dlasq1.f.html

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      SUBROUTINE <a name="DLASQ1.1"></a><a href="dlasq1.f.html#DLASQ1.1">DLASQ1</a>( N, D, E, WORK, INFO )
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  -- LAPACK routine (version 3.1) --
</span><span class="comment">*</span><span class="comment">     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
</span><span class="comment">*</span><span class="comment">     November 2006
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     .. Scalar Arguments ..
</span>      INTEGER            INFO, N
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Array Arguments ..
</span>      DOUBLE PRECISION   D( * ), E( * ), WORK( * )
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Purpose
</span><span class="comment">*</span><span class="comment">  =======
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  <a name="DLASQ1.17"></a><a href="dlasq1.f.html#DLASQ1.1">DLASQ1</a> computes the singular values of a real N-by-N bidiagonal
</span><span class="comment">*</span><span class="comment">  matrix with diagonal D and off-diagonal E. The singular values
</span><span class="comment">*</span><span class="comment">  are computed to high relative accuracy, in the absence of
</span><span class="comment">*</span><span class="comment">  denormalization, underflow and overflow. The algorithm was first
</span><span class="comment">*</span><span class="comment">  presented in
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  &quot;Accurate singular values and differential qd algorithms&quot; by K. V.
</span><span class="comment">*</span><span class="comment">  Fernando and B. N. Parlett, Numer. Math., Vol-67, No. 2, pp. 191-230,
</span><span class="comment">*</span><span class="comment">  1994,
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  and the present implementation is described in &quot;An implementation of
</span><span class="comment">*</span><span class="comment">  the dqds Algorithm (Positive Case)&quot;, LAPACK Working Note.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Arguments
</span><span class="comment">*</span><span class="comment">  =========
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  N     (input) INTEGER
</span><span class="comment">*</span><span class="comment">        The number of rows and columns in the matrix. N &gt;= 0.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  D     (input/output) DOUBLE PRECISION array, dimension (N)
</span><span class="comment">*</span><span class="comment">        On entry, D contains the diagonal elements of the
</span><span class="comment">*</span><span class="comment">        bidiagonal matrix whose SVD is desired. On normal exit,
</span><span class="comment">*</span><span class="comment">        D contains the singular values in decreasing order.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  E     (input/output) DOUBLE PRECISION array, dimension (N)
</span><span class="comment">*</span><span class="comment">        On entry, elements E(1:N-1) contain the off-diagonal elements
</span><span class="comment">*</span><span class="comment">        of the bidiagonal matrix whose SVD is desired.
</span><span class="comment">*</span><span class="comment">        On exit, E is overwritten.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  WORK  (workspace) DOUBLE PRECISION array, dimension (4*N)
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  INFO  (output) INTEGER
</span><span class="comment">*</span><span class="comment">        = 0: successful exit
</span><span class="comment">*</span><span class="comment">        &lt; 0: if INFO = -i, the i-th argument had an illegal value
</span><span class="comment">*</span><span class="comment">        &gt; 0: the algorithm failed
</span><span class="comment">*</span><span class="comment">             = 1, a split was marked by a positive value in E
</span><span class="comment">*</span><span class="comment">             = 2, current block of Z not diagonalized after 30*N
</span><span class="comment">*</span><span class="comment">                  iterations (in inner while loop)
</span><span class="comment">*</span><span class="comment">             = 3, termination criterion of outer while loop not met 
</span><span class="comment">*</span><span class="comment">                  (program created more than N unreduced blocks)
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  =====================================================================
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     .. Parameters ..
</span>      DOUBLE PRECISION   ZERO
      PARAMETER          ( ZERO = 0.0D0 )
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Local Scalars ..
</span>      INTEGER            I, IINFO
      DOUBLE PRECISION   EPS, SCALE, SAFMIN, SIGMN, SIGMX
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. External Subroutines ..
</span>      EXTERNAL           DCOPY, <a name="DLAS2.69"></a><a href="dlas2.f.html#DLAS2.1">DLAS2</a>, <a name="DLASCL.69"></a><a href="dlascl.f.html#DLASCL.1">DLASCL</a>, <a name="DLASQ2.69"></a><a href="dlasq2.f.html#DLASQ2.1">DLASQ2</a>, <a name="DLASRT.69"></a><a href="dlasrt.f.html#DLASRT.1">DLASRT</a>, <a name="XERBLA.69"></a><a href="xerbla.f.html#XERBLA.1">XERBLA</a>
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. External Functions ..
</span>      DOUBLE PRECISION   <a name="DLAMCH.72"></a><a href="dlamch.f.html#DLAMCH.1">DLAMCH</a>
      EXTERNAL           <a name="DLAMCH.73"></a><a href="dlamch.f.html#DLAMCH.1">DLAMCH</a>
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Intrinsic Functions ..
</span>      INTRINSIC          ABS, MAX, SQRT
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Executable Statements ..
</span><span class="comment">*</span><span class="comment">
</span>      INFO = 0
      IF( N.LT.0 ) THEN
         INFO = -2
         CALL <a name="XERBLA.83"></a><a href="xerbla.f.html#XERBLA.1">XERBLA</a>( <span class="string">'<a name="DLASQ1.83"></a><a href="dlasq1.f.html#DLASQ1.1">DLASQ1</a>'</span>, -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 <a name="DLAS2.91"></a><a href="dlas2.f.html#DLAS2.1">DLAS2</a>( D( 1 ), E( 1 ), D( 2 ), SIGMN, SIGMX )
         D( 1 ) = SIGMX
         D( 2 ) = SIGMN
         RETURN
      END IF
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     Estimate the largest singular value.
</span><span class="comment">*</span><span class="comment">
</span>      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 ) )
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     Early return if SIGMX is zero (matrix is already diagonal).
</span><span class="comment">*</span><span class="comment">
</span>      IF( SIGMX.EQ.ZERO ) THEN
         CALL <a name="DLASRT.109"></a><a href="dlasrt.f.html#DLASRT.1">DLASRT</a>( <span class="string">'D'</span>, N, D, IINFO )
         RETURN
      END IF
<span class="comment">*</span><span class="comment">
</span>      DO 20 I = 1, N
         SIGMX = MAX( SIGMX, D( I ) )
   20 CONTINUE
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     Copy D and E into WORK (in the Z format) and scale (squaring the
</span><span class="comment">*</span><span class="comment">     input data makes scaling by a power of the radix pointless).
</span><span class="comment">*</span><span class="comment">
</span>      EPS = <a name="DLAMCH.120"></a><a href="dlamch.f.html#DLAMCH.1">DLAMCH</a>( <span class="string">'Precision'</span> )
      SAFMIN = <a name="DLAMCH.121"></a><a href="dlamch.f.html#DLAMCH.1">DLAMCH</a>( <span class="string">'Safe minimum'</span> )
      SCALE = SQRT( EPS / SAFMIN )
      CALL DCOPY( N, D, 1, WORK( 1 ), 2 )
      CALL DCOPY( N-1, E, 1, WORK( 2 ), 2 )
      CALL <a name="DLASCL.125"></a><a href="dlascl.f.html#DLASCL.1">DLASCL</a>( <span class="string">'G'</span>, 0, 0, SIGMX, SCALE, 2*N-1, 1, WORK, 2*N-1,
     $             IINFO )
<span class="comment">*</span><span class="comment">         
</span><span class="comment">*</span><span class="comment">     Compute the q's and e's.
</span><span class="comment">*</span><span class="comment">
</span>      DO 30 I = 1, 2*N - 1
         WORK( I ) = WORK( I )**2
   30 CONTINUE
      WORK( 2*N ) = ZERO
<span class="comment">*</span><span class="comment">
</span>      CALL <a name="DLASQ2.135"></a><a href="dlasq2.f.html#DLASQ2.1">DLASQ2</a>( N, WORK, INFO )
<span class="comment">*</span><span class="comment">
</span>      IF( INFO.EQ.0 ) THEN
         DO 40 I = 1, N
            D( I ) = SQRT( WORK( I ) )
   40    CONTINUE
         CALL <a name="DLASCL.141"></a><a href="dlascl.f.html#DLASCL.1">DLASCL</a>( <span class="string">'G'</span>, 0, 0, SCALE, SIGMX, N, 1, D, N, IINFO )
      END IF
<span class="comment">*</span><span class="comment">
</span>      RETURN
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     End of <a name="DLASQ1.146"></a><a href="dlasq1.f.html#DLASQ1.1">DLASQ1</a>
</span><span class="comment">*</span><span class="comment">
</span>      END

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