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Solving a system of equations in Hessenberg form with one offdiagonal and one right-hand side

**[Specification]**
**[Arguments]**
**[Method]**
**[References]**
**[Comments]**
**[Example]**
**Purpose**

To solve a system of equations in Hessenberg form with one
offdiagonal and one right-hand side.

**Specification**
SUBROUTINE SB04NY( RC, UL, M, A, LDA, LAMBDA, D, TOL, IWORK,
$ DWORK, LDDWOR, INFO )
C .. Scalar Arguments ..
CHARACTER RC, UL
INTEGER INFO, LDA, LDDWOR, M
DOUBLE PRECISION LAMBDA, TOL
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), D(*), DWORK(LDDWOR,*)

**Arguments**
**Mode Parameters**

RC CHARACTER*1
Indicates processing by columns or rows, as follows:
= 'R': Row transformations are applied;
= 'C': Column transformations are applied.
UL CHARACTER*1
Indicates whether AB is upper or lower Hessenberg matrix,
as follows:
= 'U': AB is upper Hessenberg;
= 'L': AB is lower Hessenberg.

**Input/Output Parameters**
M (input) INTEGER
The order of the matrix A. M >= 0.
A (input) DOUBLE PRECISION array, dimension (LDA,M)
The leading M-by-M part of this array must contain a
matrix A in Hessenberg form.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,M).
LAMBDA (input) DOUBLE PRECISION
This variable must contain the value to be added to the
diagonal elements of A.
D (input/output) DOUBLE PRECISION array, dimension (M)
On entry, this array must contain the right-hand side
vector of the Hessenberg system.
On exit, if INFO = 0, this array contains the solution
vector of the Hessenberg system.

**Tolerances**
TOL DOUBLE PRECISION
The tolerance to be used to test for near singularity of
the triangular factor R of the Hessenberg matrix. A matrix
whose estimated condition number is less than 1/TOL is
considered to be nonsingular.

**Workspace**
IWORK INTEGER array, dimension (M)
DWORK DOUBLE PRECISION array, dimension (LDDWOR,M+3)
The leading M-by-M part of this array is used for
computing the triangular factor of the QR decomposition
of the Hessenberg matrix. The remaining 3*M elements are
used as workspace for the computation of the reciprocal
condition estimate.
LDDWOR INTEGER
The leading dimension of array DWORK. LDDWOR >= MAX(1,M).

**Error Indicator**
INFO INTEGER
= 0: successful exit;
= 1: if the Hessenberg matrix is (numerically) singular.
That is, its estimated reciprocal condition number
is less than or equal to TOL.

**Numerical Aspects**
None.

**Further Comments**
None

**Example**
**Program Text**

None

**Program Data**
None

**Program Results**
None

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