--- rpl/lapack/lapack/dgesvj.f 2011/11/21 20:42:52 1.7
+++ rpl/lapack/lapack/dgesvj.f 2020/05/21 21:45:57 1.20
@@ -2,25 +2,25 @@
*
* =========== DOCUMENTATION ===========
*
-* Online html documentation available at
-* http://www.netlib.org/lapack/explore-html/
+* Online html documentation available at
+* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
-*> Download DGESVJ + dependencies
-*>
-*> [TGZ]
-*>
-*> [ZIP]
-*>
+*> Download DGESVJ + dependencies
+*>
+*> [TGZ]
+*>
+*> [ZIP]
+*>
*> [TXT]
-*> \endhtmlonly
+*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGESVJ( JOBA, JOBU, JOBV, M, N, A, LDA, SVA, MV, V,
* LDV, WORK, LWORK, INFO )
-*
+*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, LDV, LWORK, M, MV, N
* CHARACTER*1 JOBA, JOBU, JOBV
@@ -29,7 +29,7 @@
* DOUBLE PRECISION A( LDA, * ), SVA( N ), V( LDV, * ),
* $ WORK( LWORK )
* ..
-*
+*
*
*> \par Purpose:
* =============
@@ -45,6 +45,8 @@
*> matrix, and V is an N-by-N orthogonal matrix. The diagonal elements
*> of SIGMA are the singular values of A. The columns of U and V are the
*> left and the right singular vectors of A, respectively.
+*> DGESVJ can sometimes compute tiny singular values and their singular vectors much
+*> more accurately than other SVD routines, see below under Further Details.
*> \endverbatim
*
* Arguments:
@@ -52,7 +54,7 @@
*
*> \param[in] JOBA
*> \verbatim
-*> JOBA is CHARACTER* 1
+*> JOBA is CHARACTER*1
*> Specifies the structure of A.
*> = 'L': The input matrix A is lower triangular;
*> = 'U': The input matrix A is upper triangular;
@@ -88,20 +90,20 @@
*> JOBV is CHARACTER*1
*> Specifies whether to compute the right singular vectors, that
*> is, the matrix V:
-*> = 'V' : the matrix V is computed and returned in the array V
-*> = 'A' : the Jacobi rotations are applied to the MV-by-N
+*> = 'V': the matrix V is computed and returned in the array V
+*> = 'A': the Jacobi rotations are applied to the MV-by-N
*> array V. In other words, the right singular vector
*> matrix V is not computed explicitly, instead it is
*> applied to an MV-by-N matrix initially stored in the
*> first MV rows of V.
-*> = 'N' : the matrix V is not computed and the array V is not
+*> = 'N': the matrix V is not computed and the array V is not
*> referenced
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
-*> The number of rows of the input matrix A. 1/DLAMCH('E') > M >= 0.
+*> The number of rows of the input matrix A. 1/DLAMCH('E') > M >= 0.
*> \endverbatim
*>
*> \param[in] N
@@ -116,8 +118,8 @@
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the M-by-N matrix A.
*> On exit :
-*> If JOBU .EQ. 'U' .OR. JOBU .EQ. 'C' :
-*> If INFO .EQ. 0 :
+*> If JOBU = 'U' .OR. JOBU = 'C' :
+*> If INFO = 0 :
*> RANKA orthonormal columns of U are returned in the
*> leading RANKA columns of the array A. Here RANKA <= N
*> is the number of computed singular values of A that are
@@ -127,9 +129,9 @@
*> in the array WORK as RANKA=NINT(WORK(2)). Also see the
*> descriptions of SVA and WORK. The computed columns of U
*> are mutually numerically orthogonal up to approximately
-*> TOL=DSQRT(M)*EPS (default); or TOL=CTOL*EPS (JOBU.EQ.'C'),
+*> TOL=DSQRT(M)*EPS (default); or TOL=CTOL*EPS (JOBU = 'C'),
*> see the description of JOBU.
-*> If INFO .GT. 0 :
+*> If INFO > 0 :
*> the procedure DGESVJ did not converge in the given number
*> of iterations (sweeps). In that case, the computed
*> columns of U may not be orthogonal up to TOL. The output
@@ -138,8 +140,8 @@
*> input matrix A in the sense that the residual
*> ||A-SCALE*U*SIGMA*V^T||_2 / ||A||_2 is small.
*>
-*> If JOBU .EQ. 'N' :
-*> If INFO .EQ. 0 :
+*> If JOBU = 'N' :
+*> If INFO = 0 :
*> Note that the left singular vectors are 'for free' in the
*> one-sided Jacobi SVD algorithm. However, if only the
*> singular values are needed, the level of numerical
@@ -148,7 +150,7 @@
*> numerically orthogonal up to approximately M*EPS. Thus,
*> on exit, A contains the columns of U scaled with the
*> corresponding singular values.
-*> If INFO .GT. 0 :
+*> If INFO > 0 :
*> the procedure DGESVJ did not converge in the given number
*> of iterations (sweeps).
*> \endverbatim
@@ -163,9 +165,9 @@
*> \verbatim
*> SVA is DOUBLE PRECISION array, dimension (N)
*> On exit :
-*> If INFO .EQ. 0 :
+*> If INFO = 0 :
*> depending on the value SCALE = WORK(1), we have:
-*> If SCALE .EQ. ONE :
+*> If SCALE = ONE :
*> SVA(1:N) contains the computed singular values of A.
*> During the computation SVA contains the Euclidean column
*> norms of the iterated matrices in the array A.
@@ -173,7 +175,7 @@
*> The singular values of A are SCALE*SVA(1:N), and this
*> factored representation is due to the fact that some of the
*> singular values of A might underflow or overflow.
-*> If INFO .GT. 0 :
+*> If INFO > 0 :
*> the procedure DGESVJ did not converge in the given number of
*> iterations (sweeps) and SCALE*SVA(1:N) may not be accurate.
*> \endverbatim
@@ -181,7 +183,7 @@
*> \param[in] MV
*> \verbatim
*> MV is INTEGER
-*> If JOBV .EQ. 'A', then the product of Jacobi rotations in DGESVJ
+*> If JOBV = 'A', then the product of Jacobi rotations in DGESVJ
*> is applied to the first MV rows of V. See the description of JOBV.
*> \endverbatim
*>
@@ -199,16 +201,16 @@
*> \param[in] LDV
*> \verbatim
*> LDV is INTEGER
-*> The leading dimension of the array V, LDV .GE. 1.
-*> If JOBV .EQ. 'V', then LDV .GE. max(1,N).
-*> If JOBV .EQ. 'A', then LDV .GE. max(1,MV) .
+*> The leading dimension of the array V, LDV >= 1.
+*> If JOBV = 'V', then LDV >= max(1,N).
+*> If JOBV = 'A', then LDV >= max(1,MV) .
*> \endverbatim
*>
*> \param[in,out] WORK
*> \verbatim
-*> WORK is DOUBLE PRECISION array, dimension max(4,M+N).
+*> WORK is DOUBLE PRECISION array, dimension (LWORK)
*> On entry :
-*> If JOBU .EQ. 'C' :
+*> If JOBU = 'C' :
*> WORK(1) = CTOL, where CTOL defines the threshold for convergence.
*> The process stops if all columns of A are mutually
*> orthogonal up to CTOL*EPS, EPS=DLAMCH('E').
@@ -228,7 +230,7 @@
*> WORK(5) = max_{i.NE.j} |COS(A(:,i),A(:,j))| in the last sweep.
*> This is useful information in cases when DGESVJ did
*> not converge, as it can be used to estimate whether
-*> the output is stil useful and for post festum analysis.
+*> the output is still useful and for post festum analysis.
*> WORK(6) = the largest absolute value over all sines of the
*> Jacobi rotation angles in the last sweep. It can be
*> useful for a post festum analysis.
@@ -243,9 +245,9 @@
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
-*> = 0 : successful exit.
-*> < 0 : if INFO = -i, then the i-th argument had an illegal value
-*> > 0 : DGESVJ did not converge in the maximal allowed number (30)
+*> = 0: successful exit.
+*> < 0: if INFO = -i, then the i-th argument had an illegal value
+*> > 0: DGESVJ did not converge in the maximal allowed number (30)
*> of sweeps. The output may still be useful. See the
*> description of WORK.
*> \endverbatim
@@ -253,12 +255,12 @@
* Authors:
* ========
*
-*> \author Univ. of Tennessee
-*> \author Univ. of California Berkeley
-*> \author Univ. of Colorado Denver
-*> \author NAG Ltd.
+*> \author Univ. of Tennessee
+*> \author Univ. of California Berkeley
+*> \author Univ. of Colorado Denver
+*> \author NAG Ltd.
*
-*> \date November 2011
+*> \date June 2017
*
*> \ingroup doubleGEcomputational
*
@@ -335,10 +337,10 @@
SUBROUTINE DGESVJ( JOBA, JOBU, JOBV, M, N, A, LDA, SVA, MV, V,
$ LDV, WORK, LWORK, INFO )
*
-* -- LAPACK computational routine (version 3.4.0) --
+* -- LAPACK computational routine (version 3.7.1) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
-* November 2011
+* June 2017
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, LDV, LWORK, M, MV, N
@@ -352,9 +354,8 @@
* =====================================================================
*
* .. Local Parameters ..
- DOUBLE PRECISION ZERO, HALF, ONE, TWO
- PARAMETER ( ZERO = 0.0D0, HALF = 0.5D0, ONE = 1.0D0,
- $ TWO = 2.0D0 )
+ DOUBLE PRECISION ZERO, HALF, ONE
+ PARAMETER ( ZERO = 0.0D0, HALF = 0.5D0, ONE = 1.0D0)
INTEGER NSWEEP
PARAMETER ( NSWEEP = 30 )
* ..
@@ -375,7 +376,7 @@
DOUBLE PRECISION FASTR( 5 )
* ..
* .. Intrinsic Functions ..
- INTRINSIC DABS, DMAX1, DMIN1, DBLE, MIN0, DSIGN, DSQRT
+ INTRINSIC DABS, MAX, MIN, DBLE, DSIGN, DSQRT
* ..
* .. External Functions ..
* ..
@@ -429,7 +430,7 @@
INFO = -11
ELSE IF( UCTOL .AND. ( WORK( 1 ).LE.ONE ) ) THEN
INFO = -12
- ELSE IF( LWORK.LT.MAX0( M+N, 6 ) ) THEN
+ ELSE IF( LWORK.LT.MAX( M+N, 6 ) ) THEN
INFO = -13
ELSE
INFO = 0
@@ -595,8 +596,8 @@
AAPP = ZERO
AAQQ = BIG
DO 4781 p = 1, N
- IF( SVA( p ).NE.ZERO )AAQQ = DMIN1( AAQQ, SVA( p ) )
- AAPP = DMAX1( AAPP, SVA( p ) )
+ IF( SVA( p ).NE.ZERO )AAQQ = MIN( AAQQ, SVA( p ) )
+ AAPP = MAX( AAPP, SVA( p ) )
4781 CONTINUE
*
* #:) Quick return for zero matrix
@@ -637,19 +638,19 @@
TEMP1 = DSQRT( BIG / DBLE( N ) )
IF( ( AAPP.LE.SN ) .OR. ( AAQQ.GE.TEMP1 ) .OR.
$ ( ( SN.LE.AAQQ ) .AND. ( AAPP.LE.TEMP1 ) ) ) THEN
- TEMP1 = DMIN1( BIG, TEMP1 / AAPP )
+ TEMP1 = MIN( BIG, TEMP1 / AAPP )
* AAQQ = AAQQ*TEMP1
* AAPP = AAPP*TEMP1
ELSE IF( ( AAQQ.LE.SN ) .AND. ( AAPP.LE.TEMP1 ) ) THEN
- TEMP1 = DMIN1( SN / AAQQ, BIG / ( AAPP*DSQRT( DBLE( N ) ) ) )
+ TEMP1 = MIN( SN / AAQQ, BIG / ( AAPP*DSQRT( DBLE( N ) ) ) )
* AAQQ = AAQQ*TEMP1
* AAPP = AAPP*TEMP1
ELSE IF( ( AAQQ.GE.SN ) .AND. ( AAPP.GE.TEMP1 ) ) THEN
- TEMP1 = DMAX1( SN / AAQQ, TEMP1 / AAPP )
+ TEMP1 = MAX( SN / AAQQ, TEMP1 / AAPP )
* AAQQ = AAQQ*TEMP1
* AAPP = AAPP*TEMP1
ELSE IF( ( AAQQ.LE.SN ) .AND. ( AAPP.GE.TEMP1 ) ) THEN
- TEMP1 = DMIN1( SN / AAQQ, BIG / ( DSQRT( DBLE( N ) )*AAPP ) )
+ TEMP1 = MIN( SN / AAQQ, BIG / ( DSQRT( DBLE( N ) )*AAPP ) )
* AAQQ = AAQQ*TEMP1
* AAPP = AAPP*TEMP1
ELSE
@@ -690,7 +691,7 @@
* The boundaries are determined dynamically, based on the number of
* pivots above a threshold.
*
- KBL = MIN0( 8, N )
+ KBL = MIN( 8, N )
*[TP] KBL is a tuning parameter that defines the tile size in the
* tiling of the p-q loops of pivot pairs. In general, an optimal
* value of KBL depends on the matrix dimensions and on the
@@ -702,7 +703,7 @@
BLSKIP = KBL**2
*[TP] BLKSKIP is a tuning parameter that depends on SWBAND and KBL.
*
- ROWSKIP = MIN0( 5, KBL )
+ ROWSKIP = MIN( 5, KBL )
*[TP] ROWSKIP is a tuning parameter.
*
LKAHEAD = 1
@@ -713,7 +714,7 @@
* invokes cubic convergence. Big part of this cycle is done inside
* canonical subspaces of dimensions less than M.
*
- IF( ( LOWER .OR. UPPER ) .AND. ( N.GT.MAX0( 64, 4*KBL ) ) ) THEN
+ IF( ( LOWER .OR. UPPER ) .AND. ( N.GT.MAX( 64, 4*KBL ) ) ) THEN
*[TP] The number of partition levels and the actual partition are
* tuning parameters.
N4 = N / 4
@@ -811,11 +812,11 @@
*
igl = ( ibr-1 )*KBL + 1
*
- DO 1002 ir1 = 0, MIN0( LKAHEAD, NBL-ibr )
+ DO 1002 ir1 = 0, MIN( LKAHEAD, NBL-ibr )
*
igl = igl + ir1*KBL
*
- DO 2001 p = igl, MIN0( igl+KBL-1, N-1 )
+ DO 2001 p = igl, MIN( igl+KBL-1, N-1 )
*
* .. de Rijk's pivoting
*
@@ -864,7 +865,7 @@
*
PSKIPPED = 0
*
- DO 2002 q = p + 1, MIN0( igl+KBL-1, N )
+ DO 2002 q = p + 1, MIN( igl+KBL-1, N )
*
AAQQ = SVA( q )
*
@@ -903,7 +904,7 @@
END IF
END IF
*
- MXAAPQ = DMAX1( MXAAPQ, DABS( AAPQ ) )
+ MXAAPQ = MAX( MXAAPQ, DABS( AAPQ ) )
*
* TO rotate or NOT to rotate, THAT is the question ...
*
@@ -936,11 +937,11 @@
$ V( 1, p ), 1,
$ V( 1, q ), 1,
$ FASTR )
- SVA( q ) = AAQQ*DSQRT( DMAX1( ZERO,
+ SVA( q ) = AAQQ*DSQRT( MAX( ZERO,
$ ONE+T*APOAQ*AAPQ ) )
- AAPP = AAPP*DSQRT( DMAX1( ZERO,
+ AAPP = AAPP*DSQRT( MAX( ZERO,
$ ONE-T*AQOAP*AAPQ ) )
- MXSINJ = DMAX1( MXSINJ, DABS( T ) )
+ MXSINJ = MAX( MXSINJ, DABS( T ) )
*
ELSE
*
@@ -952,10 +953,10 @@
CS = DSQRT( ONE / ( ONE+T*T ) )
SN = T*CS
*
- MXSINJ = DMAX1( MXSINJ, DABS( SN ) )
- SVA( q ) = AAQQ*DSQRT( DMAX1( ZERO,
+ MXSINJ = MAX( MXSINJ, DABS( SN ) )
+ SVA( q ) = AAQQ*DSQRT( MAX( ZERO,
$ ONE+T*APOAQ*AAPQ ) )
- AAPP = AAPP*DSQRT( DMAX1( ZERO,
+ AAPP = AAPP*DSQRT( MAX( ZERO,
$ ONE-T*AQOAP*AAPQ ) )
*
APOAQ = WORK( p ) / WORK( q )
@@ -1069,9 +1070,9 @@
$ A( 1, q ), 1 )
CALL DLASCL( 'G', 0, 0, ONE, AAQQ, M,
$ 1, A( 1, q ), LDA, IERR )
- SVA( q ) = AAQQ*DSQRT( DMAX1( ZERO,
+ SVA( q ) = AAQQ*DSQRT( MAX( ZERO,
$ ONE-AAPQ*AAPQ ) )
- MXSINJ = DMAX1( MXSINJ, SFMIN )
+ MXSINJ = MAX( MXSINJ, SFMIN )
END IF
* END IF ROTOK THEN ... ELSE
*
@@ -1137,7 +1138,7 @@
ELSE
SVA( p ) = AAPP
IF( ( ir1.EQ.0 ) .AND. ( AAPP.EQ.ZERO ) )
- $ NOTROT = NOTROT + MIN0( igl+KBL-1, N ) - p
+ $ NOTROT = NOTROT + MIN( igl+KBL-1, N ) - p
END IF
*
2001 CONTINUE
@@ -1157,14 +1158,14 @@
* doing the block at ( ibr, jbc )
*
IJBLSK = 0
- DO 2100 p = igl, MIN0( igl+KBL-1, N )
+ DO 2100 p = igl, MIN( igl+KBL-1, N )
*
AAPP = SVA( p )
IF( AAPP.GT.ZERO ) THEN
*
PSKIPPED = 0
*
- DO 2200 q = jgl, MIN0( jgl+KBL-1, N )
+ DO 2200 q = jgl, MIN( jgl+KBL-1, N )
*
AAQQ = SVA( q )
IF( AAQQ.GT.ZERO ) THEN
@@ -1214,7 +1215,7 @@
END IF
END IF
*
- MXAAPQ = DMAX1( MXAAPQ, DABS( AAPQ ) )
+ MXAAPQ = MAX( MXAAPQ, DABS( AAPQ ) )
*
* TO rotate or NOT to rotate, THAT is the question ...
*
@@ -1242,11 +1243,11 @@
$ V( 1, p ), 1,
$ V( 1, q ), 1,
$ FASTR )
- SVA( q ) = AAQQ*DSQRT( DMAX1( ZERO,
+ SVA( q ) = AAQQ*DSQRT( MAX( ZERO,
$ ONE+T*APOAQ*AAPQ ) )
- AAPP = AAPP*DSQRT( DMAX1( ZERO,
+ AAPP = AAPP*DSQRT( MAX( ZERO,
$ ONE-T*AQOAP*AAPQ ) )
- MXSINJ = DMAX1( MXSINJ, DABS( T ) )
+ MXSINJ = MAX( MXSINJ, DABS( T ) )
ELSE
*
* .. choose correct signum for THETA and rotate
@@ -1257,10 +1258,10 @@
$ DSQRT( ONE+THETA*THETA ) )
CS = DSQRT( ONE / ( ONE+T*T ) )
SN = T*CS
- MXSINJ = DMAX1( MXSINJ, DABS( SN ) )
- SVA( q ) = AAQQ*DSQRT( DMAX1( ZERO,
+ MXSINJ = MAX( MXSINJ, DABS( SN ) )
+ SVA( q ) = AAQQ*DSQRT( MAX( ZERO,
$ ONE+T*APOAQ*AAPQ ) )
- AAPP = AAPP*DSQRT( DMAX1( ZERO,
+ AAPP = AAPP*DSQRT( MAX( ZERO,
$ ONE-T*AQOAP*AAPQ ) )
*
APOAQ = WORK( p ) / WORK( q )
@@ -1377,9 +1378,9 @@
CALL DLASCL( 'G', 0, 0, ONE, AAQQ,
$ M, 1, A( 1, q ), LDA,
$ IERR )
- SVA( q ) = AAQQ*DSQRT( DMAX1( ZERO,
+ SVA( q ) = AAQQ*DSQRT( MAX( ZERO,
$ ONE-AAPQ*AAPQ ) )
- MXSINJ = DMAX1( MXSINJ, SFMIN )
+ MXSINJ = MAX( MXSINJ, SFMIN )
ELSE
CALL DCOPY( M, A( 1, q ), 1,
$ WORK( N+1 ), 1 )
@@ -1395,9 +1396,9 @@
CALL DLASCL( 'G', 0, 0, ONE, AAPP,
$ M, 1, A( 1, p ), LDA,
$ IERR )
- SVA( p ) = AAPP*DSQRT( DMAX1( ZERO,
+ SVA( p ) = AAPP*DSQRT( MAX( ZERO,
$ ONE-AAPQ*AAPQ ) )
- MXSINJ = DMAX1( MXSINJ, SFMIN )
+ MXSINJ = MAX( MXSINJ, SFMIN )
END IF
END IF
* END IF ROTOK THEN ... ELSE
@@ -1467,7 +1468,7 @@
ELSE
*
IF( AAPP.EQ.ZERO )NOTROT = NOTROT +
- $ MIN0( jgl+KBL-1, N ) - jgl + 1
+ $ MIN( jgl+KBL-1, N ) - jgl + 1
IF( AAPP.LT.ZERO )NOTROT = 0
*
END IF
@@ -1478,7 +1479,7 @@
* end of the jbc-loop
2011 CONTINUE
*2011 bailed out of the jbc-loop
- DO 2012 p = igl, MIN0( igl+KBL-1, N )
+ DO 2012 p = igl, MIN( igl+KBL-1, N )
SVA( p ) = DABS( SVA( p ) )
2012 CONTINUE
***
@@ -1574,11 +1575,11 @@
END IF
*
* Undo scaling, if necessary (and possible).
- IF( ( ( SKL.GT.ONE ) .AND. ( SVA( 1 ).LT.( BIG /
- $ SKL) ) ) .OR. ( ( SKL.LT.ONE ) .AND. ( SVA( N2 ).GT.
+ IF( ( ( SKL.GT.ONE ) .AND. ( SVA( 1 ).LT.( BIG / SKL) ) )
+ $ .OR. ( ( SKL.LT.ONE ) .AND. ( SVA( MAX( N2, 1 ) ) .GT.
$ ( SFMIN / SKL) ) ) ) THEN
DO 2400 p = 1, N
- SVA( p ) = SKL*SVA( p )
+ SVA( P ) = SKL*SVA( P )
2400 CONTINUE
SKL= ONE
END IF