TODO
```

TODO

```c
TODO
```

</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
TODO
```

</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

[dspmv]: https://www.netlib.org/lapack/explore-html/d0/d4b/group__hpmv_ga739f8dc2316523832bde2b237fcad8a6.html#ga739f8dc2316523832bde2b237fcad8a6

[mdn-float64array]: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Float64Array

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

</section>

<!-- /.links -->
