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SUBROUTINE DSTERF( N, D, E, INFO )
*
* -- LAPACK routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
INTEGER INFO, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION D( * ), E( * )
* ..
*
* Purpose
* =======
*
* DSTERF computes all eigenvalues of a symmetric tridiagonal matrix
* using the Pal-Walker-Kahan variant of the QL or QR algorithm.
*
* Arguments
* =========
*
* N (input) INTEGER
* The order of the matrix. N >= 0.
*
* D (input/output) DOUBLE PRECISION array, dimension (N)
* On entry, the n diagonal elements of the tridiagonal matrix.
* On exit, if INFO = 0, the eigenvalues in ascending order.
*
* E (input/output) DOUBLE PRECISION array, dimension (N-1)
* On entry, the (n-1) subdiagonal elements of the tridiagonal
* matrix.
* On exit, E has been destroyed.
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
* > 0: the algorithm failed to find all of the eigenvalues in
* a total of 30*N iterations; if INFO = i, then i
* elements of E have not converged to zero.
*
* =====================================================================
*
* .. 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, ISCALE, JTOT, L, L1, LEND, LENDSV, LSV, M,
$ NMAXIT
DOUBLE PRECISION ALPHA, ANORM, BB, C, EPS, EPS2, GAMMA, OLDC,
$ OLDGAM, P, R, RT1, RT2, RTE, S, SAFMAX, SAFMIN,
$ SIGMA, SSFMAX, SSFMIN
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH, DLANST, DLAPY2
EXTERNAL DLAMCH, DLANST, DLAPY2
* ..
* .. External Subroutines ..
EXTERNAL DLAE2, DLASCL, DLASRT, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, SIGN, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
*
* Quick return if possible
*
IF( N.LT.0 ) THEN
INFO = -1
CALL XERBLA( 'DSTERF', -INFO )
RETURN
END IF
IF( N.LE.1 )
$ RETURN
*
* Determine the unit roundoff for this environment.
*
EPS = DLAMCH( 'E' )
EPS2 = EPS**2
SAFMIN = DLAMCH( 'S' )
SAFMAX = ONE / SAFMIN
SSFMAX = SQRT( SAFMAX ) / THREE
SSFMIN = SQRT( SAFMIN ) / EPS2
*
* Compute the eigenvalues of the tridiagonal matrix.
*
NMAXIT = N*MAXIT
SIGMA = ZERO
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
*
10 CONTINUE
IF( L1.GT.N )
$ GO TO 170
IF( L1.GT.1 )
$ E( L1-1 ) = ZERO
DO 20 M = L1, N - 1
IF( ABS( E( M ) ).LE.( SQRT( ABS( D( M ) ) )*SQRT( ABS( D( M+
$ 1 ) ) ) )*EPS ) THEN
E( M ) = ZERO
GO TO 30
END IF
20 CONTINUE
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( 'I', LEND-L+1, D( L ), E( L ) )
ISCALE = 0
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
*
DO 40 I = L, LEND - 1
E( I ) = E( I )**2
40 CONTINUE
*
* Choose between QL and QR iteration
*
IF( ABS( D( LEND ) ).LT.ABS( D( L ) ) ) THEN
LEND = LSV
L = LENDSV
END IF
*
IF( LEND.GE.L ) THEN
*
* QL Iteration
*
* Look for small subdiagonal element.
*
50 CONTINUE
IF( L.NE.LEND ) THEN
DO 60 M = L, LEND - 1
IF( ABS( E( M ) ).LE.EPS2*ABS( D( M )*D( M+1 ) ) )
$ GO TO 70
60 CONTINUE
END IF
M = LEND
*
70 CONTINUE
IF( M.LT.LEND )
$ E( M ) = ZERO
P = D( L )
IF( M.EQ.L )
$ GO TO 90
*
* If remaining matrix is 2 by 2, use DLAE2 to compute its
* eigenvalues.
*
IF( M.EQ.L+1 ) THEN
RTE = SQRT( E( L ) )
CALL DLAE2( D( L ), RTE, D( L+1 ), RT1, RT2 )
D( L ) = RT1
D( L+1 ) = RT2
E( L ) = ZERO
L = L + 2
IF( L.LE.LEND )
$ GO TO 50
GO TO 150
END IF
*
IF( JTOT.EQ.NMAXIT )
$ GO TO 150
JTOT = JTOT + 1
*
* Form shift.
*
RTE = SQRT( E( L ) )
SIGMA = ( D( L+1 )-P ) / ( TWO*RTE )
R = DLAPY2( SIGMA, ONE )
SIGMA = P - ( RTE / ( SIGMA+SIGN( R, SIGMA ) ) )
*
C = ONE
S = ZERO
GAMMA = D( M ) - SIGMA
P = GAMMA*GAMMA
*
* Inner loop
*
DO 80 I = M - 1, L, -1
BB = E( I )
R = P + BB
IF( I.NE.M-1 )
$ E( I+1 ) = S*R
OLDC = C
C = P / R
S = BB / R
OLDGAM = GAMMA
ALPHA = D( I )
GAMMA = C*( ALPHA-SIGMA ) - S*OLDGAM
D( I+1 ) = OLDGAM + ( ALPHA-GAMMA )
IF( C.NE.ZERO ) THEN
P = ( GAMMA*GAMMA ) / C
ELSE
P = OLDC*BB
END IF
80 CONTINUE
*
E( L ) = S*P
D( L ) = SIGMA + GAMMA
GO TO 50
*
* Eigenvalue found.
*
90 CONTINUE
D( L ) = P
*
L = L + 1
IF( L.LE.LEND )
$ GO TO 50
GO TO 150
*
ELSE
*
* QR Iteration
*
* Look for small superdiagonal element.
*
100 CONTINUE
DO 110 M = L, LEND + 1, -1
IF( ABS( E( M-1 ) ).LE.EPS2*ABS( D( M )*D( M-1 ) ) )
$ GO TO 120
110 CONTINUE
M = LEND
*
120 CONTINUE
IF( M.GT.LEND )
$ E( M-1 ) = ZERO
P = D( L )
IF( M.EQ.L )
$ GO TO 140
*
* If remaining matrix is 2 by 2, use DLAE2 to compute its
* eigenvalues.
*
IF( M.EQ.L-1 ) THEN
RTE = SQRT( E( L-1 ) )
CALL DLAE2( D( L ), RTE, D( L-1 ), RT1, RT2 )
D( L ) = RT1
D( L-1 ) = RT2
E( L-1 ) = ZERO
L = L - 2
IF( L.GE.LEND )
$ GO TO 100
GO TO 150
END IF
*
IF( JTOT.EQ.NMAXIT )
$ GO TO 150
JTOT = JTOT + 1
*
* Form shift.
*
RTE = SQRT( E( L-1 ) )
SIGMA = ( D( L-1 )-P ) / ( TWO*RTE )
R = DLAPY2( SIGMA, ONE )
SIGMA = P - ( RTE / ( SIGMA+SIGN( R, SIGMA ) ) )
*
C = ONE
S = ZERO
GAMMA = D( M ) - SIGMA
P = GAMMA*GAMMA
*
* Inner loop
*
DO 130 I = M, L - 1
BB = E( I )
R = P + BB
IF( I.NE.M )
$ E( I-1 ) = S*R
OLDC = C
C = P / R
S = BB / R
OLDGAM = GAMMA
ALPHA = D( I+1 )
GAMMA = C*( ALPHA-SIGMA ) - S*OLDGAM
D( I ) = OLDGAM + ( ALPHA-GAMMA )
IF( C.NE.ZERO ) THEN
P = ( GAMMA*GAMMA ) / C
ELSE
P = OLDC*BB
END IF
130 CONTINUE
*
E( L-1 ) = S*P
D( L ) = SIGMA + GAMMA
GO TO 100
*
* Eigenvalue found.
*
140 CONTINUE
D( L ) = P
*
L = L - 1
IF( L.GE.LEND )
$ GO TO 100
GO TO 150
*
END IF
*
* Undo scaling if necessary
*
150 CONTINUE
IF( ISCALE.EQ.1 )
$ CALL DLASCL( 'G', 0, 0, SSFMAX, ANORM, LENDSV-LSV+1, 1,
$ D( LSV ), N, INFO )
IF( ISCALE.EQ.2 )
$ CALL DLASCL( 'G', 0, 0, SSFMIN, ANORM, LENDSV-LSV+1, 1,
$ D( LSV ), N, INFO )
*
* Check for no convergence to an eigenvalue after a total
* of N*MAXIT iterations.
*
IF( JTOT.LT.NMAXIT )
$ GO TO 10
DO 160 I = 1, N - 1
IF( E( I ).NE.ZERO )
$ INFO = INFO + 1
160 CONTINUE
GO TO 180
*
* Sort eigenvalues in increasing order.
*
170 CONTINUE
CALL DLASRT( 'I', N, D, INFO )
*
180 CONTINUE
RETURN
*
* End of DSTERF
*
END
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