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SUBROUTINE <a name="DLAHQR.1"></a><a href="dlahqr.f.html#DLAHQR.1">DLAHQR</a>( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI,
$ ILOZ, IHIZ, Z, LDZ, INFO )
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> -- LAPACK auxiliary routine (version 3.1) --
</span><span class="comment">*</span><span class="comment"> Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
</span><span class="comment">*</span><span class="comment"> November 2006
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> .. Scalar Arguments ..
</span> INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, N
LOGICAL WANTT, WANTZ
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. Array Arguments ..
</span> DOUBLE PRECISION H( LDH, * ), WI( * ), WR( * ), Z( LDZ, * )
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Purpose
</span><span class="comment">*</span><span class="comment"> =======
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> <a name="DLAHQR.19"></a><a href="dlahqr.f.html#DLAHQR.1">DLAHQR</a> is an auxiliary routine called by <a name="DHSEQR.19"></a><a href="dhseqr.f.html#DHSEQR.1">DHSEQR</a> to update the
</span><span class="comment">*</span><span class="comment"> eigenvalues and Schur decomposition already computed by <a name="DHSEQR.20"></a><a href="dhseqr.f.html#DHSEQR.1">DHSEQR</a>, by
</span><span class="comment">*</span><span class="comment"> dealing with the Hessenberg submatrix in rows and columns ILO to
</span><span class="comment">*</span><span class="comment"> IHI.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Arguments
</span><span class="comment">*</span><span class="comment"> =========
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> WANTT (input) LOGICAL
</span><span class="comment">*</span><span class="comment"> = .TRUE. : the full Schur form T is required;
</span><span class="comment">*</span><span class="comment"> = .FALSE.: only eigenvalues are required.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> WANTZ (input) LOGICAL
</span><span class="comment">*</span><span class="comment"> = .TRUE. : the matrix of Schur vectors Z is required;
</span><span class="comment">*</span><span class="comment"> = .FALSE.: Schur vectors are not required.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> N (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The order of the matrix H. N >= 0.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> ILO (input) INTEGER
</span><span class="comment">*</span><span class="comment"> IHI (input) INTEGER
</span><span class="comment">*</span><span class="comment"> It is assumed that H is already upper quasi-triangular in
</span><span class="comment">*</span><span class="comment"> rows and columns IHI+1:N, and that H(ILO,ILO-1) = 0 (unless
</span><span class="comment">*</span><span class="comment"> ILO = 1). <a name="DLAHQR.42"></a><a href="dlahqr.f.html#DLAHQR.1">DLAHQR</a> works primarily with the Hessenberg
</span><span class="comment">*</span><span class="comment"> submatrix in rows and columns ILO to IHI, but applies
</span><span class="comment">*</span><span class="comment"> transformations to all of H if WANTT is .TRUE..
</span><span class="comment">*</span><span class="comment"> 1 <= ILO <= max(1,IHI); IHI <= N.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> H (input/output) DOUBLE PRECISION array, dimension (LDH,N)
</span><span class="comment">*</span><span class="comment"> On entry, the upper Hessenberg matrix H.
</span><span class="comment">*</span><span class="comment"> On exit, if INFO is zero and if WANTT is .TRUE., H is upper
</span><span class="comment">*</span><span class="comment"> quasi-triangular in rows and columns ILO:IHI, with any
</span><span class="comment">*</span><span class="comment"> 2-by-2 diagonal blocks in standard form. If INFO is zero
</span><span class="comment">*</span><span class="comment"> and WANTT is .FALSE., the contents of H are unspecified on
</span><span class="comment">*</span><span class="comment"> exit. The output state of H if INFO is nonzero is given
</span><span class="comment">*</span><span class="comment"> below under the description of INFO.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> LDH (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The leading dimension of the array H. LDH >= max(1,N).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> WR (output) DOUBLE PRECISION array, dimension (N)
</span><span class="comment">*</span><span class="comment"> WI (output) DOUBLE PRECISION array, dimension (N)
</span><span class="comment">*</span><span class="comment"> The real and imaginary parts, respectively, of the computed
</span><span class="comment">*</span><span class="comment"> eigenvalues ILO to IHI are stored in the corresponding
</span><span class="comment">*</span><span class="comment"> elements of WR and WI. If two eigenvalues are computed as a
</span><span class="comment">*</span><span class="comment"> complex conjugate pair, they are stored in consecutive
</span><span class="comment">*</span><span class="comment"> elements of WR and WI, say the i-th and (i+1)th, with
</span><span class="comment">*</span><span class="comment"> WI(i) > 0 and WI(i+1) < 0. If WANTT is .TRUE., the
</span><span class="comment">*</span><span class="comment"> eigenvalues are stored in the same order as on the diagonal
</span><span class="comment">*</span><span class="comment"> of the Schur form returned in H, with WR(i) = H(i,i), and, if
</span><span class="comment">*</span><span class="comment"> H(i:i+1,i:i+1) is a 2-by-2 diagonal block,
</span><span class="comment">*</span><span class="comment"> WI(i) = sqrt(H(i+1,i)*H(i,i+1)) and WI(i+1) = -WI(i).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> ILOZ (input) INTEGER
</span><span class="comment">*</span><span class="comment"> IHIZ (input) INTEGER
</span><span class="comment">*</span><span class="comment"> Specify the rows of Z to which transformations must be
</span><span class="comment">*</span><span class="comment"> applied if WANTZ is .TRUE..
</span><span class="comment">*</span><span class="comment"> 1 <= ILOZ <= ILO; IHI <= IHIZ <= N.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Z (input/output) DOUBLE PRECISION array, dimension (LDZ,N)
</span><span class="comment">*</span><span class="comment"> If WANTZ is .TRUE., on entry Z must contain the current
</span><span class="comment">*</span><span class="comment"> matrix Z of transformations accumulated by <a name="DHSEQR.80"></a><a href="dhseqr.f.html#DHSEQR.1">DHSEQR</a>, and on
</span><span class="comment">*</span><span class="comment"> exit Z has been updated; transformations are applied only to
</span><span class="comment">*</span><span class="comment"> the submatrix Z(ILOZ:IHIZ,ILO:IHI).
</span><span class="comment">*</span><span class="comment"> If WANTZ is .FALSE., Z is not referenced.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> LDZ (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The leading dimension of the array Z. LDZ >= max(1,N).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> INFO (output) INTEGER
</span><span class="comment">*</span><span class="comment"> = 0: successful exit
</span><span class="comment">*</span><span class="comment"> .GT. 0: If INFO = i, <a name="DLAHQR.90"></a><a href="dlahqr.f.html#DLAHQR.1">DLAHQR</a> failed to compute all the
</span><span class="comment">*</span><span class="comment"> eigenvalues ILO to IHI in a total of 30 iterations
</span><span class="comment">*</span><span class="comment"> per eigenvalue; elements i+1:ihi of WR and WI
</span><span class="comment">*</span><span class="comment"> contain those eigenvalues which have been
</span><span class="comment">*</span><span class="comment"> successfully computed.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> If INFO .GT. 0 and WANTT is .FALSE., then on exit,
</span><span class="comment">*</span><span class="comment"> the remaining unconverged eigenvalues are the
</span><span class="comment">*</span><span class="comment"> eigenvalues of the upper Hessenberg matrix rows
</span><span class="comment">*</span><span class="comment"> and columns ILO thorugh INFO of the final, output
</span><span class="comment">*</span><span class="comment"> value of H.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> If INFO .GT. 0 and WANTT is .TRUE., then on exit
</span><span class="comment">*</span><span class="comment"> (*) (initial value of H)*U = U*(final value of H)
</span><span class="comment">*</span><span class="comment"> where U is an orthognal matrix. The final
</span><span class="comment">*</span><span class="comment"> value of H is upper Hessenberg and triangular in
</span><span class="comment">*</span><span class="comment"> rows and columns INFO+1 through IHI.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> If INFO .GT. 0 and WANTZ is .TRUE., then on exit
</span><span class="comment">*</span><span class="comment"> (final value of Z) = (initial value of Z)*U
</span><span class="comment">*</span><span class="comment"> where U is the orthogonal matrix in (*)
</span><span class="comment">*</span><span class="comment"> (regardless of the value of WANTT.)
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Further Details
</span><span class="comment">*</span><span class="comment"> ===============
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> 02-96 Based on modifications by
</span><span class="comment">*</span><span class="comment"> David Day, Sandia National Laboratory, USA
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> 12-04 Further modifications by
</span><span class="comment">*</span><span class="comment"> Ralph Byers, University of Kansas, USA
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> This is a modified version of <a name="DLAHQR.122"></a><a href="dlahqr.f.html#DLAHQR.1">DLAHQR</a> from LAPACK version 3.0.
</span><span class="comment">*</span><span class="comment"> It is (1) more robust against overflow and underflow and
</span><span class="comment">*</span><span class="comment"> (2) adopts the more conservative Ahues & Tisseur stopping
</span><span class="comment">*</span><span class="comment"> criterion (LAWN 122, 1997).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> =========================================================
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> .. Parameters ..
</span> INTEGER ITMAX
PARAMETER ( ITMAX = 30 )
DOUBLE PRECISION ZERO, ONE, TWO
PARAMETER ( ZERO = 0.0d0, ONE = 1.0d0, TWO = 2.0d0 )
DOUBLE PRECISION DAT1, DAT2
PARAMETER ( DAT1 = 3.0d0 / 4.0d0, DAT2 = -0.4375d0 )
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. Local Scalars ..
</span> DOUBLE PRECISION AA, AB, BA, BB, CS, DET, H11, H12, H21, H21S,
$ H22, RT1I, RT1R, RT2I, RT2R, RTDISC, S, SAFMAX,
$ SAFMIN, SMLNUM, SN, SUM, T1, T2, T3, TR, TST,
$ ULP, V2, V3
INTEGER I, I1, I2, ITS, J, K, L, M, NH, NR, NZ
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. Local Arrays ..
</span> DOUBLE PRECISION V( 3 )
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. External Functions ..
</span> DOUBLE PRECISION <a name="DLAMCH.148"></a><a href="dlamch.f.html#DLAMCH.1">DLAMCH</a>
EXTERNAL <a name="DLAMCH.149"></a><a href="dlamch.f.html#DLAMCH.1">DLAMCH</a>
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. External Subroutines ..
</span> EXTERNAL DCOPY, <a name="DLABAD.152"></a><a href="dlabad.f.html#DLABAD.1">DLABAD</a>, <a name="DLANV2.152"></a><a href="dlanv2.f.html#DLANV2.1">DLANV2</a>, <a name="DLARFG.152"></a><a href="dlarfg.f.html#DLARFG.1">DLARFG</a>, DROT
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. Intrinsic Functions ..
</span> INTRINSIC ABS, DBLE, MAX, MIN, SQRT
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. Executable Statements ..
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