feat: add lapack/base/dlatrs

lib/node_modules/@stdlib/lapack/base/dlatrs/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.
+
+-->
+
+# dlatrs
+
+> LAPACK routine to solve a triangular system of equations with the scale factor set to prevent overflow.
+
+<section class="intro">
+
+The `dlatrs` routine solves a triangular system of linear equations with a scaling factor to prevent overflow. It can handle either the original system or its transpose:
+
+<!-- <equation class="equation" label="eq:triangular_system_linear_equations" align="center" raw="A X = s B \quad \text{or} \quad A^{T} X = s B" alt="Triangular system of linear equations solved by `dlatrs`."> -->
+
+```math
+A X = s B \quad \text{or} \quad A^{T} X = s B
+```
+
+<!-- </equation> -->
+
+where:
+
+-   `A` is an upper or lower triangular matrix,
+-   `X` is the solution vector,
+-   `B` is the right-hand side vector,
+-   `s` is a scaling factor ( 0 < `s` < 1 ) chosen to avoid numerical overflow.
+
+The routine aims to compute a solution `x` such that the magnitude of all elements in `X` remains safely below the machine overflow threshold.
+
+If it determines that solving the unscaled system would not lead to overflow, it calls the Level 2 BLAS routine `dtrsv` for a standard triangular solve. Otherwise, `dlatrs` applies additional scaling and iterative techniques to avoid overflow and maintain numerical stability.
+
+In the presence of a singular matrix `A` (i.e., `A( j, j )` = 0 for some `j`), the scaling factor `s` is set to zero, and a non trivial solution to the homogeneous system:
+
+<!-- <equation class="equation" label="eq:homogeneous_system" align="center" raw="A X = 0" alt="Non trivial solution to homogeneous system."> -->
+
+```math
+A X = 0
+```
+
+<!-- </equation> -->
+
+is returned instead.
+
+To ensure stability during the solve, `dlatrs` relies on several internal estimates, including:
+
+-   the off diagonal column norms,
+-   the scaling factor `s`,
+-   and an estimate of the maximum element in the intermediate vector `X`.
+
+The algorithm may scale `X` at various stages using a factor `scale`, such that the final computed result satisfies:
+
+<!-- <equation class="equation" label="eq:scale_factor" align="center" raw="x \leftarrow scale \cdot x" alt="Vector `X` being scaled by a scale factor `s`."> -->
+
+```math
+x \leftarrow scale \cdot x
+```
+
+<!-- </equation> -->
+
+This ensures both correctness and numerical safety when solving the triangular system under limited-precision arithmetic.
+
+</section>
+
+<!-- /.intro -->
+
+<section class="usage">
+
+## Usage
+
+```javascript
+var dlatrs = require( '@stdlib/lapack/base/dlatrs' );
+```
+
+#### dlatrs( order, uplo, trans, diag, normin, N, A, LDA, X, CNORM )
+
+Solves a triangular system of equations with the scale factor set to prevent overflow.
+
+```javascript
+var Float64Array = require( '@stdlib/array/float64' );
+
+var A = new Float64Array( [ 2.0, 1.0, -1.0, 0.0, 3.0, 2.0, 0.0, 0.0, 4.0 ] ); // => [ [ 2.0, 1.0, -1.0 ], [ 0.0, 3.0, 2.0 ], [ 0.0, 0.0, 4.0 ] ]
+var X = new Float64Array( [ 5.0, 10.0, 20.0 ] );
+var CNORM = new Float64Array( 3 );
+
+var scale = dlatrs( 'row-major', 'upper', 'no-transpose', 'non-unit', 'no', 3, A, 3, X, CNORM );
+// returns 1.0
+// X => <Float64Array>[ 5.0, 0.0, 5.0 ]
+// CNORM => <Float64Array>[ 0.0, 1.0, 3.0 ]
+```
+
+The function has the following parameters:
+
+-   **order**: storage layout.
+-   **uplo**: specifies whether `A` is an upper or lower triangular matrix.
+-   **trans**: specifies whether `A` should be transposed, conjugate-transposed, or not transposed.
+-   **diag**: specifies whether `A` has a unit diagonal.
+-   **normin**: specifies whether `CNORM` has the column norms on entry or not, should be either `yes` or `no`.
+-   **N**: number of elements along each dimension of `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`).
+-   **X**: input vector [`Float64Array`][mdn-float64array] describing the right hand side `B`, should have `N` elements.
+-   **CNORM**: [`Float64Array`][mdn-float64array] used to store the column norms of `A`, should have `N` elements.
+
+`X` is overwritten by the solution vector on the left hand side. If `normin` is `yes` then `CNORM` is an input parameter and it should contain the off diagonal column norms of `A`, else `CNORM` is overwritten by the off diagonal column norms computed by the routine
+
+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' );
+var dlatrs = require( '@stdlib/lapack/base/dlatrs' );
+
+// Initial arrays...
+var A0 = new Float64Array( [ 9999.0, 2.0, 1.0, -1.0, 0.0, 3.0, 2.0, 0.0, 0.0, 4.0 ] ); // => [ [ 2.0, 1.0, -1.0 ], [ 0.0, 3.0, 2.0 ], [ 0.0, 0.0, 4.0 ] ]
+var X0 = new Float64Array( [ 9999.0, 5.0, 10.0, 20.0 ] );
+var CNORM0 = new Float64Array( 4 );
+
+// Create offset views...
+var A = new Float64Array( A0.buffer, A0.BYTES_PER_ELEMENT*1 ); // start at 2nd element
+var X = new Float64Array( X0.buffer, X0.BYTES_PER_ELEMENT*1 ); // start at 2nd element
+var CNORM = new Float64Array( CNORM0.buffer, CNORM0.BYTES_PER_ELEMENT*1 ); // start at 2nd element
+
+var scale = dlatrs( 'row-major', 'upper', 'no-transpose', 'non-unit', 'no', 3, A, 3, X, CNORM );
+// returns 1.0
+// X0 => <Float64Array>[ 9999.0, 5.0, 0.0, 5.0 ]
+// CNORM0 => <Float64Array>[ 0.0, 0.0, 1.0, 3.0 ]
+```
+
+<!-- lint disable maximum-heading-length -->
+
+#### dlatrs.ndarray( uplo, trans, diag, normin, N, A, sa1, sa2, oa, X, sx, ox, CNORM, sc, oc )
+
+Solves a triangular system of equations with the scale factor set to prevent overflow using alternative indexing semantics.
+
+```javascript
+var Float64Array = require( '@stdlib/array/float64' );
+
+var A = new Float64Array( [ 2.0, 1.0, -1.0, 0.0, 3.0, 2.0, 0.0, 0.0, 4.0 ] ); // => [ [ 2.0, 1.0, -1.0 ], [ 0.0, 3.0, 2.0 ], [ 0.0, 0.0, 4.0 ] ]
+var X = new Float64Array( [ 5.0, 10.0, 20.0 ] );
+var CNORM = new Float64Array( 3 );
+
+var scale = dlatrs.ndarray( 'upper', 'no-transpose', 'non-unit', 'no', 3, A, 3, 1, 0, X, 1, 0, CNORM, 1, 0 );
+// returns 1.0
+// X => <Float64Array>[ 5.0, 0.0, 5.0 ]
+// CNORM => <Float64Array>[ 0.0, 1.0, 3.0 ]
+```
+
+The function has the following additional parameters:
+
+-   **sa1**: stride of the first dimensiosn of `A`.
+-   **sa2**: stride of the second dimension of `A`.
+-   **oa**: starting index for `A`.
+-   **sx**: stride length for `X`.
+-   **ox**:  starting index for `X`.
+-   **sc**: stride length for `CNORM`.
+-   **oc**:  starting index for `CNORM`.
+
+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 A = new Float64Array( [ 9999.0, 2.0, 1.0, -1.0, 0.0, 3.0, 2.0, 0.0, 0.0, 4.0 ] ); // => [ [ 2.0, 1.0, -1.0 ], [ 0.0, 3.0, 2.0 ], [ 0.0, 0.0, 4.0 ] ]
+var X = new Float64Array( [ 9999.0, 5.0, 10.0, 20.0 ] );
+var CNORM = new Float64Array( 4 );
+
+var scale = dlatrs.ndarray( 'upper', 'no-transpose', 'non-unit', 'no', 3, A, 3, 1, 1, X, 1, 1, CNORM, 1, 1 );
+// returns 1.0
+// X => <Float64Array>[ 9999.0, 5.0, 0.0, 5.0 ]
+// CNORM => <Float64Array>[ 0.0, 0.0, 1.0, 3.0 ]
+```
+
+</section>
+
+<!-- /.usage -->
+
+<section class="notes">
+
+## Notes
+
+-   Both functions mutate the input arrays `X` and `CNORM` (if `normin` = `no`).
+-   `dlatrs()` corresponds to the [LAPACK][LAPACK] routine [`dlatrs`][lapack-dlatrs].
+
+</section>
+
+<!-- /.notes -->
+
+<section class="examples">
+
+## Examples
+
+<!-- eslint no-undef: "error" -->
+
+```javascript
+var Float64Array = require( '@stdlib/array/float64' );
+var numel = require( '@stdlib/ndarray/base/numel' );
+var uniform = require( '@stdlib/random/array/uniform' );
+var ndarray2array = require( '@stdlib/ndarray/base/to-array' );
+var dlatrs = require( '@stdlib/lapack/base/dlatrs' );
+
+// Specify matrix meta data:
+var shape = [ 3, 3 ];
+var strides = [ 3, 1 ];
+var offset = 0;
+var N = numel( shape );
+var order = 'row-major';
+
+// Create a matrix stored in linear memory:
+var A = uniform( N, 2.0, 10.0, {
+    'dtype': 'float64'
+});
+
+console.log( ndarray2array( A, shape, strides, offset, order ) );
+
+var X = uniform( shape[ 0 ], 0.0, 10.0, {
+    'dtype': 'float64'
+});
+
+var CNORM = new Float64Array( shape[ 0 ] );
+
+var scale = dlatrs( order, 'upper', 'no-transpose', 'non-unit', 'no', shape[ 0 ], A, strides[ 0 ], X, CNORM );
+console.log( ndarray2array( A, shape, strides, offset, order ) );
+console.log( 'scaling factor: ', 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-dlatrs]: https://www.netlib.org/lapack/explore-html-3.6.1/d8/dc3/dlatrs_8f_aab36439db8d657622a1d296b89d4acc0.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 -->