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*> \brief \b DSTEQR
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSTEQR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsteqr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsteqr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsteqr.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSTEQR( COMPZ, N, D, E, Z, LDZ, WORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER COMPZ
* INTEGER INFO, LDZ, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION D( * ), E( * ), WORK( * ), Z( LDZ, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSTEQR computes all eigenvalues and, optionally, eigenvectors of a
*> symmetric tridiagonal matrix using the implicit QL or QR method.
*> The eigenvectors of a full or band symmetric matrix can also be found
*> if DSYTRD or DSPTRD or DSBTRD has been used to reduce this matrix to
*> tridiagonal form.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] COMPZ
*> \verbatim
*> COMPZ is CHARACTER*1
*> = 'N': Compute eigenvalues only.
*> = 'V': Compute eigenvalues and eigenvectors of the original
*> symmetric matrix. On entry, Z must contain the
*> orthogonal matrix used to reduce the original matrix
*> to tridiagonal form.
*> = 'I': Compute eigenvalues and eigenvectors of the
*> tridiagonal matrix. Z is initialized to the identity
*> matrix.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix. N >= 0.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> On entry, the diagonal elements of the tridiagonal matrix.
*> On exit, if INFO = 0, the eigenvalues in ascending order.
*> \endverbatim
*>
*> \param[in,out] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (N-1)
*> On entry, the (n-1) subdiagonal elements of the tridiagonal
*> matrix.
*> On exit, E has been destroyed.
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (LDZ, N)
*> On entry, if COMPZ = 'V', then Z contains the orthogonal
*> matrix used in the reduction to tridiagonal form.
*> On exit, if INFO = 0, then if COMPZ = 'V', Z contains the
*> orthonormal eigenvectors of the original symmetric matrix,
*> and if COMPZ = 'I', Z contains the orthonormal eigenvectors
*> of the symmetric tridiagonal matrix.
*> If COMPZ = 'N', then Z is not referenced.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*> LDZ is INTEGER
*> The leading dimension of the array Z. LDZ >= 1, and if
*> eigenvectors are desired, then LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (max(1,2*N-2))
*> If COMPZ = 'N', then WORK is not referenced.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: the algorithm has failed to find all the eigenvalues in
*> a total of 30*N iterations; if INFO = i, then i
*> elements of E have not converged to zero; on exit, D
*> and E contain the elements of a symmetric tridiagonal
*> matrix which is orthogonally similar to the original
*> matrix.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup auxOTHERcomputational
*
* =====================================================================
SUBROUTINE DSTEQR( COMPZ, N, D, E, Z, LDZ, WORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER COMPZ
INTEGER INFO, LDZ, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION D( * ), E( * ), WORK( * ), Z( LDZ, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE, TWO, THREE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0,
$ THREE = 3.0D0 )
INTEGER MAXIT
PARAMETER ( MAXIT = 30 )
* ..
* .. Local Scalars ..
INTEGER I, ICOMPZ, II, ISCALE, J, JTOT, K, L, L1, LEND,
$ LENDM1, LENDP1, LENDSV, LM1, LSV, M, MM, MM1,
$ NM1, NMAXIT
DOUBLE PRECISION ANORM, B, C, EPS, EPS2, F, G, P, R, RT1, RT2,
$ S, SAFMAX, SAFMIN, SSFMAX, SSFMIN, TST
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH, DLANST, DLAPY2
EXTERNAL LSAME, DLAMCH, DLANST, DLAPY2
* ..
* .. External Subroutines ..
EXTERNAL DLAE2, DLAEV2, DLARTG, DLASCL, DLASET, DLASR,
$ DLASRT, DSWAP, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, SIGN, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
*
IF( LSAME( COMPZ, 'N' ) ) THEN
ICOMPZ = 0
ELSE IF( LSAME( COMPZ, 'V' ) ) THEN
ICOMPZ = 1
ELSE IF( LSAME( COMPZ, 'I' ) ) THEN
ICOMPZ = 2
ELSE
ICOMPZ = -1
END IF
IF( ICOMPZ.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( ( LDZ.LT.1 ) .OR. ( ICOMPZ.GT.0 .AND. LDZ.LT.MAX( 1,
$ N ) ) ) THEN
INFO = -6
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSTEQR', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
IF( N.EQ.1 ) THEN
IF( ICOMPZ.EQ.2 )
$ Z( 1, 1 ) = ONE
RETURN
END IF
*
* Determine the unit roundoff and over/underflow thresholds.
*
EPS = DLAMCH( 'E' )
EPS2 = EPS**2
SAFMIN = DLAMCH( 'S' )
SAFMAX = ONE / SAFMIN
SSFMAX = SQRT( SAFMAX ) / THREE
SSFMIN = SQRT( SAFMIN ) / EPS2
*
* Compute the eigenvalues and eigenvectors of the tridiagonal
* matrix.
*
IF( ICOMPZ.EQ.2 )
$ CALL DLASET( 'Full', N, N, ZERO, ONE, Z, LDZ )
*
NMAXIT = N*MAXIT
JTOT = 0
*
* Determine where the matrix splits and choose QL or QR iteration
* for each block, according to whether top or bottom diagonal
* element is smaller.
*
L1 = 1
NM1 = N - 1
*
10 CONTINUE
IF( L1.GT.N )
$ GO TO 160
IF( L1.GT.1 )
$ E( L1-1 ) = ZERO
IF( L1.LE.NM1 ) THEN
DO 20 M = L1, NM1
TST = ABS( E( M ) )
IF( TST.EQ.ZERO )
$ GO TO 30
IF( TST.LE.( SQRT( ABS( D( M ) ) )*SQRT( ABS( D( M+
$ 1 ) ) ) )*EPS ) THEN
E( M ) = ZERO
GO TO 30
END IF
20 CONTINUE
END IF
M = N
*
30 CONTINUE
L = L1
LSV = L
LEND = M
LENDSV = LEND
L1 = M + 1
IF( LEND.EQ.L )
$ GO TO 10
*
* Scale submatrix in rows and columns L to LEND
*
ANORM = DLANST( 'M', LEND-L+1, D( L ), E( L ) )
ISCALE = 0
IF( ANORM.EQ.ZERO )
$ GO TO 10
IF( ANORM.GT.SSFMAX ) THEN
ISCALE = 1
CALL DLASCL( 'G', 0, 0, ANORM, SSFMAX, LEND-L+1, 1, D( L ), N,
$ INFO )
CALL DLASCL( 'G', 0, 0, ANORM, SSFMAX, LEND-L, 1, E( L ), N,
$ INFO )
ELSE IF( ANORM.LT.SSFMIN ) THEN
ISCALE = 2
CALL DLASCL( 'G', 0, 0, ANORM, SSFMIN, LEND-L+1, 1, D( L ), N,
$ INFO )
CALL DLASCL( 'G', 0, 0, ANORM, SSFMIN, LEND-L, 1, E( L ), N,
$ INFO )
END IF
*
* Choose between QL and QR iteration
*
IF( ABS( D( LEND ) ).LT.ABS( D( L ) ) ) THEN
LEND = LSV
L = LENDSV
END IF
*
IF( LEND.GT.L ) THEN
*
* QL Iteration
*
* Look for small subdiagonal element.
*
40 CONTINUE
IF( L.NE.LEND ) THEN
LENDM1 = LEND - 1
DO 50 M = L, LENDM1
TST = ABS( E( M ) )**2
IF( TST.LE.( EPS2*ABS( D( M ) ) )*ABS( D( M+1 ) )+
$ SAFMIN )GO TO 60
50 CONTINUE
END IF
*
M = LEND
*
60 CONTINUE
IF( M.LT.LEND )
$ E( M ) = ZERO
P = D( L )
IF( M.EQ.L )
$ GO TO 80
*
* If remaining matrix is 2-by-2, use DLAE2 or SLAEV2
* to compute its eigensystem.
*
IF( M.EQ.L+1 ) THEN
IF( ICOMPZ.GT.0 ) THEN
CALL DLAEV2( D( L ), E( L ), D( L+1 ), RT1, RT2, C, S )
WORK( L ) = C
WORK( N-1+L ) = S
CALL DLASR( 'R', 'V', 'B', N, 2, WORK( L ),
$ WORK( N-1+L ), Z( 1, L ), LDZ )
ELSE
CALL DLAE2( D( L ), E( L ), D( L+1 ), RT1, RT2 )
END IF
D( L ) = RT1
D( L+1 ) = RT2
E( L ) = ZERO
L = L + 2
IF( L.LE.LEND )
$ GO TO 40
GO TO 140
END IF
*
IF( JTOT.EQ.NMAXIT )
$ GO TO 140
JTOT = JTOT + 1
*
* Form shift.
*
G = ( D( L+1 )-P ) / ( TWO*E( L ) )
R = DLAPY2( G, ONE )
G = D( M ) - P + ( E( L ) / ( G+SIGN( R, G ) ) )
*
S = ONE
C = ONE
P = ZERO
*
* Inner loop
*
MM1 = M - 1
DO 70 I = MM1, L, -1
F = S*E( I )
B = C*E( I )
CALL DLARTG( G, F, C, S, R )
IF( I.NE.M-1 )
$ E( I+1 ) = R
G = D( I+1 ) - P
R = ( D( I )-G )*S + TWO*C*B
P = S*R
D( I+1 ) = G + P
G = C*R - B
*
* If eigenvectors are desired, then save rotations.
*
IF( ICOMPZ.GT.0 ) THEN
WORK( I ) = C
WORK( N-1+I ) = -S
END IF
*
70 CONTINUE
*
* If eigenvectors are desired, then apply saved rotations.
*
IF( ICOMPZ.GT.0 ) THEN
MM = M - L + 1
CALL DLASR( 'R', 'V', 'B', N, MM, WORK( L ), WORK( N-1+L ),
$ Z( 1, L ), LDZ )
END IF
*
D( L ) = D( L ) - P
E( L ) = G
GO TO 40
*
* Eigenvalue found.
*
80 CONTINUE
D( L ) = P
*
L = L + 1
IF( L.LE.LEND )
$ GO TO 40
GO TO 140
*
ELSE
*
* QR Iteration
*
* Look for small superdiagonal element.
*
90 CONTINUE
IF( L.NE.LEND ) THEN
LENDP1 = LEND + 1
DO 100 M = L, LENDP1, -1
TST = ABS( E( M-1 ) )**2
IF( TST.LE.( EPS2*ABS( D( M ) ) )*ABS( D( M-1 ) )+
$ SAFMIN )GO TO 110
100 CONTINUE
END IF
*
M = LEND
*
110 CONTINUE
IF( M.GT.LEND )
$ E( M-1 ) = ZERO
P = D( L )
IF( M.EQ.L )
$ GO TO 130
*
* If remaining matrix is 2-by-2, use DLAE2 or SLAEV2
* to compute its eigensystem.
*
IF( M.EQ.L-1 ) THEN
IF( ICOMPZ.GT.0 ) THEN
CALL DLAEV2( D( L-1 ), E( L-1 ), D( L ), RT1, RT2, C, S )
WORK( M ) = C
WORK( N-1+M ) = S
CALL DLASR( 'R', 'V', 'F', N, 2, WORK( M ),
$ WORK( N-1+M ), Z( 1, L-1 ), LDZ )
ELSE
CALL DLAE2( D( L-1 ), E( L-1 ), D( L ), RT1, RT2 )
END IF
D( L-1 ) = RT1
D( L ) = RT2
E( L-1 ) = ZERO
L = L - 2
IF( L.GE.LEND )
$ GO TO 90
GO TO 140
END IF
*
IF( JTOT.EQ.NMAXIT )
$ GO TO 140
JTOT = JTOT + 1
*
* Form shift.
*
G = ( D( L-1 )-P ) / ( TWO*E( L-1 ) )
R = DLAPY2( G, ONE )
G = D( M ) - P + ( E( L-1 ) / ( G+SIGN( R, G ) ) )
*
S = ONE
C = ONE
P = ZERO
*
* Inner loop
*
LM1 = L - 1
DO 120 I = M, LM1
F = S*E( I )
B = C*E( I )
CALL DLARTG( G, F, C, S, R )
IF( I.NE.M )
$ E( I-1 ) = R
G = D( I ) - P
R = ( D( I+1 )-G )*S + TWO*C*B
P = S*R
D( I ) = G + P
G = C*R - B
*
* If eigenvectors are desired, then save rotations.
*
IF( ICOMPZ.GT.0 ) THEN
WORK( I ) = C
WORK( N-1+I ) = S
END IF
*
120 CONTINUE
*
* If eigenvectors are desired, then apply saved rotations.
*
IF( ICOMPZ.GT.0 ) THEN
MM = L - M + 1
CALL DLASR( 'R', 'V', 'F', N, MM, WORK( M ), WORK( N-1+M ),
$ Z( 1, M ), LDZ )
END IF
*
D( L ) = D( L ) - P
E( LM1 ) = G
GO TO 90
*
* Eigenvalue found.
*
130 CONTINUE
D( L ) = P
*
L = L - 1
IF( L.GE.LEND )
$ GO TO 90
GO TO 140
*
END IF
*
* Undo scaling if necessary
*
140 CONTINUE
IF( ISCALE.EQ.1 ) THEN
CALL DLASCL( 'G', 0, 0, SSFMAX, ANORM, LENDSV-LSV+1, 1,
$ D( LSV ), N, INFO )
CALL DLASCL( 'G', 0, 0, SSFMAX, ANORM, LENDSV-LSV, 1, E( LSV ),
$ N, INFO )
ELSE IF( ISCALE.EQ.2 ) THEN
CALL DLASCL( 'G', 0, 0, SSFMIN, ANORM, LENDSV-LSV+1, 1,
$ D( LSV ), N, INFO )
CALL DLASCL( 'G', 0, 0, SSFMIN, ANORM, LENDSV-LSV, 1, E( LSV ),
$ N, INFO )
END IF
*
* Check for no convergence to an eigenvalue after a total
* of N*MAXIT iterations.
*
IF( JTOT.LT.NMAXIT )
$ GO TO 10
DO 150 I = 1, N - 1
IF( E( I ).NE.ZERO )
$ INFO = INFO + 1
150 CONTINUE
GO TO 190
*
* Order eigenvalues and eigenvectors.
*
160 CONTINUE
IF( ICOMPZ.EQ.0 ) THEN
*
* Use Quick Sort
*
CALL DLASRT( 'I', N, D, INFO )
*
ELSE
*
* Use Selection Sort to minimize swaps of eigenvectors
*
DO 180 II = 2, N
I = II - 1
K = I
P = D( I )
DO 170 J = II, N
IF( D( J ).LT.P ) THEN
K = J
P = D( J )
END IF
170 CONTINUE
IF( K.NE.I ) THEN
D( K ) = D( I )
D( I ) = P
CALL DSWAP( N, Z( 1, I ), 1, Z( 1, K ), 1 )
END IF
180 CONTINUE
END IF
*
190 CONTINUE
RETURN
*
* End of DSTEQR
*
END