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SUBROUTINE DLASQ1( N, D, E, WORK, INFO ) |
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* |
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* -- LAPACK routine (version 3.2) -- |
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* |
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* -- Contributed by Osni Marques of the Lawrence Berkeley National -- |
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* -- Laboratory and Beresford Parlett of the Univ. of California at -- |
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* -- Berkeley -- |
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* -- November 2008 -- |
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* |
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* -- LAPACK is a software package provided by Univ. of Tennessee, -- |
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* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- |
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* |
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* .. Scalar Arguments .. |
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INTEGER INFO, N |
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* .. |
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* .. Array Arguments .. |
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DOUBLE PRECISION D( * ), E( * ), WORK( * ) |
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* .. |
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* |
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* Purpose |
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* ======= |
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* |
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* DLASQ1 computes the singular values of a real N-by-N bidiagonal |
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* matrix with diagonal D and off-diagonal E. The singular values |
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* are computed to high relative accuracy, in the absence of |
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* denormalization, underflow and overflow. The algorithm was first |
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* presented in |
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* |
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* "Accurate singular values and differential qd algorithms" by K. V. |
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* Fernando and B. N. Parlett, Numer. Math., Vol-67, No. 2, pp. 191-230, |
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* 1994, |
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* |
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* and the present implementation is described in "An implementation of |
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* the dqds Algorithm (Positive Case)", LAPACK Working Note. |
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* |
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* Arguments |
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* ========= |
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* |
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* N (input) INTEGER |
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* The number of rows and columns in the matrix. N >= 0. |
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* |
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* D (input/output) DOUBLE PRECISION array, dimension (N) |
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* On entry, D contains the diagonal elements of the |
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* bidiagonal matrix whose SVD is desired. On normal exit, |
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* D contains the singular values in decreasing order. |
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* |
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* E (input/output) DOUBLE PRECISION array, dimension (N) |
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* On entry, elements E(1:N-1) contain the off-diagonal elements |
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* of the bidiagonal matrix whose SVD is desired. |
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* On exit, E is overwritten. |
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* |
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* WORK (workspace) DOUBLE PRECISION array, dimension (4*N) |
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* |
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* INFO (output) INTEGER |
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* = 0: successful exit |
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* < 0: if INFO = -i, the i-th argument had an illegal value |
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* > 0: the algorithm failed |
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* = 1, a split was marked by a positive value in E |
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* = 2, current block of Z not diagonalized after 30*N |
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* iterations (in inner while loop) |
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* = 3, termination criterion of outer while loop not met |
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* (program created more than N unreduced blocks) |
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* |
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* ===================================================================== |
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* |
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* .. Parameters .. |
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DOUBLE PRECISION ZERO |
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PARAMETER ( ZERO = 0.0D0 ) |
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* .. |
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* .. Local Scalars .. |
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INTEGER I, IINFO |
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DOUBLE PRECISION EPS, SCALE, SAFMIN, SIGMN, SIGMX |
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* .. |
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* .. External Subroutines .. |
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EXTERNAL DCOPY, DLAS2, DLASCL, DLASQ2, DLASRT, XERBLA |
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* .. |
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* .. External Functions .. |
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DOUBLE PRECISION DLAMCH |
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EXTERNAL DLAMCH |
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* .. |
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* .. Intrinsic Functions .. |
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INTRINSIC ABS, MAX, SQRT |
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* .. |
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* .. Executable Statements .. |
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* |
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INFO = 0 |
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IF( N.LT.0 ) THEN |
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INFO = -2 |
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CALL XERBLA( 'DLASQ1', -INFO ) |
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RETURN |
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ELSE IF( N.EQ.0 ) THEN |
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RETURN |
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ELSE IF( N.EQ.1 ) THEN |
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D( 1 ) = ABS( D( 1 ) ) |
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RETURN |
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ELSE IF( N.EQ.2 ) THEN |
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CALL DLAS2( D( 1 ), E( 1 ), D( 2 ), SIGMN, SIGMX ) |
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D( 1 ) = SIGMX |
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D( 2 ) = SIGMN |
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RETURN |
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END IF |
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* |
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* Estimate the largest singular value. |
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* |
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SIGMX = ZERO |
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DO 10 I = 1, N - 1 |
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D( I ) = ABS( D( I ) ) |
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SIGMX = MAX( SIGMX, ABS( E( I ) ) ) |
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10 CONTINUE |
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D( N ) = ABS( D( N ) ) |
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* |
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* Early return if SIGMX is zero (matrix is already diagonal). |
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* |
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IF( SIGMX.EQ.ZERO ) THEN |
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CALL DLASRT( 'D', N, D, IINFO ) |
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RETURN |
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END IF |
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* |
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DO 20 I = 1, N |
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SIGMX = MAX( SIGMX, D( I ) ) |
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20 CONTINUE |
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* |
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* Copy D and E into WORK (in the Z format) and scale (squaring the |
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* input data makes scaling by a power of the radix pointless). |
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* |
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EPS = DLAMCH( 'Precision' ) |
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SAFMIN = DLAMCH( 'Safe minimum' ) |
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SCALE = SQRT( EPS / SAFMIN ) |
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CALL DCOPY( N, D, 1, WORK( 1 ), 2 ) |
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CALL DCOPY( N-1, E, 1, WORK( 2 ), 2 ) |
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CALL DLASCL( 'G', 0, 0, SIGMX, SCALE, 2*N-1, 1, WORK, 2*N-1, |
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$ IINFO ) |
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* |
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* Compute the q's and e's. |
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* |
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DO 30 I = 1, 2*N - 1 |
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WORK( I ) = WORK( I )**2 |
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30 CONTINUE |
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WORK( 2*N ) = ZERO |
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* |
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CALL DLASQ2( N, WORK, INFO ) |
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* |
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IF( INFO.EQ.0 ) THEN |
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DO 40 I = 1, N |
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D( I ) = SQRT( WORK( I ) ) |
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40 CONTINUE |
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CALL DLASCL( 'G', 0, 0, SCALE, SIGMX, N, 1, D, N, IINFO ) |
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END IF |
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* |
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RETURN |
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* |
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* End of DLASQ1 |
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* |
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END |