# larrk - Man Page

larrk: step in stemr, compute one eigval

## Synopsis

### Functions

subroutine **dlarrk** (n, iw, gl, gu, d, e2, pivmin, reltol, w, werr, info)**DLARRK** computes one eigenvalue of a symmetric tridiagonal matrix T to suitable accuracy.

subroutine **slarrk** (n, iw, gl, gu, d, e2, pivmin, reltol, w, werr, info)**SLARRK** computes one eigenvalue of a symmetric tridiagonal matrix T to suitable accuracy.

## Detailed Description

## Function Documentation

### subroutine dlarrk (integer n, integer iw, double precision gl, double precision gu, double precision, dimension( * ) d, double precision, dimension( * ) e2, double precision pivmin, double precision reltol, double precision w, double precision werr, integer info)

**DLARRK** computes one eigenvalue of a symmetric tridiagonal matrix T to suitable accuracy.

**Purpose:**

DLARRK computes one eigenvalue of a symmetric tridiagonal matrix T to suitable accuracy. This is an auxiliary code to be called from DSTEMR. To avoid overflow, the matrix must be scaled so that its largest element is no greater than overflow**(1/2) * underflow**(1/4) in absolute value, and for greatest accuracy, it should not be much smaller than that. See W. Kahan 'Accurate Eigenvalues of a Symmetric Tridiagonal Matrix', Report CS41, Computer Science Dept., Stanford University, July 21, 1966.

**Parameters***N*N is INTEGER The order of the tridiagonal matrix T. N >= 0.

*IW*IW is INTEGER The index of the eigenvalues to be returned.

*GL*GL is DOUBLE PRECISION

*GU*GU is DOUBLE PRECISION An upper and a lower bound on the eigenvalue.

*D*D is DOUBLE PRECISION array, dimension (N) The n diagonal elements of the tridiagonal matrix T.

*E2*E2 is DOUBLE PRECISION array, dimension (N-1) The (n-1) squared off-diagonal elements of the tridiagonal matrix T.

*PIVMIN*PIVMIN is DOUBLE PRECISION The minimum pivot allowed in the Sturm sequence for T.

*RELTOL*RELTOL is DOUBLE PRECISION The minimum relative width of an interval. When an interval is narrower than RELTOL times the larger (in magnitude) endpoint, then it is considered to be sufficiently small, i.e., converged. Note: this should always be at least radix*machine epsilon.

*W*W is DOUBLE PRECISION

*WERR*WERR is DOUBLE PRECISION The error bound on the corresponding eigenvalue approximation in W.

*INFO*INFO is INTEGER = 0: Eigenvalue converged = -1: Eigenvalue did NOT converge

**Internal Parameters:**

FUDGE DOUBLE PRECISION, default = 2 A 'fudge factor' to widen the Gershgorin intervals.

**Author**Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line **143** of file **dlarrk.f**.

### subroutine slarrk (integer n, integer iw, real gl, real gu, real, dimension( * ) d, real, dimension( * ) e2, real pivmin, real reltol, real w, real werr, integer info)

**SLARRK** computes one eigenvalue of a symmetric tridiagonal matrix T to suitable accuracy.

**Purpose:**

SLARRK computes one eigenvalue of a symmetric tridiagonal matrix T to suitable accuracy. This is an auxiliary code to be called from SSTEMR. To avoid overflow, the matrix must be scaled so that its largest element is no greater than overflow**(1/2) * underflow**(1/4) in absolute value, and for greatest accuracy, it should not be much smaller than that. See W. Kahan 'Accurate Eigenvalues of a Symmetric Tridiagonal Matrix', Report CS41, Computer Science Dept., Stanford University, July 21, 1966.

**Parameters***N*N is INTEGER The order of the tridiagonal matrix T. N >= 0.

*IW*IW is INTEGER The index of the eigenvalues to be returned.

*GL*GL is REAL

*GU*GU is REAL An upper and a lower bound on the eigenvalue.

*D*D is REAL array, dimension (N) The n diagonal elements of the tridiagonal matrix T.

*E2*E2 is REAL array, dimension (N-1) The (n-1) squared off-diagonal elements of the tridiagonal matrix T.

*PIVMIN*PIVMIN is REAL The minimum pivot allowed in the Sturm sequence for T.

*RELTOL*RELTOL is REAL The minimum relative width of an interval. When an interval is narrower than RELTOL times the larger (in magnitude) endpoint, then it is considered to be sufficiently small, i.e., converged. Note: this should always be at least radix*machine epsilon.

*W*W is REAL

*WERR*WERR is REAL The error bound on the corresponding eigenvalue approximation in W.

*INFO*INFO is INTEGER = 0: Eigenvalue converged = -1: Eigenvalue did NOT converge

**Internal Parameters:**

FUDGE REAL , default = 2 A 'fudge factor' to widen the Gershgorin intervals.

**Author**Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line **143** of file **slarrk.f**.

## Author

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