 LAPACK  3.8.0 LAPACK: Linear Algebra PACKage

## ◆ dlaqr1()

 subroutine dlaqr1 ( integer N, double precision, dimension( ldh, * ) H, integer LDH, double precision SR1, double precision SI1, double precision SR2, double precision SI2, double precision, dimension( * ) V )

DLAQR1 sets a scalar multiple of the first column of the product of 2-by-2 or 3-by-3 matrix H and specified shifts.

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Purpose:
```      Given a 2-by-2 or 3-by-3 matrix H, DLAQR1 sets v to a
scalar multiple of the first column of the product

(*)  K = (H - (sr1 + i*si1)*I)*(H - (sr2 + i*si2)*I)

scaling to avoid overflows and most underflows. It
is assumed that either

1) sr1 = sr2 and si1 = -si2
or
2) si1 = si2 = 0.

This is useful for starting double implicit shift bulges
in the QR algorithm.```
Parameters
 [in] N ``` N is INTEGER Order of the matrix H. N must be either 2 or 3.``` [in] H ``` H is DOUBLE PRECISION array, dimension (LDH,N) The 2-by-2 or 3-by-3 matrix H in (*).``` [in] LDH ``` LDH is INTEGER The leading dimension of H as declared in the calling procedure. LDH.GE.N``` [in] SR1 ` SR1 is DOUBLE PRECISION` [in] SI1 ` SI1 is DOUBLE PRECISION` [in] SR2 ` SR2 is DOUBLE PRECISION` [in] SI2 ``` SI2 is DOUBLE PRECISION The shifts in (*).``` [out] V ``` V is DOUBLE PRECISION array, dimension (N) A scalar multiple of the first column of the matrix K in (*).```
Date
June 2017
Contributors:
Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA

Definition at line 123 of file dlaqr1.f.

123 *
124 * -- LAPACK auxiliary routine (version 3.7.1) --
125 * -- LAPACK is a software package provided by Univ. of Tennessee, --
126 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
127 * June 2017
128 *
129 * .. Scalar Arguments ..
130  DOUBLE PRECISION si1, si2, sr1, sr2
131  INTEGER ldh, n
132 * ..
133 * .. Array Arguments ..
134  DOUBLE PRECISION h( ldh, * ), v( * )
135 * ..
136 *
137 * ================================================================
138 *
139 * .. Parameters ..
140  DOUBLE PRECISION zero
141  parameter( zero = 0.0d0 )
142 * ..
143 * .. Local Scalars ..
144  DOUBLE PRECISION h21s, h31s, s
145 * ..
146 * .. Intrinsic Functions ..
147  INTRINSIC abs
148 * ..
149 * .. Executable Statements ..
150  IF( n.EQ.2 ) THEN
151  s = abs( h( 1, 1 )-sr2 ) + abs( si2 ) + abs( h( 2, 1 ) )
152  IF( s.EQ.zero ) THEN
153  v( 1 ) = zero
154  v( 2 ) = zero
155  ELSE
156  h21s = h( 2, 1 ) / s
157  v( 1 ) = h21s*h( 1, 2 ) + ( h( 1, 1 )-sr1 )*
158  \$ ( ( h( 1, 1 )-sr2 ) / s ) - si1*( si2 / s )
159  v( 2 ) = h21s*( h( 1, 1 )+h( 2, 2 )-sr1-sr2 )
160  END IF
161  ELSE
162  s = abs( h( 1, 1 )-sr2 ) + abs( si2 ) + abs( h( 2, 1 ) ) +
163  \$ abs( h( 3, 1 ) )
164  IF( s.EQ.zero ) THEN
165  v( 1 ) = zero
166  v( 2 ) = zero
167  v( 3 ) = zero
168  ELSE
169  h21s = h( 2, 1 ) / s
170  h31s = h( 3, 1 ) / s
171  v( 1 ) = ( h( 1, 1 )-sr1 )*( ( h( 1, 1 )-sr2 ) / s ) -
172  \$ si1*( si2 / s ) + h( 1, 2 )*h21s + h( 1, 3 )*h31s
173  v( 2 ) = h21s*( h( 1, 1 )+h( 2, 2 )-sr1-sr2 ) +
174  \$ h( 2, 3 )*h31s
175  v( 3 ) = h31s*( h( 1, 1 )+h( 3, 3 )-sr1-sr2 ) +
176  \$ h21s*h( 3, 2 )
177  END IF
178  END IF
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