feat: add lapack/base/dgebal

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

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+<!--
+
+@license Apache-2.0
+
+Copyright (c) 2025 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.
+
+-->
+
+# dgebal
+
+> Balances a general real matrix `A`.
+
+<section class="usage">
+
+## Usage
+
+```javascript
+var dgebal = require( '@stdlib/lapack/base/dgebal' );
+```
+
+#### dgebal( order, job, N, A, LDA, out, scale )
+
+Balances a general real matrix `A`.
+
+```javascript
+var Float64Array = require( '@stdlib/array/float64' );
+var Int32Array = require( '@stdlib/array/int32' );
+
+var A = new Float64Array( [ 1.0, 100.0, 0.0, 2.0, 200.0, 0.0, 0.0, 0.0, 3.0 ] );
+var out = new Int32Array( 2 );
+var scale = new Float64Array( 3 );
+
+/*
+    A = [
+        [ 1.0, 100.0, 0.0 ],
+        [ 2.0, 200.0, 0.0 ],
+        [ 0.0,  0.0,  3.0 ],
+    ]
+*/
+
+dgebal( 'row-major', 'both', 3, A, 3, out, scale );
+// A => <Float64Array>[ 1.0, 12.5, 0.0, 16.0, 200.0, 0.0, 0.0, 0.0, 3.0 ]
+// out => <Int32Array>[ 0, 1 ]
+// scale => <Float64Array>[ 8.0, 1.0, 2.0 ]
+```
+
+The function has the following parameters:
+
+-   **order**: storage layout.
+-   **job**: indicates the operations to be performed. The `job` parameter can be one of the following. `none` (do nothing), `permute` (permute only), `scale` (scale only) and `both` (both permute and scale).
+-   **N**: number of row/columns in `A`.
+-   **A**: input [`Float64Array`][mdn-float64array].
+-   **LDA**: stride of the first dimension of `A` (a.k.a., leading dimension of the matrix `A`).
+-   **out**: stores the first and last row/column of the balanced submatrix as a [`Int32Array`][mdn-int32array].
+-   **scale**: scale factors are stored in the `scale` array as a [`Float64Array`][mdn-float64array]. Where the indices `0` to `out[ 0 ] - 1` and `out[ 1 ] + 1` to `N-1` contain the index of the interchanged row/column (zero based), and the indices `out[ 0 ]` to `out[ 1 ]` contains the scaling factor applied.
+
+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, max-len -->
+
+```javascript
+var Float64Array = require( '@stdlib/array/float64' );
+var Int32Array = require( '@stdlib/array/int32' );
+
+// Initial arrays...
+var A0 = new Float64Array( [ 0.0, 1.0, 100.0, 0.0, 2.0, 200.0, 0.0, 0.0, 0.0, 3.0 ] );
+var out0 = new Int32Array( 3 );
+var scale0 = new Float64Array( 4 );
+
+// Create offset views...
+var A1 = new Float64Array( A0.buffer, A0.BYTES_PER_ELEMENT*1 ); // start at 2nd element
+var out1 = new Int32Array( out0.buffer, out0.BYTES_PER_ELEMENT*1 ); // start at 2nd element
+var scale1 = new Float64Array( scale0.buffer, scale0.BYTES_PER_ELEMENT*1 ); // start at 2nd element
+
+/*
+    A = [
+        [ 1.0, 100.0, 0.0 ],
+        [ 2.0, 200.0, 0.0 ],
+        [ 0.0,  0.0,  3.0 ],
+    ]
+*/
+
+dgebal( 'row-major', 'both', 3, A1, 3, out1, scale1 );
+// A0 => <Float64Array>[ 0.0, 1.0, 12.5, 0.0, 16.0, 200.0, 0.0, 0.0, 0.0, 3.0 ]
+// out0 => <Int32Array>[ 0, 0, 1 ]
+// scale0 => <Float64Array>[ 0.0, 8.0, 1.0, 2.0 ]
+```
+
+#### dgebal.ndarray( uplo, M, N, A, sa1, sa2, oa, B, sb1, sb2, ob )
+
+Balances a general real matrix `A` using alternative indexing semantics.
+
+```javascript
+var Float64Array = require( '@stdlib/array/float64' );
+var Int32Array = require( '@stdlib/array/int32' );
+
+var A = new Float64Array( [ 1.0, 100.0, 0.0, 2.0, 200.0, 0.0, 0.0, 0.0, 3.0 ] );
+var out = new Int32Array( 2 );
+var scale = new Float64Array( 3 );
+
+/*
+    A = [
+        [ 1.0, 100.0, 0.0 ],
+        [ 2.0, 200.0, 0.0 ],
+        [ 0.0,  0.0,  3.0 ],
+    ]
+*/
+
+dgebal.ndarray( 'both', 3, A, 3, 1, 0, out, 1, 0, scale, 1, 0 );
+// A => <Float64Array>[ 1.0, 12.5, 0.0, 16.0, 200.0, 0.0, 0.0, 0.0, 3.0 ]
+// out => <Int32Array>[ 0, 1 ]
+// scale => <Float64Array>[ 8.0, 1.0, 2.0 ]
+```
+
+The function has the following parameters:
+
+-   **job**: indicates the operations to be performed. The `job` parameter can be one of the following. `none` (do nothing), `permute` (permute only), `scale` (scale only) and `both` (both permute and scale).
+-   **N**: number of row/columns in `A`.
+-   **A**: input [`Float64Array`][mdn-float64array].
+-   **strideA1**: stride of the first dimension of `A`.
+-   **strideA2**: stride of the second dimension of `A`.
+-   **offsetA**: starting index for `A`.
+-   **out**: stores the first and last row/column of the balanced submatrix as a [`Int32Array`][mdn-int32array].
+-   **strideOut**: stride length of `out`.
+-   **offsetOut**: starting index for `out`.
+-   **scale**: scale factors are stored in the `scale` array as a [`Float64Array`][mdn-float64array]. Where the indices `0` to `out[ 0 ] - 1` and `out[ 1 ] + 1` to `N-1` contain the index of the interchanged row/column (zero based), and the indices `out[ 0 ]` to `out[ 1 ]` contains the scaling factor applied.
+-   **strideScale**: stride length for `scale`.
+-   **offsetScale**: starting index for `scale`.
+
+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,
+
+<!-- eslint-disable max-len -->
+
+```javascript
+var Float64Array = require( '@stdlib/array/float64' );
+var Int32Array = require( '@stdlib/array/int32' );
+
+var A = new Float64Array( [ 0.0, 1.0, 100.0, 0.0, 2.0, 200.0, 0.0, 0.0, 0.0, 3.0 ] );
+var out = new Int32Array( 3 );
+var scale = new Float64Array( 4 );
+
+/*
+    A = [
+        [ 1.0, 100.0, 0.0 ],
+        [ 2.0, 200.0, 0.0 ],
+        [ 0.0,  0.0,  3.0 ],
+    ]
+*/
+
+dgebal.ndarray( 'both', 3, A, 3, 1, 1, out, 1, 1, scale, 1, 1 );
+// A => <Float64Array>[ 0.0, 1.0, 12.5, 0.0, 16.0, 200.0, 0.0, 0.0, 0.0, 3.0 ]
+// out => <Int32Array>[ 0, 0, 1 ]
+// scale => <Float64Array>[ 0.0, 8.0, 1.0, 2.0 ]
+```
+
+</section>
+
+<!-- /.usage -->
+
+<section class="notes">
+
+## Notes
+
+-   `dgebal()` corresponds to the [LAPACK][lapack] routine [`dgebal`][lapack-dgebal].
+
+</section>
+
+<!-- /.notes -->
+
+<section class="examples">
+
+## Examples
+
+<!-- eslint no-undef: "error" -->
+
+```javascript
+var Float64Array = require( '@stdlib/array/float64' );
+var Int32Array = require( '@stdlib/array/int32' );
+var ndarray2array = require( '@stdlib/ndarray/base/to-array' );
+var dgebal = require( '@stdlib/lapack/base/dgebal' );
+
+// Specify matrix meta data:
+var shape = [ 3, 3 ];
+var strides = [ 3, 1 ];
+var offset = 0;
+var order = 'row-major';
+
+// Create a matrix stored in linear memory:
+var A = new Float64Array( [ 1.0, 100.0, 0.0, 2.0, 200.0, 0.0, 0.0, 0.0, 3.0 ] );
+console.log( ndarray2array( A, shape, strides, offset, order ) );
+
+// Define arrays to store the scaling factors
+var out = new Int32Array( 2 );
+var scale = new Float64Array( 3 );
+
+// Permute and scale the matrix:
+dgebal( order, 'both', shape[ 0 ], A, strides[ 0 ], out, scale );
+
+console.log( ndarray2array( A, shape, strides, offset, order ) );
+console.log( out );
+console.log( scale );
+```
+
+</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-dgebal]: https://www.netlib.org/lapack/explore-html-3.6.1/dd/d9a/group__double_g_ecomputational_ga411292dd693c20ff9c27650fb7bddf85.html
+
+[mdn-float64array]: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Float64Array
+
+[mdn-int32array]: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Int32Array
+
+[mdn-typed-array]: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/TypedArray
+
+</section>
+
+<!-- /.links -->