lapack-base-dgttrf

Compute an `LU` factorization of a real tridiagonal matrix `A` using elimination with partial pivoting and row interchanges

https://github.com/stdlib-js/lapack-base-dgttrf

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algebra array cholesky dgttrf double factorization float64 float64array javascript lapack linear math mathematics ndarray node node-js nodejs stdlib subroutines

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Compute an `LU` factorization of a real tridiagonal matrix `A` using elimination with partial pivoting and row interchanges

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algebra array cholesky dgttrf double factorization float64 float64array javascript lapack linear math mathematics ndarray node node-js nodejs stdlib subroutines

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dgttrf

Compute an LU factorization of a real tridiagonal matrix A using elimination with partial pivoting and row interchanges.

The `dgttrf` routine computes an LU factorization of a real N-by-N tridiagonal matrix `A` using elimination with partial pivoting and row interchanges. The factorization has the form: ```math A = L U ``` where `L` is a product of permutation and unit lower bidiagonal matrices and `U` is upper triangular with non-zeros in only the main diagonal and first two superdiagonals. For a 5-by-5 tridiagonal matrix `A`, elements are stored in three arrays: ```math A = \left[ \begin{array}{rrrrr} d_1 & du_1 & 0 & 0 & 0 \\ dl_1 & d_2 & du_2 & 0 & 0 \\ 0 & dl_2 & d_3 & du_3 & 0 \\ 0 & 0 & dl_3 & d_4 & du_4 \\ 0 & 0 & 0 & dl_4 & d_5 \end{array} \right] ``` where: - `dl` contains the subdiagonal elements. - `d` contains the diagonal elements. - `du` contains the superdiagonal elements. After factorization, the elements of `L` and `U` are used to update the input arrays, where: - `dl` contains the multipliers that define unit lower bidiagonal matrix `L`. - `d` contains the diagonal elements of `U`. - `du` and `du2` contain the elements of `U` on the first and second superdiagonals. The resulting `L` and `U` matrices have the following structure: ```math L = \left[ \begin{array}{rrrrr} 1 & 0 & 0 & 0 & 0 \\ l_1 & 1 & 0 & 0 & 0 \\ 0 & l_2 & 1 & 0 & 0 \\ 0 & 0 & l_3 & 1 & 0 \\ 0 & 0 & 0 & l_4 & 1 \end{array} \right] ``` ```math U = \left[ \begin{array}{rrrrr} u_{1,1} & u_{1,2} & u_{1,3} & 0 & 0 \\ 0 & u_{2,2} & u_{2,3} & u_{2,4} & 0 \\ 0 & 0 & u_{3,3} & u_{3,4} & u_{3,5} \\ 0 & 0 & 0 & u_{4,4} & u_{4,5} \\ 0 & 0 & 0 & 0 & u_{5,5} \end{array} \right] ``` where the `l(i)` values are stored in `DL`, the diagonal elements `u(i,i)` are stored in `D`, and the superdiagonal elements `u(i,i+1)` and `u(i,i+2)` are stored in `DU` and `DU2`, respectively.

## Installation ```bash npm install @stdlib/lapack-base-dgttrf ``` Alternatively, - To load the package in a website via a `script` tag without installation and bundlers, use the [ES Module][es-module] available on the [`esm`][esm-url] branch (see [README][esm-readme]). - If you are using Deno, visit the [`deno`][deno-url] branch (see [README][deno-readme] for usage intructions). - For use in Observable, or in browser/node environments, use the [Universal Module Definition (UMD)][umd] build available on the [`umd`][umd-url] branch (see [README][umd-readme]). The [branches.md][branches-url] file summarizes the available branches and displays a diagram illustrating their relationships. To view installation and usage instructions specific to each branch build, be sure to explicitly navigate to the respective README files on each branch, as linked to above.

## Usage ```javascript var dgttrf = require( '@stdlib/lapack-base-dgttrf' ); ``` #### dgttrf( N, DL, D, DU, DU2, IPIV ) Computes an `LU` factorization of a real tridiagonal matrix `A` using elimination with partial pivoting and row interchanges. ```javascript var Float64Array = require( '@stdlib/array-float64' ); var Int32Array = require( '@stdlib/array-int32' ); var dgttrf = require( '@stdlib/lapack-base-dgttrf' ); var DL = new Float64Array( [ 6.0, 6.0 ] ); var D = new Float64Array( [ 20.0, 30.0, 10.0 ] ); var DU = new Float64Array( [ 8.0, 8.0 ] ); var DU2 = new Float64Array( [ 0.0 ] ); var IPIV = new Int32Array( [ 0, 0, 0 ] ); /* A = [ [ 20.0, 8.0, 0.0 ], [ 6.0, 30.0, 8.0 ], [ 0.0, 6.0, 10.0 ] ] */ dgttrf( 3, DL, D, DU, DU2, IPIV ); // DL => [ 0.3, ~0.22 ] // D => [ 20.0, 27.6, ~8.26 ] // DU => [ 8.0, 8.0 ] // DU2 => [ 0.0 ] // IPIV => [ 0, 1, 2 ] ``` The function has the following parameters: - **N**: number of rows/columns in `A`. - **DL**: the first sub diagonal of `A` as a [`Float64Array`][mdn-float64array]. Should have `N-1` indexed elements. `DL` is overwritten by the multipliers that define the matrix `L` from the `LU` factorization of `A`. - **D**: the diagonal of `A` as a [`Float64Array`][mdn-float64array]. Should have `N` indexed elements. `D` is overwritten by the diagonal elements of the upper triangular matrix `U` from the `LU` factorization of `A`. - **DU**: the first super-diagonal of `A` as a [`Float64Array`][mdn-float64array]. Should have `N-1` indexed elements. `DU` is overwritten by the elements of the first super-diagonal of `U`. - **DU2**: the second super-diagonal of `U` as a [`Float64Array`][mdn-float64array]. Should have `N-2` indexed elements. `DU2` is overwritten by the elements of the second super-diagonal of `U` as a [`Float64Array`][mdn-float64array]. - **IPIV**: vector of pivot indices as a [`Int32Array`][mdn-int32array]. Should have `N` indexed elements. Note that indexing is relative to the first index. To introduce an offset, use [`typed array`][mdn-typed-array] views. ```javascript var Float64Array = require( '@stdlib/array-float64' ); var Int32Array = require( '@stdlib/array-int32' ); var dgttrf = require( '@stdlib/lapack-base-dgttrf' ); // Initial arrays... var DL0 = new Float64Array( [ 0.0, 6.0, 6.0 ] ); var D0 = new Float64Array( [ 0.0, 20.0, 30.0, 10.0 ] ); var DU0 = new Float64Array( [ 0.0, 8.0, 8.0 ] ); var DU20 = new Float64Array( [ 0.0, 0.0 ] ); var IPIV0 = new Int32Array( [ 0, 0, 0, 0 ] ); /* A = [ [ 20.0, 8.0, 0.0 ], [ 6.0, 30.0, 8.0 ], [ 0.0, 6.0, 10.0 ] ] */ // Create offset views... var DL = new Float64Array( DL0.buffer, DL0.BYTES_PER_ELEMENT*1 ); // start at 2nd element var D = new Float64Array( D0.buffer, D0.BYTES_PER_ELEMENT*1 ); // start at 2nd element var DU = new Float64Array( DU0.buffer, DU0.BYTES_PER_ELEMENT*1 ); // start at 2nd element var DU2 = new Float64Array( DU20.buffer, DU20.BYTES_PER_ELEMENT*1 ); // start at 2nd element var IPIV = new Int32Array( IPIV0.buffer, IPIV0.BYTES_PER_ELEMENT*1 ); // start at 2nd element dgttrf( 3, DL, D, DU, DU2, IPIV ); // DL0 => [ 0.0, 0.3, ~0.22 ] // D0 => [ 0.0, 20.0, 27.6, ~8.26 ] // DU0 => [ 0.0, 8.0, 8.0 ] // DU20 => [ 0.0, 0.0 ] // IPIV0 => [ 0, 0, 1, 2 ] ``` #### dgttrf.ndarray( N, DL, sdl, odl, D, sd, od, DU, sdu, odu, DU2, sdu2, odu2, IPIV, si, oi ) Computes an `LU` factorization of a real tridiagonal matrix `A` using elimination with partial pivoting and row interchanges and alternative indexing semantics. ```javascript var Float64Array = require( '@stdlib/array-float64' ); var Int32Array = require( '@stdlib/array-int32' ); var dgttrf = require( '@stdlib/lapack-base-dgttrf' ); var DL = new Float64Array( [ 6.0, 6.0 ] ); var D = new Float64Array( [ 20.0, 30.0, 10.0 ] ); var DU = new Float64Array( [ 8.0, 8.0 ] ); var DU2 = new Float64Array( [ 0.0 ] ); var IPIV = new Int32Array( [ 0, 0, 0 ] ); /* A = [ [ 20.0, 8.0, 0.0 ], [ 6.0, 30.0, 8.0 ], [ 0.0, 6.0, 10.0 ] ] */ dgttrf.ndarray( 3, DL, 1, 0, D, 1, 0, DU, 1, 0, DU2, 1, 0, IPIV, 1, 0 ); // DL => [ 0.3, ~0.22 ] // D => [ 20.0, 27.6, ~8.26 ] // DU => [ 8.0, 8.0 ] // DU2 => [ 0.0 ] // IPIV => [ 0, 1, 2 ] ``` The function has the following additional parameters: - **sdl**: stride length for `DL`. - **odl**: starting index for `DL`. - **sd**: stride length for `D`. - **od**: starting index for `D`. - **sdu**: stride length for `DU`. - **odu**: starting index for `DU`. - **sdu2**: stride length for `DU2`. - **odu2**: starting index for `DU2`. - **si**: stride length for `IPIV`. - **oi**: starting index for `IPIV`. 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, ```javascript var Float64Array = require( '@stdlib/array-float64' ); var Int32Array = require( '@stdlib/array-int32' ); var dgttrf = require( '@stdlib/lapack-base-dgttrf' ); var DL = new Float64Array( [ 0.0, 6.0, 6.0 ] ); var D = new Float64Array( [ 0.0, 20.0, 30.0, 10.0 ] ); var DU = new Float64Array( [ 0.0, 8.0, 8.0 ] ); var DU2 = new Float64Array( [ 0.0, 0.0 ] ); var IPIV = new Int32Array( [ 0, 0, 0, 0 ] ); /* A = [ [ 20.0, 8.0, 0.0 ], [ 6.0, 30.0, 8.0 ], [ 0.0, 6.0, 10.0 ] ] */ dgttrf.ndarray( 3, DL, 1, 1, D, 1, 1, DU, 1, 1, DU2, 1, 1, IPIV, 1, 1 ); // DL => [ 0.0, 0.3, ~0.22 ] // D => [ 0.0, 20.0, 27.6, ~8.26 ] // DU => [ 0.0, 8.0, 8.0 ] // DU2 => [ 0.0, 0.0 ] // IPIV => [ 0, 0, 1, 2 ] ```

## Notes - Both functions mutate the input arrays `DL`, `D`, `DU`, `DU2`, and `IPIV`. - Both functions return a status code indicating success or failure. The status code indicates the following conditions: - `0`: factorization was successful. - `>0`: `U(k, k)` is exactly zero the factorization has been completed, but the factor `U` is exactly singular, and division by zero will occur if it is used to solve a system of equations where `k` equals the status code value. - `dgttrf()` corresponds to the [LAPACK][LAPACK] routine [`dgttrf`][lapack-dgttrf].

## Examples ```javascript var Int32Array = require( '@stdlib/array-int32' ); var Float64Array = require( '@stdlib/array-float64' ); var dgttrf = require( '@stdlib/lapack-base-dgttrf' ); var N = 9; var DL = new Float64Array( [ 3.0, 3.0, 3.0, 3.0, 3.0, 3.0, 3.0, 3.0 ] ); var D = new Float64Array( [ 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0 ] ); var DU = new Float64Array( [ 4.0, 4.0, 4.0, 4.0, 4.0, 4.0, 4.0, 4.0 ] ); var DU2 = new Float64Array( N-2 ); var IPIV = new Int32Array( N ); /* A = [ [ 1.0, 4.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0 ], [ 3.0, 1.0, 4.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0 ], [ 0.0, 3.0, 1.0, 4.0, 0.0, 0.0, 0.0, 0.0, 0.0 ], [ 0.0, 0.0, 3.0, 1.0, 4.0, 0.0, 0.0, 0.0, 0.0 ], [ 0.0, 0.0, 0.0, 3.0, 1.0, 4.0, 0.0, 0.0, 0.0 ], [ 0.0, 0.0, 0.0, 0.0, 3.0, 1.0, 4.0, 0.0, 0.0 ], [ 0.0, 0.0, 0.0, 0.0, 0.0, 3.0, 1.0, 4.0, 0.0 ], [ 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 3.0, 1.0, 4.0 ], [ 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 3.0, 1.0 ], ] */ // Perform the `A = LU` factorization: var info = dgttrf( N, DL, D, DU, DU2, IPIV ); console.log( DL ); console.log( D ); console.log( DU ); console.log( DU2 ); console.log( IPIV ); console.log( info ); ```

## C APIs

### Usage ```c TODO ``` #### TODO TODO. ```c TODO ``` TODO ```c TODO ```

### Examples ```c TODO ```

* * * ## Notice This package is part of [stdlib][stdlib], a standard library for JavaScript and Node.js, with an emphasis on numerical and scientific computing. The library provides a collection of robust, high performance libraries for mathematics, statistics, streams, utilities, and more. For more information on the project, filing bug reports and feature requests, and guidance on how to develop [stdlib][stdlib], see the main project [repository][stdlib]. #### Community [![Chat][chat-image]][chat-url] --- ## License See [LICENSE][stdlib-license]. ## Copyright Copyright © 2016-2025. The Stdlib [Authors][stdlib-authors].

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