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dgttrf

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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:

$$A = L U$$

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:

$$A = \left[\begin{array}{rrrrr}d_1 & du_1 & 0 & 0 & 0 \\dl_1 & d_2 & du_2 & 0 & 0 \\0 & dl_2 & d_3 & du_3 & 0 \\0 & 0 & dl_3 & d_4 & du_4 \\0 & 0 & 0 & dl_4 & d_5\end{array}\right]$$

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:

$$L = \left[\begin{array}{rrrrr}1 & 0 & 0 & 0 & 0 \\l_1 & 1 & 0 & 0 & 0 \\0 & l_2 & 1 & 0 & 0 \\0 & 0 & l_3 & 1 & 0 \\0 & 0 & 0 & l_4 & 1\end{array}\right]$$ $$U = \left[\begin{array}{rrrrr}u_{1,1} & u_{1,2} & u_{1,3} & 0 & 0 \\0 & u_{2,2} & u_{2,3} & u_{2,4} & 0 \\0 & 0 & u_{3,3} & u_{3,4} & u_{3,5} \\0 & 0 & 0 & u_{4,4} & u_{4,5} \\0 & 0 & 0 & 0 & u_{5,5}\end{array}\right]$$

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

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 available on the esm branch (see README).
  • If you are using Deno, visit the deno branch (see README for usage intructions).
  • For use in Observable, or in browser/node environments, use the Universal Module Definition (UMD) build available on the umd branch (see README).

The branches.md 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

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.

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 => <Float64Array>[ 0.3, ~0.22 ]
// D => <Float64Array>[ 20.0, 27.6, ~8.26 ]
// DU => <Float64Array>[ 8.0, 8.0 ]
// DU2 => <Float64Array>[ 0.0 ]
// IPIV => <Int32Array>[ 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. 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. 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. 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. Should have N-2 indexed elements. DU2 is overwritten by the elements of the second super-diagonal of U as a Float64Array.
  • IPIV: vector of pivot indices as a Int32Array. Should have N indexed elements.

Note that indexing is relative to the first index. To introduce an offset, use typed array views.

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 => <Float64Array>[ 0.0, 0.3, ~0.22 ]
// D0 => <Float64Array>[ 0.0, 20.0, 27.6, ~8.26 ]
// DU0 => <Float64Array>[ 0.0, 8.0, 8.0 ]
// DU20 => <Float64Array>[ 0.0, 0.0 ]
// IPIV0 => <Int32Array>[ 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.

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 => <Float64Array>[ 0.3, ~0.22 ]
// D => <Float64Array>[ 20.0, 27.6, ~8.26 ]
// DU => <Float64Array>[ 8.0, 8.0 ]
// DU2 => <Float64Array>[ 0.0 ]
// IPIV => <Int32Array>[ 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 views mandate a view offset based on the underlying buffer, the offset parameters support indexing semantics based on starting indices. For example,

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 => <Float64Array>[ 0.0, 0.3, ~0.22 ]
// D => <Float64Array>[ 0.0, 20.0, 27.6, ~8.26 ]
// DU => <Float64Array>[ 0.0, 8.0, 8.0 ]
// DU2 => <Float64Array>[ 0.0, 0.0 ]
// IPIV => <Int32Array>[ 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 routine dgttrf.

Examples

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

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This package is part of 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, see the main project repository.

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Copyright © 2016-2026. The Stdlib Authors.

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

github.com/stdlib-js/stdlib

Topics

algebraarraycholeskydgttrfdoublefactorizationfloat64float64arrayjavascriptlapacklinearmathmathematicsndarraynodenode-jsnodejsstdlibsubroutines

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