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+ SUBROUTINE DLASQ1( 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 ..
+ DOUBLE PRECISION D( * ), E( * ), WORK( * )
+* ..
+*
+* Purpose
+* =======
+*
+* DLASQ1 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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 ..
+ DOUBLE PRECISION ZERO
+ PARAMETER ( ZERO = 0.0D0 )
+* ..
+* .. Local Scalars ..
+ INTEGER I, IINFO
+ DOUBLE PRECISION EPS, SCALE, SAFMIN, SIGMN, SIGMX
+* ..
+* .. External Subroutines ..
+ EXTERNAL DCOPY, DLAS2, DLASCL, DLASQ2, DLASRT, XERBLA
+* ..
+* .. External Functions ..
+ DOUBLE PRECISION DLAMCH
+ EXTERNAL DLAMCH
+* ..
+* .. Intrinsic Functions ..
+ INTRINSIC ABS, MAX, SQRT
+* ..
+* .. Executable Statements ..
+*
+ INFO = 0
+ IF( N.LT.0 ) THEN
+ INFO = -2
+ CALL XERBLA( 'DLASQ1', -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 DLAS2( 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 DLASRT( '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 = DLAMCH( 'Precision' )
+ SAFMIN = DLAMCH( 'Safe minimum' )
+ SCALE = SQRT( EPS / SAFMIN )
+ CALL DCOPY( N, D, 1, WORK( 1 ), 2 )
+ CALL DCOPY( N-1, E, 1, WORK( 2 ), 2 )
+ CALL DLASCL( '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 DLASQ2( N, WORK, INFO )
+*
+ IF( INFO.EQ.0 ) THEN
+ DO 40 I = 1, N
+ D( I ) = SQRT( WORK( I ) )
+ 40 CONTINUE
+ CALL DLASCL( 'G', 0, 0, SCALE, SIGMX, N, 1, D, N, IINFO )
+ END IF
+*
+ RETURN
+*
+* End of DLASQ1
+*
+ END