Compute an `LU` factorization of a real tridiagonal matrix `A` using elimination with partial pivoting and row interchanges
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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:
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:
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:
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.
``bash`
npm install @stdlib/lapack-base-dgttrf
`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 =>
// D =>
// DU =>
// DU2 =>
// IPIV =>
`
The function has the following parameters:
- N: number of rows/columns in A.A
- DL: the first sub diagonal of 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.A
- D: the diagonal of 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.A
- DU: the first super-diagonal of as a [Float64Array][mdn-float64array]. Should have N-1 indexed elements. DU is overwritten by the elements of the first super-diagonal of U.U
- DU2: the second super-diagonal of 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].Int32Array
- IPIV: vector of pivot indices as a [][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 =>
// D0 =>
// DU0 =>
// DU20 =>
// IPIV0 =>
`
#### 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 =>
// D =>
// DU =>
// DU2 =>
// IPIV =>
`
The function has the following additional parameters:
- sdl: stride length for DL.DL
- odl: starting index for .D
- sd: stride length for .D
- od: starting index for .DU
- sdu: stride length for .DU
- odu: starting index for .DU2
- sdu2: stride length for .DU2
- odu2: starting index for .IPIV
- si: stride length for .IPIV
- oi: starting index for .
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 =>
// D =>
// DU =>
// DU2 =>
// IPIV =>
`
- 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].
`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 );
`
*
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#### TODO
TODO.
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*
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[mdn-float64array]: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Float64Array
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