--- rpl/lapack/lapack/zlaqr4.f 2011/11/21 22:19:53 1.9
+++ rpl/lapack/lapack/zlaqr4.f 2023/08/07 08:39:30 1.19
@@ -1,26 +1,26 @@
-*> \brief \b ZLAQR4
+*> \brief \b ZLAQR4 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition.
*
* =========== DOCUMENTATION ===========
*
-* Online html documentation available at
-* http://www.netlib.org/lapack/explore-html/
+* Online html documentation available at
+* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
-*> Download ZLAQR4 + dependencies
-*>
-*> [TGZ]
-*>
-*> [ZIP]
-*>
+*> Download ZLAQR4 + dependencies
+*>
+*> [TGZ]
+*>
+*> [ZIP]
+*>
*> [TXT]
-*> \endhtmlonly
+*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE ZLAQR4( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ,
* IHIZ, Z, LDZ, WORK, LWORK, INFO )
-*
+*
* .. Scalar Arguments ..
* INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, LWORK, N
* LOGICAL WANTT, WANTZ
@@ -28,7 +28,7 @@
* .. Array Arguments ..
* COMPLEX*16 H( LDH, * ), W( * ), WORK( * ), Z( LDZ, * )
* ..
-*
+*
*
*> \par Purpose:
* =============
@@ -73,7 +73,7 @@
*> \param[in] N
*> \verbatim
*> N is INTEGER
-*> The order of the matrix H. N .GE. 0.
+*> The order of the matrix H. N >= 0.
*> \endverbatim
*>
*> \param[in] ILO
@@ -85,12 +85,12 @@
*> \verbatim
*> IHI is INTEGER
*> It is assumed that H is already upper triangular in rows
-*> and columns 1:ILO-1 and IHI+1:N and, if ILO.GT.1,
+*> and columns 1:ILO-1 and IHI+1:N and, if ILO > 1,
*> H(ILO,ILO-1) is zero. ILO and IHI are normally set by a
*> previous call to ZGEBAL, and then passed to ZGEHRD when the
*> matrix output by ZGEBAL is reduced to Hessenberg form.
*> Otherwise, ILO and IHI should be set to 1 and N,
-*> respectively. If N.GT.0, then 1.LE.ILO.LE.IHI.LE.N.
+*> respectively. If N > 0, then 1 <= ILO <= IHI <= N.
*> If N = 0, then ILO = 1 and IHI = 0.
*> \endverbatim
*>
@@ -102,17 +102,17 @@
*> contains the upper triangular matrix T from the Schur
*> decomposition (the Schur form). If INFO = 0 and WANT is
*> .FALSE., then the contents of H are unspecified on exit.
-*> (The output value of H when INFO.GT.0 is given under the
+*> (The output value of H when INFO > 0 is given under the
*> description of INFO below.)
*>
-*> This subroutine may explicitly set H(i,j) = 0 for i.GT.j and
+*> This subroutine may explicitly set H(i,j) = 0 for i > j and
*> j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N.
*> \endverbatim
*>
*> \param[in] LDH
*> \verbatim
*> LDH is INTEGER
-*> The leading dimension of the array H. LDH .GE. max(1,N).
+*> The leading dimension of the array H. LDH >= max(1,N).
*> \endverbatim
*>
*> \param[out] W
@@ -134,7 +134,7 @@
*> IHIZ is INTEGER
*> Specify the rows of Z to which transformations must be
*> applied if WANTZ is .TRUE..
-*> 1 .LE. ILOZ .LE. ILO; IHI .LE. IHIZ .LE. N.
+*> 1 <= ILOZ <= ILO; IHI <= IHIZ <= N.
*> \endverbatim
*>
*> \param[in,out] Z
@@ -144,7 +144,7 @@
*> If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is
*> replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the
*> orthogonal Schur factor of H(ILO:IHI,ILO:IHI).
-*> (The output value of Z when INFO.GT.0 is given under
+*> (The output value of Z when INFO > 0 is given under
*> the description of INFO below.)
*> \endverbatim
*>
@@ -152,7 +152,7 @@
*> \verbatim
*> LDZ is INTEGER
*> The leading dimension of the array Z. if WANTZ is .TRUE.
-*> then LDZ.GE.MAX(1,IHIZ). Otherwize, LDZ.GE.1.
+*> then LDZ >= MAX(1,IHIZ). Otherwise, LDZ >= 1.
*> \endverbatim
*>
*> \param[out] WORK
@@ -165,7 +165,7 @@
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
-*> The dimension of the array WORK. LWORK .GE. max(1,N)
+*> The dimension of the array WORK. LWORK >= max(1,N)
*> is sufficient, but LWORK typically as large as 6*N may
*> be required for optimal performance. A workspace query
*> to determine the optimal workspace size is recommended.
@@ -182,18 +182,18 @@
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
-*> .GT. 0: if INFO = i, ZLAQR4 failed to compute all of
+*> > 0: if INFO = i, ZLAQR4 failed to compute all of
*> the eigenvalues. Elements 1:ilo-1 and i+1:n of WR
*> and WI contain those eigenvalues which have been
*> successfully computed. (Failures are rare.)
*>
-*> If INFO .GT. 0 and WANT is .FALSE., then on exit,
+*> If INFO > 0 and WANT is .FALSE., then on exit,
*> the remaining unconverged eigenvalues are the eigen-
*> values of the upper Hessenberg matrix rows and
*> columns ILO through INFO of the final, output
*> value of H.
*>
-*> If INFO .GT. 0 and WANTT is .TRUE., then on exit
+*> If INFO > 0 and WANTT is .TRUE., then on exit
*>
*> (*) (initial value of H)*U = U*(final value of H)
*>
@@ -201,7 +201,7 @@
*> value of H is upper Hessenberg and triangular in
*> rows and columns INFO+1 through IHI.
*>
-*> If INFO .GT. 0 and WANTZ is .TRUE., then on exit
+*> If INFO > 0 and WANTZ is .TRUE., then on exit
*>
*> (final value of Z(ILO:IHI,ILOZ:IHIZ)
*> = (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U
@@ -209,19 +209,17 @@
*> where U is the unitary matrix in (*) (regard-
*> less of the value of WANTT.)
*>
-*> If INFO .GT. 0 and WANTZ is .FALSE., then Z is not
+*> If INFO > 0 and WANTZ is .FALSE., then Z is not
*> accessed.
*> \endverbatim
*
* Authors:
* ========
*
-*> \author Univ. of Tennessee
-*> \author Univ. of California Berkeley
-*> \author Univ. of Colorado Denver
-*> \author NAG Ltd.
-*
-*> \date November 2011
+*> \author Univ. of Tennessee
+*> \author Univ. of California Berkeley
+*> \author Univ. of Colorado Denver
+*> \author NAG Ltd.
*
*> \ingroup complex16OTHERauxiliary
*
@@ -247,10 +245,9 @@
SUBROUTINE ZLAQR4( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ,
$ IHIZ, Z, LDZ, WORK, LWORK, INFO )
*
-* -- LAPACK auxiliary routine (version 3.4.0) --
+* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
-* November 2011
*
* .. Scalar Arguments ..
INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, LWORK, N
@@ -268,7 +265,7 @@
* . ZLAHQR because of insufficient subdiagonal scratch space.
* . (This is a hard limit.) ====
INTEGER NTINY
- PARAMETER ( NTINY = 11 )
+ PARAMETER ( NTINY = 15 )
*
* ==== Exceptional deflation windows: try to cure rare
* . slow convergence by varying the size of the
@@ -363,22 +360,22 @@
END IF
*
* ==== NWR = recommended deflation window size. At this
-* . point, N .GT. NTINY = 11, so there is enough
+* . point, N .GT. NTINY = 15, so there is enough
* . subdiagonal workspace for NWR.GE.2 as required.
* . (In fact, there is enough subdiagonal space for
-* . NWR.GE.3.) ====
+* . NWR.GE.4.) ====
*
NWR = ILAENV( 13, 'ZLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
NWR = MAX( 2, NWR )
NWR = MIN( IHI-ILO+1, ( N-1 ) / 3, NWR )
*
* ==== NSR = recommended number of simultaneous shifts.
-* . At this point N .GT. NTINY = 11, so there is at
+* . At this point N .GT. NTINY = 15, so there is at
* . enough subdiagonal workspace for NSR to be even
* . and greater than or equal to two as required. ====
*
NSR = ILAENV( 15, 'ZLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
- NSR = MIN( NSR, ( N+6 ) / 9, IHI-ILO )
+ NSR = MIN( NSR, ( N-3 ) / 6, IHI-ILO )
NSR = MAX( 2, NSR-MOD( NSR, 2 ) )
*
* ==== Estimate optimal workspace ====
@@ -426,7 +423,7 @@
* ==== NSMAX = the Largest number of simultaneous shifts
* . for which there is sufficient workspace. ====
*
- NSMAX = MIN( ( N+6 ) / 9, 2*LWORK / 3 )
+ NSMAX = MIN( ( N-3 ) / 6, 2*LWORK / 3 )
NSMAX = NSMAX - MOD( NSMAX, 2 )
*
* ==== NDFL: an iteration count restarted at deflation. ====
@@ -566,7 +563,7 @@
*
* ==== Got NS/2 or fewer shifts? Use ZLAHQR
* . on a trailing principal submatrix to
-* . get more. (Since NS.LE.NSMAX.LE.(N+6)/9,
+* . get more. (Since NS.LE.NSMAX.LE.(N-3)/6,
* . there is enough space below the subdiagonal
* . to fit an NS-by-NS scratch array.) ====
*
@@ -641,7 +638,7 @@
END IF
END IF
*
-* ==== Use up to NS of the the smallest magnatiude
+* ==== Use up to NS of the the smallest magnitude
* . shifts. If there aren't NS shifts available,
* . then use them all, possibly dropping one to
* . make the number of shifts even. ====
@@ -661,7 +658,7 @@
* . (NVE-by-KDU) vertical work WV arrow along
* . the left-hand-edge. ====
*
- KDU = 3*NS - 3
+ KDU = 2*NS
KU = N - KDU + 1
KWH = KDU + 1
NHO = ( N-KDU+1-4 ) - ( KDU+1 ) + 1