Computes the LU factorization of a general m-by-n band matrix.
Syntax
FORTRAN 77:
call sgbtrf( m, n, kl, ku, ab, ldab, ipiv, info )
call dgbtrf( m, n, kl, ku, ab, ldab, ipiv, info )
call cgbtrf( m, n, kl, ku, ab, ldab, ipiv, info )
call zgbtrf( m, n, kl, ku, ab, ldab, ipiv, info )
FORTRAN 95:
call gbtrf( ab [,kl] [,m] [,ipiv] [,info] )
C:
lapack_int LAPACKE_<?>gbtrf( int matrix_order, lapack_int m, lapack_int n, lapack_int kl, lapack_int ku, <datatype>* ab, lapack_int ldab, lapack_int* ipiv );
Description
The routine forms the LU factorization of a general m-by-n band matrix A with kl non-zero subdiagonals and ku non-zero superdiagonals, that is,
A = P*L*U,
where P is a permutation matrix; L is lower triangular with unit diagonal elements and at most kl non-zero elements in each column; U is an upper triangular band matrix with kl + ku superdiagonals. The routine uses partial pivoting, with row interchanges (which creates the additional kl superdiagonals in U).
Note
This routine supports the Progress Routine feature. See Progress Function for details.
Input Parameters
The data types are given for the Fortran interface. A <datatype> placeholder, if present, is used for the C interface data types in the C interface section above. See C Interface Conventions for the C interface principal conventions and type definitions.
m |
INTEGER. The number of rows in matrix A; m ≥ 0. |
n |
INTEGER. The number of columns in matrix A; n ≥ 0. |
kl |
INTEGER. The number of subdiagonals within the band of A; kl ≥ 0. |
ku |
INTEGER. The number of superdiagonals within the band of A; ku ≥ 0. |
ab |
REAL for sgbtrf DOUBLE PRECISION for dgbtrf COMPLEX for cgbtrf DOUBLE COMPLEX for zgbtrf. Array, DIMENSION (ldab,*). The array ab contains the matrix A in band storage, in rows kl + 1 to 2*kl + ku + 1; rows 1 to kl of the array need not be set. The j-th column of A is stored in the j-th column of the array ab as follows: ab(kl + ku + 1 + i - j, j) = a(i,j) for max(1,j-ku) ≤ i ≤ min(m,j+kl). |
ldab |
INTEGER. The leading dimension of the array ab. (ldab ≥ 2*kl + ku + 1) |
Output Parameters
ab |
Overwritten by L and U. U is stored as an upper triangular band matrix with kl + ku superdiagonals in rows 1 to kl + ku + 1, and the multipliers used during the factorization are stored in rows kl + ku + 2 to 2*kl + ku + 1. See Application Notes below for further details. |
ipiv |
INTEGER. Array, DIMENSION at least max(1,min(m, n)). The pivot indices; for 1 ≤ i ≤ min(m, n) , row i was interchanged with row ipiv(i). . |
info |
INTEGER. If info = 0, the execution is successful. If info = -i, the i-th parameter had an illegal value. If info = i, uii is 0. The factorization has been completed, but U is exactly singular. Division by 0 will occur if you use the factor U for solving a system of linear equations. |
Fortran 95 Interface Notes
Routines in Fortran 95 interface have fewer arguments in the calling sequence than their FORTRAN 77 counterparts. For general conventions applied to skip redundant or reconstructible arguments, see Fortran 95 Interface Conventions.
Specific details for the routine gbtrf interface are as follows:
ab |
Holds the array A of size (2*kl+ku+1,n). |
ipiv |
Holds the vector of length min(m,n). |
kl |
If omitted, assumed kl = ku. |
ku |
Restored as ku = lda-2*kl-1. |
m |
If omitted, assumed m = n. |
Application Notes
The computed L and U are the exact factors of a perturbed matrix A + E, where
|E| ≤ c(kl+ku+1) ε P|L||U|
c(k) is a modest linear function of k, and ε is the machine precision.
The total number of floating-point operations for real flavors varies between approximately 2n(ku+1)kl and 2n(kl+ku+1)kl. The number of operations for complex flavors is four times greater. All these estimates assume that kl and ku are much less than min(m,n).
The band storage scheme is illustrated by the following example, when m = n = 6, kl = 2, ku = 1:
Array elements marked * are not used by the routine; elements marked + need not be set on entry, but are required by the routine to store elements ofU because of fill-in resulting from the row interchanges.
After calling this routine with m = n, you can call the following routines:
to solve A*X = B or AT*X = B or AH*X = B |
|
to estimate the condition number of A. |