.TH "lagtf" 3 "Wed Feb 7 2024 11:30:40" "Version 3.12.0" "LAPACK" \" -*- nroff -*- .ad l .nh .SH NAME lagtf \- lagtf: LU factor of (T - λI) .SH SYNOPSIS .br .PP .SS "Functions" .in +1c .ti -1c .RI "subroutine \fBdlagtf\fP (n, a, lambda, b, c, tol, d, in, info)" .br .RI "\fBDLAGTF\fP computes an LU factorization of a matrix T-λI, where T is a general tridiagonal matrix, and λ a scalar, using partial pivoting with row interchanges\&. " .ti -1c .RI "subroutine \fBslagtf\fP (n, a, lambda, b, c, tol, d, in, info)" .br .RI "\fBSLAGTF\fP computes an LU factorization of a matrix T-λI, where T is a general tridiagonal matrix, and λ a scalar, using partial pivoting with row interchanges\&. " .in -1c .SH "Detailed Description" .PP .SH "Function Documentation" .PP .SS "subroutine dlagtf (integer n, double precision, dimension( * ) a, double precision lambda, double precision, dimension( * ) b, double precision, dimension( * ) c, double precision tol, double precision, dimension( * ) d, integer, dimension( * ) in, integer info)" .PP \fBDLAGTF\fP computes an LU factorization of a matrix T-λI, where T is a general tridiagonal matrix, and λ a scalar, using partial pivoting with row interchanges\&. .PP \fBPurpose:\fP .RS 4 .PP .nf DLAGTF factorizes the matrix (T - lambda*I), where T is an n by n tridiagonal matrix and lambda is a scalar, as T - lambda*I = PLU, where P is a permutation matrix, L is a unit lower tridiagonal matrix with at most one non-zero sub-diagonal elements per column and U is an upper triangular matrix with at most two non-zero super-diagonal elements per column\&. The factorization is obtained by Gaussian elimination with partial pivoting and implicit row scaling\&. The parameter LAMBDA is included in the routine so that DLAGTF may be used, in conjunction with DLAGTS, to obtain eigenvectors of T by inverse iteration\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIN\fP .PP .nf N is INTEGER The order of the matrix T\&. .fi .PP .br \fIA\fP .PP .nf A is DOUBLE PRECISION array, dimension (N) On entry, A must contain the diagonal elements of T\&. On exit, A is overwritten by the n diagonal elements of the upper triangular matrix U of the factorization of T\&. .fi .PP .br \fILAMBDA\fP .PP .nf LAMBDA is DOUBLE PRECISION On entry, the scalar lambda\&. .fi .PP .br \fIB\fP .PP .nf B is DOUBLE PRECISION array, dimension (N-1) On entry, B must contain the (n-1) super-diagonal elements of T\&. On exit, B is overwritten by the (n-1) super-diagonal elements of the matrix U of the factorization of T\&. .fi .PP .br \fIC\fP .PP .nf C is DOUBLE PRECISION array, dimension (N-1) On entry, C must contain the (n-1) sub-diagonal elements of T\&. On exit, C is overwritten by the (n-1) sub-diagonal elements of the matrix L of the factorization of T\&. .fi .PP .br \fITOL\fP .PP .nf TOL is DOUBLE PRECISION On entry, a relative tolerance used to indicate whether or not the matrix (T - lambda*I) is nearly singular\&. TOL should normally be chose as approximately the largest relative error in the elements of T\&. For example, if the elements of T are correct to about 4 significant figures, then TOL should be set to about 5*10**(-4)\&. If TOL is supplied as less than eps, where eps is the relative machine precision, then the value eps is used in place of TOL\&. .fi .PP .br \fID\fP .PP .nf D is DOUBLE PRECISION array, dimension (N-2) On exit, D is overwritten by the (n-2) second super-diagonal elements of the matrix U of the factorization of T\&. .fi .PP .br \fIIN\fP .PP .nf IN is INTEGER array, dimension (N) On exit, IN contains details of the permutation matrix P\&. If an interchange occurred at the kth step of the elimination, then IN(k) = 1, otherwise IN(k) = 0\&. The element IN(n) returns the smallest positive integer j such that abs( u(j,j) ) <= norm( (T - lambda*I)(j) )*TOL, where norm( A(j) ) denotes the sum of the absolute values of the jth row of the matrix A\&. If no such j exists then IN(n) is returned as zero\&. If IN(n) is returned as positive, then a diagonal element of U is small, indicating that (T - lambda*I) is singular or nearly singular, .fi .PP .br \fIINFO\fP .PP .nf INFO is INTEGER = 0: successful exit < 0: if INFO = -k, the kth argument had an illegal value .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP .SS "subroutine slagtf (integer n, real, dimension( * ) a, real lambda, real, dimension( * ) b, real, dimension( * ) c, real tol, real, dimension( * ) d, integer, dimension( * ) in, integer info)" .PP \fBSLAGTF\fP computes an LU factorization of a matrix T-λI, where T is a general tridiagonal matrix, and λ a scalar, using partial pivoting with row interchanges\&. .PP \fBPurpose:\fP .RS 4 .PP .nf SLAGTF factorizes the matrix (T - lambda*I), where T is an n by n tridiagonal matrix and lambda is a scalar, as T - lambda*I = PLU, where P is a permutation matrix, L is a unit lower tridiagonal matrix with at most one non-zero sub-diagonal elements per column and U is an upper triangular matrix with at most two non-zero super-diagonal elements per column\&. The factorization is obtained by Gaussian elimination with partial pivoting and implicit row scaling\&. The parameter LAMBDA is included in the routine so that SLAGTF may be used, in conjunction with SLAGTS, to obtain eigenvectors of T by inverse iteration\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIN\fP .PP .nf N is INTEGER The order of the matrix T\&. .fi .PP .br \fIA\fP .PP .nf A is REAL array, dimension (N) On entry, A must contain the diagonal elements of T\&. On exit, A is overwritten by the n diagonal elements of the upper triangular matrix U of the factorization of T\&. .fi .PP .br \fILAMBDA\fP .PP .nf LAMBDA is REAL On entry, the scalar lambda\&. .fi .PP .br \fIB\fP .PP .nf B is REAL array, dimension (N-1) On entry, B must contain the (n-1) super-diagonal elements of T\&. On exit, B is overwritten by the (n-1) super-diagonal elements of the matrix U of the factorization of T\&. .fi .PP .br \fIC\fP .PP .nf C is REAL array, dimension (N-1) On entry, C must contain the (n-1) sub-diagonal elements of T\&. On exit, C is overwritten by the (n-1) sub-diagonal elements of the matrix L of the factorization of T\&. .fi .PP .br \fITOL\fP .PP .nf TOL is REAL On entry, a relative tolerance used to indicate whether or not the matrix (T - lambda*I) is nearly singular\&. TOL should normally be chose as approximately the largest relative error in the elements of T\&. For example, if the elements of T are correct to about 4 significant figures, then TOL should be set to about 5*10**(-4)\&. If TOL is supplied as less than eps, where eps is the relative machine precision, then the value eps is used in place of TOL\&. .fi .PP .br \fID\fP .PP .nf D is REAL array, dimension (N-2) On exit, D is overwritten by the (n-2) second super-diagonal elements of the matrix U of the factorization of T\&. .fi .PP .br \fIIN\fP .PP .nf IN is INTEGER array, dimension (N) On exit, IN contains details of the permutation matrix P\&. If an interchange occurred at the kth step of the elimination, then IN(k) = 1, otherwise IN(k) = 0\&. The element IN(n) returns the smallest positive integer j such that abs( u(j,j) ) <= norm( (T - lambda*I)(j) )*TOL, where norm( A(j) ) denotes the sum of the absolute values of the jth row of the matrix A\&. If no such j exists then IN(n) is returned as zero\&. If IN(n) is returned as positive, then a diagonal element of U is small, indicating that (T - lambda*I) is singular or nearly singular, .fi .PP .br \fIINFO\fP .PP .nf INFO is INTEGER = 0: successful exit < 0: if INFO = -k, the kth argument had an illegal value .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP .SH "Author" .PP Generated automatically by Doxygen for LAPACK from the source code\&.