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:

$$A = L U$$

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.

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 );

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dgttrf

Usage

dgttrf( N, DL, D, DU, DU2, IPIV )

dgttrf.ndarray( N, DL, sdl, odl, D, sd, od, DU, sdu, odu, DU2, sdu2, odu2, IPIV, si, oi )

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Usage

dgttrf( N, DL, D, DU, DU2, IPIV )

dgttrf.ndarray( N, DL, sdl, odl, D, sd, od, DU, sdu, odu, DU2, sdu2, odu2, IPIV, si, oi )

Notes

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