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‎lib/node_modules/@stdlib/lapack/base/dlaqr1/README.md

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<!--
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@license Apache-2.0
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Copyright (c) 2025 The Stdlib Authors.
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Licensed under the Apache License, Version 2.0 (the "License");
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you may not use this file except in compliance with the License.
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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
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distributed under the License is distributed on an "AS IS" BASIS,
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WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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See the License for the specific language governing permissions and
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limitations under the License.
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-->
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# dlaqr1
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> Given a 2-by-2 or a 3-by-3 matrix, this LAPACK routine sets `V` to a scalar multiple of the first column of `K` where `K = (H - (sr1 + i*si1)*I)*(H - (sr2 + i*si2)*I)`.
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<section class="usage">
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## Usage
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```javascript
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var dlaqr1 = require( '@stdlib/lapack/base/dlaqr1' );
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```
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#### dlaqr1( order, N, H, LDH, sr1, si1, sr2, si2, V )
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Given a 2-by-2 or a 3-by-3 matrix, 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)`.
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```javascript
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var Float64Array = require( '@stdlib/array/float64' );
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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 ] ]
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var V = new Float64Array( 3 );
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var out = dlaqr1( 'row-major', 3, H, 3, 1.5, 0.0, 2.5, 0.0, V );
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// returns <Float64Array>[ ~1.93, ~0.57, ~2.86 ]
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```
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The function has the following parameters:
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- **order**: storage layout.
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- **N**: number of row/columns in `H`.
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- **H**: input matrix stored in linear memory as a [`Float64Array`][mdn-float64array].
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- **LDH**: stride of the first dimension of `H` (a.k.a., leading dimension of the matrix `H`).
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- **sr1**: real part of the first conjugate complex shift.
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- **si1**: imaginary part of the first conjugate complex shift.
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- **sr2**: real part of the second conjugate complex shift.
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- **si2**: imaginary part of the second conjugate complex shift.
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- **V**: output [`Float64Array`][mdn-float64array].
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Note that indexing is relative to the first index. To introduce an offset, use [`typed array`][mdn-typed-array] views.
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<!-- eslint-disable max-len -->
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```javascript
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var Float64Array = require( '@stdlib/array/float64' );
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// Initial arrays...
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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 ] ]
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var V = new Float64Array( 4 );
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// Create offset views...
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var H1 = new Float64Array( H.buffer, H.BYTES_PER_ELEMENT*1 ); // start at 2nd element
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var V1 = new Float64Array( V.buffer, V.BYTES_PER_ELEMENT*1 ); // start at 2nd element
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dlaqr1( 'row-major', 3, H1, 3, 1.5, 0.0, 2.5, 0.0, V1 );
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// V => <Float64Array>[ 0.0, ~1.93, ~0.57, ~2.86 ]
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```
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#### dlaqr1.ndarray( N, H, sh1, sh2, oh, sr1, si1, sr2, si2, V, sv, ov )
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Given a 2-by-2 or a 3-by-3 matrix, 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.
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```javascript
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var Float64Array = require( '@stdlib/array/float64' );
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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 ] ]
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var V = new Float64Array( 3 );
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var out = dlaqr1.ndarray( 3, H, 3, 1, 0, 1.5, 0.0, 2.5, 0.0, V, 1, 0 );
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// returns <Float64Array>[ ~1.93, ~0.57, ~2.86 ]
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```
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The function has the following additional parameters:
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- **N**: number of row/columns in `H`.
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- **H**: input matrix stored in linear memory as a [`Float64Array`][mdn-float64array].
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- **sh1**: stride of the first dimension of `H`.
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- **sh2**: stride of the second dimension of `H`.
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- **oh**: index offset for `H`.
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- **sr1**: real part of the first conjugate complex shift.
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- **si1**: imaginary part of the first conjugate complex shift.
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- **sr2**: real part of the second conjugate complex shift.
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- **si2**: imaginary part of the second conjugate complex shift.
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- **V**: output [`Float64Array`][mdn-float64array].
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- **sv**: stride length for `V`.
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- **ov**: index offset for `V`.
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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,
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<!-- eslint-disable max-len -->
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```javascript
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var Float64Array = require( '@stdlib/array/float64' );
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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 ] ]
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var V = new Float64Array( 4 );
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var out = dlaqr1.ndarray( 3, H, 3, 1, 1, 1.5, 0.0, 2.5, 0.0, V, 1, 1 );
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// returns <Float64Array>[ 0.0, ~1.93, ~0.57, ~2.86 ]
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```
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</section>
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<!-- /.usage -->
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<section class="notes">
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## Notes
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- It is expected that either `sr1 = sr2` and `si1 + si2 = 0` or `si1 = si2 = 0` (i.e., they represent complex conjugate values).
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- This is useful for starting double implicit shift bulges in the QR algorithm.
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- `V` should have at least `N` indexed elements.
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- `dlaqr1()` corresponds to the [LAPACK][LAPACK] function [`dlaqr1`][lapack-dlaqr1].
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</section>
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<!-- /.notes -->
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<section class="examples">
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## Examples
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<!-- eslint no-undef: "error" -->
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```javascript
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var uniform = require( '@stdlib/random/array/uniform' );
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var Float64Array = require( '@stdlib/array/float64' );
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var dlaqr1 = require( '@stdlib/lapack/base/dlaqr1' );
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// Create a random 3x3 matrix:
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var H = uniform( 9, -10.0, 10.0, {
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'dtype': 'float64'
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});
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// Create an output vector:
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var V = new Float64Array( 3 );
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dlaqr1( 'row-major', 3, H, 3, 1.5, 0.0, 2.5, 0.0, V );
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// Print the result:
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console.log( V );
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```
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</section>
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<!-- /.examples -->
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<!-- C interface documentation. -->
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* * *
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<section class="c">
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## C APIs
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<!-- Section to include introductory text. Make sure to keep an empty line after the intro `section` element and another before the `/section` close. -->
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<section class="intro">
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</section>
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<!-- /.intro -->
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<!-- C usage documentation. -->
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<section class="usage">
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### Usage
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```c
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TODO
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```
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#### TODO
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TODO.
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```c
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TODO
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```
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TODO
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```c
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TODO
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```
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</section>
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<!-- /.usage -->
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<!-- C API usage notes. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->
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<section class="notes">
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</section>
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<!-- /.notes -->
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<!-- C API usage examples. -->
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<section class="examples">
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### Examples
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```c
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TODO
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```
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</section>
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<!-- /.examples -->
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</section>
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<!-- /.c -->
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<!-- Section for related `stdlib` packages. Do not manually edit this section, as it is automatically populated. -->
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<section class="related">
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</section>
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<!-- /.related -->
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<!-- Section for all links. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->
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<section class="links">
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[lapack]: https://www.netlib.org/lapack/explore-html/
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[lapack-dlaqr1]: https://www.netlib.org/lapack/explore-html/d1/d72/dlaqr1_8f.html
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[mdn-float64array]: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Float64Array
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[mdn-typed-array]: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/TypedArray
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</section>
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<!-- /.links -->

‎lib/node_modules/@stdlib/lapack/base/dlaqr1/docs/repl.txt

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{{alias}}( order, N, H, LDH, sr1, si1, sr2, si2, V )
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Given a 2-by-2 or a 3-by-3 matrix, this LAPACK routine sets `V` to a scalar
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multiple of the first column of `K` where:
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`K = (H - (sr1 + i*si1)*I)*(H - (sr2 + i*si2)*I)`.
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Indexing is relative to the first index. To introduce an offset, use typed
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array views.
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It is expected that either `sr1 = sr2` and `si1 + si2 = 0` or si1 = si2 = 0`
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(i.e., they represent complex conjugate values).
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This is used for starting double implicit shift bulges in the QR algorithm.
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Parameters
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----------
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order: string
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Storage layout.
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N: integer
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Number of row/columns in `H`.
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H: Float64Array
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Input matrix.
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LDH: integer
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Stride of the first dimension of `H` (a.k.a., leading dimension of the
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matrix `H`).
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sr1: number
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Real part of the first conjugate complex shift.
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si1: number
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Imaginary part of the first conjugate complex shift.
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sr2: number
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Real part of the second conjugate complex shift.
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si2: number
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Imaginary part of the second conjugate complex shift.
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V: Float64Array
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Output array.
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Returns
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-------
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V: Float64Array
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Mutated output array.
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Examples
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--------
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> var H = new {{alias:@stdlib/array/float64}}( [ 1.0, 2.0, 3.0, 4.0 ] );
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> var V = new {{alias:@stdlib/array/float64}}( 2 );
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> {{alias}}( 'row-major', 2, H, 2, 1.5, 1.0, 1.0, 2.0, V )
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<Float64Array>[ ~0.8, 1.5 ]
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{{alias}}.ndarray( N, H, sh1, sh2, oh, sr1, si1, sr2, si2, V, sv, ov )
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Given a 2-by-2 or a 3-by-3 matrix, this LAPACK routine sets `V` to a scalar
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multiple of the first column of `K` using alternative indexing semantics
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where:
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`K = (H - (sr1 + i*si1)*I)*(H - (sr2 + i*si2)*I)`.
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It is expected that either `sr1 = sr2` and `si1 + si2 = 0` or si1 = si2 = 0`
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(i.e., they represent complex conjugate values).
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This is used for starting double implicit shift bulges in the QR algorithm.
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While typed array views mandate a view offset based on the underlying
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buffer, the offset parameters support indexing semantics based on starting
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indices.
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Parameters
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----------
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N: integer
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Number of row/columns in `H`.
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H: Float64Array
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Input matrix.
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sh1: integer
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Stride of the first dimension of `H`.
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sh2: integer
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Stride of the second dimension of `H`.
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oh: integer
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Index offset for `H`.
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sr1: number
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Real part of the first conjugate complex shift.
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si1: number
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Imaginary part of the first conjugate complex shift.
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sr2: number
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Real part of the second conjugate complex shift.
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si2: number
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Imaginary part of the second conjugate complex shift.
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V: Float64Array
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Output array.
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sv: integer
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Stride length for `V`.
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ov: integer
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Index offset for `V`.
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Returns
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-------
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V: Float64Array
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Mutated output array.
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Examples
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--------
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> var H = new {{alias:@stdlib/array/float64}}( [ 1.0, 2.0, 3.0, 4.0 ] );
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> var V = new {{alias:@stdlib/array/float64}}( 2 );
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> {{alias}}.ndarray( 2, H, 2, 1, 0, 1.5, 1.0, 1.0, 2.0, V, 1, 0 )
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<Float64Array>[ ~0.8, 1.5 ]
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See Also
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--------

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