feat: add lapack/base/dlaqr1

lib/node_modules/@stdlib/lapack/base/dlaqr1/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.
+
+-->
+
+# dlaqr1
+
+> Compute the scalar multiple of the first column of `K` where `K = (H - Z1)*(H - Z2)` for a 2-by-2 or a 3-by-3 matrix `H` and where `Z1 = z1*I`, `Z2 = z2*I`, `z1 = a + bi`, `z2 = c + di`, and `I` is the identity matrix.
+
+<section class="intro">
+
+The `dlaqr1` routine computes a scalar multiple of the first column of a matrix `K` defined by the product of two shifted matrices. Given a matrix `H` and two complex shifts (which may be conjugates), the routine constructs:
+
+<!-- <equation class="equation" label="eq:matrix_k" align="center" raw="K = (H - (sr_1 + i \cdot si_1) \cdot I) \cdot (H - (sr_2 + i \cdot si_2) \cdot I)" alt="Definition of matrix K."> -->
+
+```math
+K = (H - (sr_1 + i \cdot si_1) \cdot I) \cdot (H - (sr_2 + i \cdot si_2) \cdot I)
+```
+
+<!-- </equation> -->
+
+where:
+
+-   `H` is an N-by-N matrix (N = 2 or 3)
+-   `sr1, si1` represent the real and imaginary parts of the first shift
+-   `sr2, si2` represent the real and imaginary parts of the second shift
+-   `I` is the identity matrix
+
+The routine extracts the first column of `K` and stores it as a scalar multiple in the output vector `V`. For a 2-by-2 matrix `H`:
+
+<!-- <equation class="equation" label="eq:matrix_h_2x2" align="center" raw="H = \left[\begin{array}{cc}h_{11} & h_{12} \\h_{21} & h_{22}\end{array}\right]" alt="2-by-2 matrix H."> -->
+
+```math
+H = \left[
+\begin{array}{cc}
+h_{11} & h_{12} \\
+h_{21} & h_{22}
+\end{array}
+\right]
+```
+
+<!-- </equation> -->
+
+For a 3-by-3 matrix `H`:
+
+<!-- <equation class="equation" label="eq:matrix_h_3x3" align="center" raw="H = \left[\begin{array}{ccc}h_{11} & h_{12} & h_{13} \\h_{21} & h_{22} & h_{23} \\h_{31} & h_{32} & h_{33}\end{array}\right]" alt="3-by-3 matrix H."> -->
+
+```math
+H = \left[
+\begin{array}{ccc}
+h_{11} & h_{12} & h_{13} \\
+h_{21} & h_{22} & h_{23} \\
+h_{31} & h_{32} & h_{33}
+\end{array}
+\right]
+```
+
+<!-- </equation> -->
+
+The shifts are typically complex conjugate pairs, meaning either:
+
+1.  `sr1 = sr2` and `si1 = -si2` (conjugate complex shifts), or
+2.  `si1 = si2 = 0` (real shifts)
+
+This routine is particularly useful in the QR algorithm for starting double implicit shift bulges, where the computed vector `V` defines the Householder reflector for the implicit shift step.
+
+</section>
+
+<!-- /.intro -->
+
+<section class="usage">
+
+## Usage
+
+```javascript
+var dlaqr1 = require( '@stdlib/lapack/base/dlaqr1' );
+```
+
+#### dlaqr1( order, N, H, LDH, sr1, si1, sr2, si2, V )
+
+Given a 2-by-2 or a 3-by-3 matrix `H`, this function sets `V` to a scalar multiple of the first column of `K` where `K = (H - (sr1 + i*si1)*I)*(H - (sr2 + i*si2)*I)`.
+
+```javascript
+var Float64Array = require( '@stdlib/array/float64' );
+
+var H = new Float64Array( [ 1.0, 3.0, 2.0, 2.0, 4.0, 6.0, 0.0, 5.0, 7.0 ] ); // => [ [ 1.0, 3.0, 2.0 ], [ 2.0, 4.0, 6.0 ], [ 0.0, 5.0, 7.0 ] ]
+var V = new Float64Array( 3 );
+
+var out = dlaqr1( 'row-major', 3, H, 3, 1.5, 0.0, 2.5, 0.0, V );
+// returns <Float64Array>[ ~1.93, ~0.57, ~2.86 ]
+```
+
+The function has the following parameters:
+
+-   **order**: storage layout.
+-   **N**: number of row/columns in `H`.
+-   **H**: input matrix stored in linear memory as a [`Float64Array`][mdn-float64array].
+-   **LDH**: stride of the first dimension of `H` (a.k.a., leading dimension of the matrix `H`).
+-   **sr1**: real part of the first conjugate complex shift.
+-   **si1**: imaginary part of the first conjugate complex shift.
+-   **sr2**: real part of the second conjugate complex shift.
+-   **si2**: imaginary part of the second conjugate complex shift.
+-   **V**: output [`Float64Array`][mdn-float64array].
+
+Note that indexing is relative to the first index. To introduce an offset, use [`typed array`][mdn-typed-array] views.
+
+<!-- eslint-disable max-len -->
+
+```javascript
+var Float64Array = require( '@stdlib/array/float64' );
+
+// Initial arrays...
+var H = new Float64Array( [ 0.0, 1.0, 3.0, 2.0, 2.0, 4.0, 6.0, 0.0, 5.0, 7.0 ] ); // => [ [ 1.0, 3.0, 2.0 ], [ 2.0, 4.0, 6.0 ], [ 0.0, 5.0, 7.0 ] ]
+var V = new Float64Array( 4 );
+
+// Create offset views...
+var H1 = new Float64Array( H.buffer, H.BYTES_PER_ELEMENT*1 ); // start at 2nd element
+var V1 = new Float64Array( V.buffer, V.BYTES_PER_ELEMENT*1 ); // start at 2nd element
+
+dlaqr1( 'row-major', 3, H1, 3, 1.5, 0.0, 2.5, 0.0, V1 );
+// V => <Float64Array>[ 0.0, ~1.93, ~0.57, ~2.86 ]
+```
+
+#### dlaqr1.ndarray( N, H, sh1, sh2, oh, sr1, si1, sr2, si2, V, sv, ov )
+
+Given a 2-by-2 or a 3-by-3 matrix `H`, this function sets `V` to a scalar multiple of the first column of `K` where `K = (H - (sr1 + i*si1)*I)*(H - (sr2 + i*si2)*I)` using alternative indexing semantics.
+
+```javascript
+var Float64Array = require( '@stdlib/array/float64' );
+
+var H = new Float64Array( [ 1.0, 3.0, 2.0, 2.0, 4.0, 6.0, 0.0, 5.0, 7.0 ] ); // => [ [ 1.0, 3.0, 2.0 ], [ 2.0, 4.0, 6.0 ], [ 0.0, 5.0, 7.0 ] ]
+var V = new Float64Array( 3 );
+
+var out = dlaqr1.ndarray( 3, H, 3, 1, 0, 1.5, 0.0, 2.5, 0.0, V, 1, 0 );
+// returns <Float64Array>[ ~1.93, ~0.57, ~2.86 ]
+```
+
+The function has the following additional parameters:
+
+-   **N**: number of row/columns in `H`.
+-   **H**: input matrix stored in linear memory as a [`Float64Array`][mdn-float64array].
+-   **sh1**: stride of the first dimension of `H`.
+-   **sh2**: stride of the second dimension of `H`.
+-   **oh**: index offset for `H`.
+-   **sr1**: real part of the first conjugate complex shift.
+-   **si1**: imaginary part of the first conjugate complex shift.
+-   **sr2**: real part of the second conjugate complex shift.
+-   **si2**: imaginary part of the second conjugate complex shift.
+-   **V**: output [`Float64Array`][mdn-float64array].
+-   **sv**: stride length for `V`.
+-   **ov**: index offset for `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 H = new Float64Array( [ 0.0, 1.0, 3.0, 2.0, 2.0, 4.0, 6.0, 0.0, 5.0, 7.0 ] ); // => [ [ 1.0, 3.0, 2.0 ], [ 2.0, 4.0, 6.0 ], [ 0.0, 5.0, 7.0 ] ]
+var V = new Float64Array( 4 );
+
+var out = dlaqr1.ndarray( 3, H, 3, 1, 1, 1.5, 0.0, 2.5, 0.0, V, 1, 1 );
+// returns <Float64Array>[ 0.0, ~1.93, ~0.57, ~2.86 ]
+```
+
+</section>
+
+<!-- /.usage -->
+
+<section class="notes">
+
+## Notes
+
+-   It is expected that either `sr1 = sr2` and `si1 + si2 = 0` or `si1 = si2 = 0` (i.e., they represent complex conjugate values).
+-   This is useful for starting double implicit shift bulges in the QR algorithm.
+-   `V` should have at least `N` indexed elements.
+-   `dlaqr1()` corresponds to the [LAPACK][LAPACK] function [`dlaqr1`][lapack-dlaqr1].
+
+</section>
+
+<!-- /.notes -->
+
+<section class="examples">
+
+## Examples
+
+<!-- eslint no-undef: "error" -->
+
+```javascript
+var uniform = require( '@stdlib/random/array/uniform' );
+var Float64Array = require( '@stdlib/array/float64' );
+var dlaqr1 = require( '@stdlib/lapack/base/dlaqr1' );
+
+// Create a random 3x3 matrix:
+var H = uniform( 9, -10.0, 10.0, {
+    'dtype': 'float64'
+});
+
+// Create an output vector:
+var V = new Float64Array( 3 );
+
+dlaqr1( 'row-major', 3, H, 3, 1.5, 0.0, 2.5, 0.0, V );
+
+// Print the result:
+console.log( V );
+```
+
+</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-dlaqr1]: https://netlib.org/lapack/explore-html/d3/d57/group__laqr1_ga72fe4989b96418ec66d6ee18b4ac23e8.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 -->