feat: add lapack/base/dlaqsy

lib/node_modules/@stdlib/lapack/base/dlaqsy/README.md

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+<!--
+
+@license Apache-2.0
+
+Copyright (c) 2026 The Stdlib Authors.
+
+Licensed under the Apache License, Version 2.0 (the "License");
+you may not use this file except in compliance with the License.
+You may obtain a copy of the License at
+
+   http://www.apache.org/licenses/LICENSE-2.0
+
+Unless required by applicable law or agreed to in writing, software
+distributed under the License is distributed on an "AS IS" BASIS,
+WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+See the License for the specific language governing permissions and
+limitations under the License.
+
+-->
+
+# dlaqsy
+
+> Equilibrate a real symmetric matrix using the scaling factors in a vector.
+
+<section class="intro">
+
+Equilibration rescales the rows and columns of a matrix in order to improve its condition number. Given a real symmetric matrix `A` and a vector of scale factors `S`, this routine overwrites `A` with
+
+<!-- <equation class="equation" label="eq:symmetric_equilibration" align="center" raw="A = \mathrm{diag}(S) \cdot A \cdot \mathrm{diag}(S)" alt="Equation for symmetric equilibration."> -->
+
+```math
+A = \mathrm{diag}(S) \cdot A \cdot \mathrm{diag}(S)
+```
+
+<!-- </equation> -->
+
+such that each referenced element is updated according to
+
+<!-- <equation class="equation" label="eq:symmetric_equilibration_element" align="center" raw="a_{ij} = s_i \cdot a_{ij} \cdot s_j" alt="Equation for updating a matrix element during symmetric equilibration."> -->
+
+```math
+a_{ij} = s_i \cdot a_{ij} \cdot s_j
+```
+
+<!-- </equation> -->
+
+Equilibration is only performed when the ratio of the smallest scale factor to the largest scale factor is less than `0.1` or when the absolute value of the largest matrix entry is either too large or too small.
+
+</section>
+
+<!-- /.intro -->
+
+<section class="usage">
+
+## Usage
+
+```javascript
+var dlaqsy = require( '@stdlib/lapack/base/dlaqsy' );
+```
+
+#### dlaqsy( order, uplo, N, A, LDA, S, scond, amax )
+
+Equilibrates a real symmetric matrix `A` using the scaling factors in the vector `S`.
+
+```javascript
+var Float64Array = require( '@stdlib/array/float64' );
+
+var A = new Float64Array( [ 1.0, 2.0, 3.0, 2.0, 5.0, 6.0, 3.0, 6.0, 9.0 ] );
+var S = new Float64Array( [ 0.25, 2.0, 4.0 ] );
+
+/*
+    A = [
+        [ 1.0, 2.0, 3.0 ],
+        [ 2.0, 5.0, 6.0 ],
+        [ 3.0, 6.0, 9.0 ]
+    ]
+*/
+
+dlaqsy( 'row-major', 'upper', 3, A, 3, S, 0.0625, 9.0 );
+// A => <Float64Array>[ 0.0625, 1.0, 3.0, 2.0, 20.0, 48.0, 3.0, 6.0, 144.0 ]
+```
+
+The function has the following parameters:
+
+-   **order**: storage layout.
+-   **uplo**: specifies whether to reference the upper or lower triangular part of the symmetric matrix `A`. Must be either `'upper'` or `'lower'`.
+-   **N**: order of the matrix `A`.
+-   **A**: input matrix stored in linear memory as a [`Float64Array`][mdn-float64array].
+-   **LDA**: stride of the first dimension of `A` (a.k.a., leading dimension of the matrix `A`). Must be at least `max(1,N)`.
+-   **S**: vector of scale factors as a [`Float64Array`][mdn-float64array]. Should have `N` elements.
+-   **scond**: ratio of the smallest scale factor to the largest scale factor.
+-   **amax**: absolute value of the largest matrix entry.
+
+The function returns a character flag indicating whether the matrix was equilibrated (`'Y'`) or not (`'N'`).
+
+Only the triangular part of `A` specified by `uplo` is referenced and updated. If equilibration is not needed, `A` is left unchanged.
+
+```javascript
+var Float64Array = require( '@stdlib/array/float64' );
+
+var A = new Float64Array( [ 1.0, 2.0, 3.0, 2.0, 5.0, 6.0, 3.0, 6.0, 9.0 ] );
+var S = new Float64Array( [ 0.5, 1.0, 2.0 ] );
+
+var equed = dlaqsy( 'row-major', 'upper', 3, A, 3, S, 0.25, 9.0 );
+// returns 'N'
+```
+
+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 A0 = new Float64Array( [ 0.0, 1.0, 2.0, 2.0, 5.0 ] );
+var S0 = new Float64Array( [ 0.0, 0.25, 4.0 ] );
+
+// Create offset views...
+var A1 = new Float64Array( A0.buffer, A0.BYTES_PER_ELEMENT*1 ); // start at 2nd element
+var S1 = new Float64Array( S0.buffer, S0.BYTES_PER_ELEMENT*1 ); // start at 2nd element
+
+dlaqsy( 'row-major', 'upper', 2, A1, 2, S1, 0.0625, 5.0 );
+// A0 => <Float64Array>[ 0.0, 0.0625, 2.0, 2.0, 80.0 ]
+```
+
+#### dlaqsy.ndarray( uplo, N, A, sa1, sa2, oa, S, ss, os, scond, amax )
+
+Equilibrates a real symmetric matrix `A` using the scaling factors in the vector `S` and alternative indexing semantics.
+
+```javascript
+var Float64Array = require( '@stdlib/array/float64' );
+
+var A = new Float64Array( [ 1.0, 2.0, 3.0, 2.0, 5.0, 6.0, 3.0, 6.0, 9.0 ] );
+var S = new Float64Array( [ 0.25, 2.0, 4.0 ] );
+
+dlaqsy.ndarray( 'upper', 3, A, 3, 1, 0, S, 1, 0, 0.0625, 9.0 );
+// A => <Float64Array>[ 0.0625, 1.0, 3.0, 2.0, 20.0, 48.0, 3.0, 6.0, 144.0 ]
+```
+
+The function has the following parameters:
+
+-   **uplo**: specifies whether to reference the upper or lower triangular part of the symmetric matrix `A`. Must be either `'upper'` or `'lower'`.
+-   **N**: order of the matrix `A`.
+-   **A**: input matrix as a [`Float64Array`][mdn-float64array].
+-   **sa1**: stride of the first dimension of `A`.
+-   **sa2**: stride of the second dimension of `A`.
+-   **oa**: starting index for `A`.
+-   **S**: vector of scale factors as a [`Float64Array`][mdn-float64array].
+-   **ss**: stride length for `S`.
+-   **os**: starting index for `S`.
+-   **scond**: ratio of the smallest scale factor to the largest scale factor.
+-   **amax**: absolute value of the largest matrix entry.
+
+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 A = new Float64Array( [ 0.0, 1.0, 2.0, 2.0, 5.0 ] );
+var S = new Float64Array( [ 0.0, 4.0, 0.0, 0.25 ] );
+
+dlaqsy.ndarray( 'upper', 2, A, 2, 1, 1, S, -2, 3, 0.0625, 5.0 );
+// A => <Float64Array>[ 0.0, 0.0625, 2.0, 2.0, 80.0 ]
+```
+
+</section>
+
+<!-- /.usage -->
+
+<section class="notes">
+
+## Notes
+
+-   `dlaqsy()` corresponds to the [LAPACK][lapack] function [`dlaqsy`][lapack-dlaqsy].
+-   Equilibration is performed when `scond < 0.1`, `amax < smallnum`, or `amax > bignum`, where `smallnum` is the ratio of the safe minimum to the machine precision and `bignum = 1/smallnum`.
+
+</section>
+
+<!-- /.notes -->
+
+<section class="examples">
+
+## Examples
+
+<!-- eslint no-undef: "error" -->
+
+```javascript
+var Float64Array = require( '@stdlib/array/float64' );
+var ndarray2array = require( '@stdlib/ndarray/base/to-array' );
+var shape2strides = require( '@stdlib/ndarray/base/shape2strides' );
+var dlaqsy = require( '@stdlib/lapack/base/dlaqsy' );
+
+// Specify matrix meta data:
+var shape = [ 3, 3 ];
+var order = 'row-major';
+var strides = shape2strides( shape, order );
+
+// Create a symmetric matrix stored in linear memory:
+var A = new Float64Array( [ 1.0, 2.0, 3.0, 2.0, 5.0, 6.0, 3.0, 6.0, 9.0 ] );
+console.log( ndarray2array( A, shape, strides, 0, order ) );
+
+// Define a vector of scale factors:
+var S = new Float64Array( [ 0.25, 2.0, 4.0 ] );
+
+// Equilibrate the upper triangular part of the matrix:
+var equed = dlaqsy( order, 'upper', 3, A, strides[ 0 ], S, 0.0625, 9.0 );
+console.log( equed );
+console.log( ndarray2array( A, shape, strides, 0, order ) );
+```
+
+</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">
+
+[lapack]: https://www.netlib.org/lapack/explore-html/
+
+[lapack-dlaqsy]: https://www.netlib.org/lapack/explore-html/df/dca/group__laqhe_gaba164109f29d549e268fb25310701918.html
+
+[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 -->