feat: add lapack/base/dgebak

lib/node_modules/@stdlib/lapack/base/dgebak/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
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+   http://www.apache.org/licenses/LICENSE-2.0
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+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.
+
+-->
+
+# dgebak
+
+> LAPACK routine to form the right or left eigenvectors of a real general matrix by backward transformation on the computed eigenvectors of the balanced matrix output by `dgebal`.
+
+<section class="usage">
+
+## Usage
+
+```javascript
+var dgebak = require( '@stdlib/lapack/base/dgebak' );
+```
+
+#### dgebak( order, job, side, N, M, ilo, ihi, scale, V, LDC )
+
+Forms the right or left eigenvectors of a real general matrix by backward transformation on the computed eigenvectors of the balanced matrix output by `dgebal`.
+
+```javascript
+var Float64Array = require( '@stdlib/array/float64' );
+
+var scale = new Float64Array( [ 0.5, 2.0, 1.0 ] );
+var V = new Float64Array( [ 1.0, 4.0, 2.0, 5.0, 3.0, 6.0 ] );
+
+/*
+    V = [
+        [ 1.0, 4.0 ],
+        [ 2.0, 5.0 ],
+        [ 3.0, 6.0 ]
+    ]
+*/
+
+var out = dgebak( 'row-major', 'scale', 'right', 3, 2, 0, 2, scale, V, 2 );
+// returns <Float64Array>[ 0.5, 2.0, 4.0, 10.0, 3.0, 6.0 ]
+```
+
+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).
+-   **side**: specifies the side of the eigenvectors in `V` (`right` or `left`).
+-   **N**: number of rows in `V`.
+-   **M**: number of columns in `V`.
+-   **ilo**: index of the leftmost element in the submatrix computed by `dgebal` (zero-based).
+-   **ihi**: index of the rightmost element in the submatrix computed by `dgebal` (zero-based).
+-   **scale**: contains the details of the permutations and scaling factors applied when balancing `V`, computed by `dgebal` as a [`Float64Array`][mdn-float64array].
+-   **V**: input matrix stored in linear memory as a [`Float64Array`][mdn-float64array].
+-   **LDV**: stride of the first dimension of `V` (a.k.a., leading dimension of the matrix `V`).
+
+
+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 scale0 = new Float64Array( [ 0.0, 0.5, 2.0, 1.0 ] );
+var V0 = new Float64Array( [ 0.0, 1.0, 4.0, 2.0, 5.0, 3.0, 6.0 ] );
+
+// Create offset views...
+var scale1 = new Float64Array( scale0.buffer, scale0.BYTES_PER_ELEMENT*1 ); // start at 2nd element
+var V1 = new Float64Array( V0.buffer, V0.BYTES_PER_ELEMENT*1 ); // start at 2nd element
+
+/*
+    V = [
+        [ 1.0, 4.0 ],
+        [ 2.0, 5.0 ],
+        [ 3.0, 6.0 ]
+    ]
+*/
+
+var out = dgebak( 'row-major', 'scale', 'right', 3, 2, 0, 2, scale1, V1, 2 );
+// V0 => <Float64Array>[ 0.0, 0.5, 2.0, 4.0, 10.0, 3.0, 6.0 ]
+```
+
+#### dgebak.ndarray( job, side, N, M, ilo, ihi, scale, ss, os, V, sv1, sv2, ov )
+
+Forms the right or left eigenvectors of a real general matrix by backward transformation on the computed eigenvectors of the balanced matrix output by `dgebal` using alternative indexing semantics.
+
+```javascript
+var Float64Array = require( '@stdlib/array/float64' );
+
+var scale = new Float64Array( [ 0.5, 2.0, 1.0 ] );
+var V = new Float64Array( [ 1.0, 4.0, 2.0, 5.0, 3.0, 6.0 ] );
+
+/*
+    V = [
+        [ 1.0, 4.0 ],
+        [ 2.0, 5.0 ],
+        [ 3.0, 6.0 ]
+    ]
+*/
+
+var out = dgebak.ndarray( 'scale', 'right', 3, 2, 0, 2, scale, 1, 0, V, 2, 1, 0 );
+// returns <Float64Array>[ 0.5, 2.0, 4.0, 10.0, 3.0, 6.0 ]
+```
+
+The function has the following additional parameters:
+
+-   **ss**: stride of `scale`.
+-   **os**: starting index of `scale`.
+-   **sv1**: stride of the first dimension of `V`.
+-   **sv2**: stride of the second dimension of `V`.
+-   **ov**: starting index of `V`.
+
+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 scale = new Float64Array( [ 0.0, 0.5, 2.0, 1.0 ] );
+var V = new Float64Array( [ 0.0, 1.0, 4.0, 2.0, 5.0, 3.0, 6.0 ] );
+
+/*
+    V = [
+        [ 1.0, 4.0 ],
+        [ 2.0, 5.0 ],
+        [ 3.0, 6.0 ]
+    ]
+*/
+
+var out = dgebak.ndarray( 'scale', 'right', 3, 2, 0, 2, scale, 1, 1, V, 2, 1, 1 );
+// returns <Float64Array>[ 0.0, 0.5, 2.0, 4.0, 10.0, 3.0, 6.0 ]
+```
+
+</section>
+
+<!-- /.usage -->
+
+<section class="notes">
+
+## Notes
+
+-   `dgebak()` corresponds to the [LAPACK][LAPACK] function [`dgebak`][lapack-dgebak].
+
+</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 dgebak = require( '@stdlib/lapack/base/dgebak' );
+
+// Specify matrix meta data:
+var shape = [ 3, 2 ];
+var strides = [ 2, 1 ];
+var offset = 0;
+var order = 'row-major';
+
+// Create a matrix stored in linear memory:
+var V = new Float64Array( [ 1.0, 4.0, 2.0, 5.0, 3.0, 6.0 ] );
+console.log( ndarray2array( V, shape, strides, offset, order ) );
+
+var scale = new Float64Array( [ 0.5, 2.0, 1.0 ] );
+
+var out = dgebak( order, 'scale', 'right', shape[ 0 ], shape[ 1 ], 0, 2, scale, V, strides[ 0 ] );
+console.log( ndarray2array( out, shape, strides, offset, 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-dgebak]: https://www.netlib.org/lapack/explore-html-3.6.1/dd/d9a/group__double_g_ecomputational_gaa16ad53e2b6d3f6310b51fb2756db98d.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 -->