SRC/slagtf.f(3)

Hey bestie 💖 welcome to the SRC/slagtf.f page from the Library Functions Manual — it’s giving “tridiagonal LU factorization but make it adorable.” Let’s get into it. ✨

🎀 NAME

SRC/slagtf.f

✨ SYNOPSIS

🌸 Functions/Subroutines

subroutine slagtf (n, a, lambda, b, c, tol, d, in, info)

SLAGTF 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.

💅 Function/Subroutine Documentation

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)

SLAGTF 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.

🦋 Purpose

Okay so here’s the main character moment: 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. 🌈

🌈 Parameters

• N — N is INTEGER. It’s the order of the matrix T. 💫
• A — 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.
• LAMBDA — LAMBDA is REAL. On entry, the scalar lambda. 🎀
• B — 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.
• C — 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.
• TOL — 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 picked 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. 🫶
• D — 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.
• IN — 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.

• INFO — INFO is INTEGER. = 0: successful exit. < 0: if INFO = -k, the kth argument had an illegal value.

👑 Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 155 of file slagtf.f.

🫶 Author

Generated automatically by Doxygen for LAPACK from the source code.

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