```      SUBROUTINE SLASQ1( N, D, E, WORK, 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 ..
REAL               D( * ), E( * ), WORK( * )
*     ..
*
*  Purpose
*  =======
*
*  SLASQ1 computes the singular values of a real N-by-N bidiagonal
*  matrix with diagonal D and off-diagonal E. The singular values
*  are computed to high relative accuracy, in the absence of
*  denormalization, underflow and overflow. The algorithm was first
*  presented in
*
*  "Accurate singular values and differential qd algorithms" by K. V.
*  Fernando and B. N. Parlett, Numer. Math., Vol-67, No. 2, pp. 191-230,
*  1994,
*
*  and the present implementation is described in "An implementation of
*  the dqds Algorithm (Positive Case)", LAPACK Working Note.
*
*  Arguments
*  =========
*
*  N     (input) INTEGER
*        The number of rows and columns in the matrix. N >= 0.
*
*  D     (input/output) REAL array, dimension (N)
*        On entry, D contains the diagonal elements of the
*        bidiagonal matrix whose SVD is desired. On normal exit,
*        D contains the singular values in decreasing order.
*
*  E     (input/output) REAL array, dimension (N)
*        On entry, elements E(1:N-1) contain the off-diagonal elements
*        of the bidiagonal matrix whose SVD is desired.
*        On exit, E is overwritten.
*
*  WORK  (workspace) REAL array, dimension (4*N)
*
*  INFO  (output) INTEGER
*        = 0: successful exit
*        < 0: if INFO = -i, the i-th argument had an illegal value
*        > 0: the algorithm failed
*             = 1, a split was marked by a positive value in E
*             = 2, current block of Z not diagonalized after 30*N
*                  iterations (in inner while loop)
*             = 3, termination criterion of outer while loop not met
*                  (program created more than N unreduced blocks)
*
*  =====================================================================
*
*     .. Parameters ..
REAL               ZERO
PARAMETER          ( ZERO = 0.0E0 )
*     ..
*     .. Local Scalars ..
INTEGER            I, IINFO
REAL               EPS, SCALE, SAFMIN, SIGMN, SIGMX
*     ..
*     .. External Subroutines ..
EXTERNAL           SCOPY, SLAS2, SLASCL, SLASQ2, SLASRT, XERBLA
*     ..
*     .. External Functions ..
REAL               SLAMCH
EXTERNAL           SLAMCH
*     ..
*     .. Intrinsic Functions ..
INTRINSIC          ABS, MAX, SQRT
*     ..
*     .. Executable Statements ..
*
INFO = 0
IF( N.LT.0 ) THEN
INFO = -2
CALL XERBLA( 'SLASQ1', -INFO )
RETURN
ELSE IF( N.EQ.0 ) THEN
RETURN
ELSE IF( N.EQ.1 ) THEN
D( 1 ) = ABS( D( 1 ) )
RETURN
ELSE IF( N.EQ.2 ) THEN
CALL SLAS2( D( 1 ), E( 1 ), D( 2 ), SIGMN, SIGMX )
D( 1 ) = SIGMX
D( 2 ) = SIGMN
RETURN
END IF
*
*     Estimate the largest singular value.
*
SIGMX = ZERO
DO 10 I = 1, N - 1
D( I ) = ABS( D( I ) )
SIGMX = MAX( SIGMX, ABS( E( I ) ) )
10 CONTINUE
D( N ) = ABS( D( N ) )
*
*     Early return if SIGMX is zero (matrix is already diagonal).
*
IF( SIGMX.EQ.ZERO ) THEN
CALL SLASRT( 'D', N, D, IINFO )
RETURN
END IF
*
DO 20 I = 1, N
SIGMX = MAX( SIGMX, D( I ) )
20 CONTINUE
*
*     Copy D and E into WORK (in the Z format) and scale (squaring the
*     input data makes scaling by a power of the radix pointless).
*
EPS = SLAMCH( 'Precision' )
SAFMIN = SLAMCH( 'Safe minimum' )
SCALE = SQRT( EPS / SAFMIN )
CALL SCOPY( N, D, 1, WORK( 1 ), 2 )
CALL SCOPY( N-1, E, 1, WORK( 2 ), 2 )
CALL SLASCL( 'G', 0, 0, SIGMX, SCALE, 2*N-1, 1, WORK, 2*N-1,
\$             IINFO )
*
*     Compute the q's and e's.
*
DO 30 I = 1, 2*N - 1
WORK( I ) = WORK( I )**2
30 CONTINUE
WORK( 2*N ) = ZERO
*
CALL SLASQ2( N, WORK, INFO )
*
IF( INFO.EQ.0 ) THEN
DO 40 I = 1, N
D( I ) = SQRT( WORK( I ) )
40    CONTINUE
CALL SLASCL( 'G', 0, 0, SCALE, SIGMX, N, 1, D, N, IINFO )
END IF
*
RETURN
*
*     End of SLASQ1
*
END

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