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Update LAPACK to 3.9.0

This commit is contained in:
Martin Kroeker 2019-12-29 23:10:31 +01:00 committed by GitHub
parent 688af253bf
commit a421ab9ce2
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GPG Key ID: 4AEE18F83AFDEB23
45 changed files with 1756 additions and 261 deletions
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2
lapack-netlib/SRC/dlaed4.f
View File

@ -82,7 +82,7 @@
*> \param[out] DELTA
*> \verbatim
*> DELTA is DOUBLE PRECISION array, dimension (N)
*> If N .GT. 2, DELTA contains (D(j) - lambda_I) in its j-th
*> If N > 2, DELTA contains (D(j) - lambda_I) in its j-th
*> component. If N = 1, then DELTA(1) = 1. If N = 2, see DLAED5
*> for detail. The vector DELTA contains the information necessary
*> to construct the eigenvectors by DLAED3 and DLAED9.

2
lapack-netlib/SRC/dlaed8.f
View File

@ -353,7 +353,7 @@
Z( I ) = W( INDX( I ) )
40 CONTINUE
*
* Calculate the allowable deflation tolerence
* Calculate the allowable deflation tolerance
*
IMAX = IDAMAX( N, Z, 1 )
JMAX = IDAMAX( N, D, 1 )

4
lapack-netlib/SRC/dlagtf.f
View File

@ -125,7 +125,7 @@
*> then IN(k) = 1, otherwise IN(k) = 0. The element IN(n)
*> returns the smallest positive integer j such that
*>
*> abs( u(j,j) ).le. norm( (T - lambda*I)(j) )*TOL,
*> abs( u(j,j) ) <= norm( (T - lambda*I)(j) )*TOL,
*>
*> where norm( A(j) ) denotes the sum of the absolute values of
*> the jth row of the matrix A. If no such j exists then IN(n)
@ -138,7 +138,7 @@
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> .lt. 0: if INFO = -k, the kth argument had an illegal value
*> < 0: if INFO = -k, the kth argument had an illegal value
*> \endverbatim
*
* Authors:

8
lapack-netlib/SRC/dlagts.f
View File

@ -122,12 +122,12 @@
*> \param[in,out] TOL
*> \verbatim
*> TOL is DOUBLE PRECISION
*> On entry, with JOB .lt. 0, TOL should be the minimum
*> On entry, with JOB < 0, TOL should be the minimum
*> perturbation to be made to very small diagonal elements of U.
*> TOL should normally be chosen as about eps*norm(U), where eps
*> is the relative machine precision, but if TOL is supplied as
*> non-positive, then it is reset to eps*max( abs( u(i,j) ) ).
*> If JOB .gt. 0 then TOL is not referenced.
*> If JOB > 0 then TOL is not referenced.
*>
*> On exit, TOL is changed as described above, only if TOL is
*> non-positive on entry. Otherwise TOL is unchanged.
@ -137,8 +137,8 @@
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> .lt. 0: if INFO = -i, the i-th argument had an illegal value
*> .gt. 0: overflow would occur when computing the INFO(th)
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: overflow would occur when computing the INFO(th)
*> element of the solution vector x. This can only occur
*> when JOB is supplied as positive and either means
*> that a diagonal element of U is very small, or that

12
lapack-netlib/SRC/dlahqr.f
View File

@ -151,25 +151,25 @@
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> .GT. 0: If INFO = i, DLAHQR failed to compute all the
*> > 0: If INFO = i, DLAHQR failed to compute all the
*> eigenvalues ILO to IHI in a total of 30 iterations
*> per eigenvalue; elements i+1:ihi of WR and WI
*> contain those eigenvalues which have been
*> successfully computed.
*>
*> If INFO .GT. 0 and WANTT is .FALSE., then on exit,
*> If INFO > 0 and WANTT is .FALSE., then on exit,
*> the remaining unconverged eigenvalues are the
*> eigenvalues of the upper Hessenberg matrix rows
*> and columns ILO thorugh INFO of the final, output
*> and columns ILO through INFO of the final, output
*> value of H.
*>
*> If INFO .GT. 0 and WANTT is .TRUE., then on exit
*> If INFO > 0 and WANTT is .TRUE., then on exit
*> (*) (initial value of H)*U = U*(final value of H)
*> where U is an orthognal matrix. The final
*> where U is an orthogonal matrix. The final
*> value of H is upper Hessenberg and triangular in
*> rows and columns INFO+1 through IHI.
*>
*> If INFO .GT. 0 and WANTZ is .TRUE., then on exit
*> If INFO > 0 and WANTZ is .TRUE., then on exit
*> (final value of Z) = (initial value of Z)*U
*> where U is the orthogonal matrix in (*)
*> (regardless of the value of WANTT.)

2
lapack-netlib/SRC/dlaln2.f
View File

@ -49,7 +49,7 @@
*> the first column of each being the real part and the second
*> being the imaginary part.
*>
*> "s" is a scaling factor (.LE. 1), computed by DLALN2, which is
*> "s" is a scaling factor (<= 1), computed by DLALN2, which is
*> so chosen that X can be computed without overflow. X is further
*> scaled if necessary to assure that norm(ca A - w D)*norm(X) is less
*> than overflow.

1
lapack-netlib/SRC/dlamswlq.f
View File

@ -1,3 +1,4 @@
*> \brief \b DLAMSWLQ
*
* Definition:
* ===========

1
lapack-netlib/SRC/dlamtsqr.f
View File

@ -1,3 +1,4 @@
*> \brief \b DLAMTSQR
*
* Definition:
* ===========

26
lapack-netlib/SRC/dlangb.f
View File

@ -129,6 +129,7 @@
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* December 2016
*
IMPLICIT NONE
* .. Scalar Arguments ..
CHARACTER NORM
INTEGER KL, KU, LDAB, N
@ -139,22 +140,24 @@
*
* =====================================================================
*
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, J, K, L
DOUBLE PRECISION SCALE, SUM, VALUE, TEMP
DOUBLE PRECISION SUM, VALUE, TEMP
* ..
* .. External Subroutines ..
EXTERNAL DLASSQ
* .. Local Arrays ..
DOUBLE PRECISION SSQ( 2 ), COLSSQ( 2 )
* ..
* .. External Functions ..
LOGICAL LSAME, DISNAN
EXTERNAL LSAME, DISNAN
* ..
* .. External Subroutines ..
EXTERNAL DLASSQ, DCOMBSSQ
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN, SQRT
* ..
@ -206,15 +209,22 @@
ELSE IF( ( LSAME( NORM, 'F' ) ) .OR. ( LSAME( NORM, 'E' ) ) ) THEN
*
* Find normF(A).
* SSQ(1) is scale
* SSQ(2) is sum-of-squares
* For better accuracy, sum each column separately.
*
SCALE = ZERO
SUM = ONE
SSQ( 1 ) = ZERO
SSQ( 2 ) = ONE
DO 90 J = 1, N
L = MAX( 1, J-KU )
K = KU + 1 - J + L
CALL DLASSQ( MIN( N, J+KL )-L+1, AB( K, J ), 1, SCALE, SUM )
COLSSQ( 1 ) = ZERO
COLSSQ( 2 ) = ONE
CALL DLASSQ( MIN( N, J+KL )-L+1, AB( K, J ), 1,
$ COLSSQ( 1 ), COLSSQ( 2 ) )
CALL DCOMBSSQ( SSQ, COLSSQ )
90 CONTINUE
VALUE = SCALE*SQRT( SUM )
VALUE = SSQ( 1 )*SQRT( SSQ( 2 ) )
END IF
*
DLANGB = VALUE

22
lapack-netlib/SRC/dlange.f
View File

@ -119,6 +119,7 @@
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* December 2016
*
IMPLICIT NONE
* .. Scalar Arguments ..
CHARACTER NORM
INTEGER LDA, M, N
@ -135,10 +136,13 @@
* ..
* .. Local Scalars ..
INTEGER I, J
DOUBLE PRECISION SCALE, SUM, VALUE, TEMP
DOUBLE PRECISION SUM, VALUE, TEMP
* ..
* .. Local Arrays ..
DOUBLE PRECISION SSQ( 2 ), COLSSQ( 2 )
* ..
* .. External Subroutines ..
EXTERNAL DLASSQ
EXTERNAL DLASSQ, DCOMBSSQ
* ..
* .. External Functions ..
LOGICAL LSAME, DISNAN
@ -194,13 +198,19 @@
ELSE IF( ( LSAME( NORM, 'F' ) ) .OR. ( LSAME( NORM, 'E' ) ) ) THEN
*
* Find normF(A).
* SSQ(1) is scale
* SSQ(2) is sum-of-squares
* For better accuracy, sum each column separately.
*
SCALE = ZERO
SUM = ONE
SSQ( 1 ) = ZERO
SSQ( 2 ) = ONE
DO 90 J = 1, N
CALL DLASSQ( M, A( 1, J ), 1, SCALE, SUM )
COLSSQ( 1 ) = ZERO
COLSSQ( 2 ) = ONE
CALL DLASSQ( M, A( 1, J ), 1, COLSSQ( 1 ), COLSSQ( 2 ) )
CALL DCOMBSSQ( SSQ, COLSSQ )
90 CONTINUE
VALUE = SCALE*SQRT( SUM )
VALUE = SSQ( 1 )*SQRT( SSQ( 2 ) )
END IF
*
DLANGE = VALUE

25
lapack-netlib/SRC/dlanhs.f
View File

@ -113,6 +113,7 @@
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* December 2016
*
IMPLICIT NONE
* .. Scalar Arguments ..
CHARACTER NORM
INTEGER LDA, N
@ -129,15 +130,18 @@
* ..
* .. Local Scalars ..
INTEGER I, J
DOUBLE PRECISION SCALE, SUM, VALUE
DOUBLE PRECISION SUM, VALUE
* ..
* .. External Subroutines ..
EXTERNAL DLASSQ
* .. Local Arrays ..
DOUBLE PRECISION SSQ( 2 ), COLSSQ( 2 )
* ..
* .. External Functions ..
LOGICAL LSAME, DISNAN
EXTERNAL LSAME, DISNAN
* ..
* .. External Subroutines ..
EXTERNAL DLASSQ, DCOMBSSQ
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MIN, SQRT
* ..
@ -188,13 +192,20 @@
ELSE IF( ( LSAME( NORM, 'F' ) ) .OR. ( LSAME( NORM, 'E' ) ) ) THEN
*
* Find normF(A).
* SSQ(1) is scale
* SSQ(2) is sum-of-squares
* For better accuracy, sum each column separately.
*
SCALE = ZERO
SUM = ONE
SSQ( 1 ) = ZERO
SSQ( 2 ) = ONE
DO 90 J = 1, N
CALL DLASSQ( MIN( N, J+1 ), A( 1, J ), 1, SCALE, SUM )
COLSSQ( 1 ) = ZERO
COLSSQ( 2 ) = ONE
CALL DLASSQ( MIN( N, J+1 ), A( 1, J ), 1,
$ COLSSQ( 1 ), COLSSQ( 2 ) )
CALL DCOMBSSQ( SSQ, COLSSQ )
90 CONTINUE
VALUE = SCALE*SQRT( SUM )
VALUE = SSQ( 1 )*SQRT( SSQ( 2 ) )
END IF
*
DLANHS = VALUE

44
lapack-netlib/SRC/dlansb.f
View File

@ -134,6 +134,7 @@
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* December 2016
*
IMPLICIT NONE
* .. Scalar Arguments ..
CHARACTER NORM, UPLO
INTEGER K, LDAB, N
@ -150,15 +151,18 @@
* ..
* .. Local Scalars ..
INTEGER I, J, L
DOUBLE PRECISION ABSA, SCALE, SUM, VALUE
DOUBLE PRECISION ABSA, SUM, VALUE
* ..
* .. External Subroutines ..
EXTERNAL DLASSQ
* .. Local Arrays ..
DOUBLE PRECISION SSQ( 2 ), COLSSQ( 2 )
* ..
* .. External Functions ..
LOGICAL LSAME, DISNAN
EXTERNAL LSAME, DISNAN
* ..
* .. External Subroutines ..
EXTERNAL DLASSQ, DCOMBSSQ
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN, SQRT
* ..
@ -225,29 +229,47 @@
ELSE IF( ( LSAME( NORM, 'F' ) ) .OR. ( LSAME( NORM, 'E' ) ) ) THEN
*
* Find normF(A).
* SSQ(1) is scale
* SSQ(2) is sum-of-squares
* For better accuracy, sum each column separately.
*
SSQ( 1 ) = ZERO
SSQ( 2 ) = ONE
*
* Sum off-diagonals
*
SCALE = ZERO
SUM = ONE
IF( K.GT.0 ) THEN
IF( LSAME( UPLO, 'U' ) ) THEN
DO 110 J = 2, N
COLSSQ( 1 ) = ZERO
COLSSQ( 2 ) = ONE
CALL DLASSQ( MIN( J-1, K ), AB( MAX( K+2-J, 1 ), J ),
$ 1, SCALE, SUM )
$ 1, COLSSQ( 1 ), COLSSQ( 2 ) )
CALL DCOMBSSQ( SSQ, COLSSQ )
110 CONTINUE
L = K + 1
ELSE
DO 120 J = 1, N - 1
CALL DLASSQ( MIN( N-J, K ), AB( 2, J ), 1, SCALE,
$ SUM )
COLSSQ( 1 ) = ZERO
COLSSQ( 2 ) = ONE
CALL DLASSQ( MIN( N-J, K ), AB( 2, J ), 1,
$ COLSSQ( 1 ), COLSSQ( 2 ) )
CALL DCOMBSSQ( SSQ, COLSSQ )
120 CONTINUE
L = 1
END IF
SUM = 2*SUM
SSQ( 2 ) = 2*SSQ( 2 )
ELSE
L = 1
END IF
CALL DLASSQ( N, AB( L, 1 ), LDAB, SCALE, SUM )
VALUE = SCALE*SQRT( SUM )
*
* Sum diagonal
*
COLSSQ( 1 ) = ZERO
COLSSQ( 2 ) = ONE
CALL DLASSQ( N, AB( L, 1 ), LDAB, COLSSQ( 1 ), COLSSQ( 2 ) )
CALL DCOMBSSQ( SSQ, COLSSQ )
VALUE = SSQ( 1 )*SQRT( SSQ( 2 ) )
END IF
*
DLANSB = VALUE

48
lapack-netlib/SRC/dlansp.f
View File

@ -119,6 +119,7 @@
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* December 2016
*
IMPLICIT NONE
* .. Scalar Arguments ..
CHARACTER NORM, UPLO
INTEGER N
@ -135,15 +136,18 @@
* ..
* .. Local Scalars ..
INTEGER I, J, K
DOUBLE PRECISION ABSA, SCALE, SUM, VALUE
DOUBLE PRECISION ABSA, SUM, VALUE
* ..
* .. External Subroutines ..
EXTERNAL DLASSQ
* .. Local Arrays ..
DOUBLE PRECISION SSQ( 2 ), COLSSQ( 2 )
* ..
* .. External Functions ..
LOGICAL LSAME, DISNAN
EXTERNAL LSAME, DISNAN
* ..
* .. External Subroutines ..
EXTERNAL DLASSQ, DCOMBSSQ
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, SQRT
* ..
@ -217,31 +221,48 @@
ELSE IF( ( LSAME( NORM, 'F' ) ) .OR. ( LSAME( NORM, 'E' ) ) ) THEN
*
* Find normF(A).
* SSQ(1) is scale
* SSQ(2) is sum-of-squares
* For better accuracy, sum each column separately.
*
SSQ( 1 ) = ZERO
SSQ( 2 ) = ONE
*
* Sum off-diagonals
*
SCALE = ZERO
SUM = ONE
K = 2
IF( LSAME( UPLO, 'U' ) ) THEN
DO 110 J = 2, N
CALL DLASSQ( J-1, AP( K ), 1, SCALE, SUM )
COLSSQ( 1 ) = ZERO
COLSSQ( 2 ) = ONE
CALL DLASSQ( J-1, AP( K ), 1, COLSSQ( 1 ), COLSSQ( 2 ) )
CALL DCOMBSSQ( SSQ, COLSSQ )
K = K + J
110 CONTINUE
ELSE
DO 120 J = 1, N - 1
CALL DLASSQ( N-J, AP( K ), 1, SCALE, SUM )
COLSSQ( 1 ) = ZERO
COLSSQ( 2 ) = ONE
CALL DLASSQ( N-J, AP( K ), 1, COLSSQ( 1 ), COLSSQ( 2 ) )
CALL DCOMBSSQ( SSQ, COLSSQ )
K = K + N - J + 1
120 CONTINUE
END IF
SUM = 2*SUM
SSQ( 2 ) = 2*SSQ( 2 )
*
* Sum diagonal
*
K = 1
COLSSQ( 1 ) = ZERO
COLSSQ( 2 ) = ONE
DO 130 I = 1, N
IF( AP( K ).NE.ZERO ) THEN
ABSA = ABS( AP( K ) )
IF( SCALE.LT.ABSA ) THEN
SUM = ONE + SUM*( SCALE / ABSA )**2
SCALE = ABSA
IF( COLSSQ( 1 ).LT.ABSA ) THEN
COLSSQ( 2 ) = ONE + COLSSQ(2)*( COLSSQ(1) / ABSA )**2
COLSSQ( 1 ) = ABSA
ELSE
SUM = SUM + ( ABSA / SCALE )**2
COLSSQ( 2 ) = COLSSQ( 2 ) + ( ABSA / COLSSQ( 1 ) )**2
END IF
END IF
IF( LSAME( UPLO, 'U' ) ) THEN
@ -250,7 +271,8 @@
K = K + N - I + 1
END IF
130 CONTINUE
VALUE = SCALE*SQRT( SUM )
CALL DCOMBSSQ( SSQ, COLSSQ )
VALUE = SSQ( 1 )*SQRT( SSQ( 2 ) )
END IF
*
DLANSP = VALUE

42
lapack-netlib/SRC/dlansy.f
View File

@ -127,6 +127,7 @@
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* December 2016
*
IMPLICIT NONE
* .. Scalar Arguments ..
CHARACTER NORM, UPLO
INTEGER LDA, N
@ -143,15 +144,18 @@
* ..
* .. Local Scalars ..
INTEGER I, J
DOUBLE PRECISION ABSA, SCALE, SUM, VALUE
DOUBLE PRECISION ABSA, SUM, VALUE
* ..
* .. External Subroutines ..
EXTERNAL DLASSQ
* .. Local Arrays ..
DOUBLE PRECISION SSQ( 2 ), COLSSQ( 2 )
* ..
* .. External Functions ..
LOGICAL LSAME, DISNAN
EXTERNAL LSAME, DISNAN
* ..
* .. External Subroutines ..
EXTERNAL DLASSQ, DCOMBSSQ
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, SQRT
* ..
@ -216,21 +220,39 @@
ELSE IF( ( LSAME( NORM, 'F' ) ) .OR. ( LSAME( NORM, 'E' ) ) ) THEN
*
* Find normF(A).
* SSQ(1) is scale
* SSQ(2) is sum-of-squares
* For better accuracy, sum each column separately.
*
SSQ( 1 ) = ZERO
SSQ( 2 ) = ONE
*
* Sum off-diagonals
*
SCALE = ZERO
SUM = ONE
IF( LSAME( UPLO, 'U' ) ) THEN
DO 110 J = 2, N
CALL DLASSQ( J-1, A( 1, J ), 1, SCALE, SUM )
COLSSQ( 1 ) = ZERO
COLSSQ( 2 ) = ONE
CALL DLASSQ( J-1, A( 1, J ), 1, COLSSQ(1), COLSSQ(2) )
CALL DCOMBSSQ( SSQ, COLSSQ )
110 CONTINUE
ELSE
DO 120 J = 1, N - 1
CALL DLASSQ( N-J, A( J+1, J ), 1, SCALE, SUM )
COLSSQ( 1 ) = ZERO
COLSSQ( 2 ) = ONE
CALL DLASSQ( N-J, A( J+1, J ), 1, COLSSQ(1), COLSSQ(2) )
CALL DCOMBSSQ( SSQ, COLSSQ )
120 CONTINUE
END IF
SUM = 2*SUM
CALL DLASSQ( N, A, LDA+1, SCALE, SUM )
VALUE = SCALE*SQRT( SUM )
SSQ( 2 ) = 2*SSQ( 2 )
*
* Sum diagonal
*
COLSSQ( 1 ) = ZERO
COLSSQ( 2 ) = ONE
CALL DLASSQ( N, A, LDA+1, COLSSQ( 1 ), COLSSQ( 2 ) )
CALL DCOMBSSQ( SSQ, COLSSQ )
VALUE = SSQ( 1 )*SQRT( SSQ( 2 ) )
END IF
*
DLANSY = VALUE

57
lapack-netlib/SRC/dlantb.f
View File

@ -145,6 +145,7 @@
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* December 2016
*
IMPLICIT NONE
* .. Scalar Arguments ..
CHARACTER DIAG, NORM, UPLO
INTEGER K, LDAB, N
@ -162,15 +163,18 @@
* .. Local Scalars ..
LOGICAL UDIAG
INTEGER I, J, L
DOUBLE PRECISION SCALE, SUM, VALUE
DOUBLE PRECISION SUM, VALUE
* ..
* .. External Subroutines ..
EXTERNAL DLASSQ
* .. Local Arrays ..
DOUBLE PRECISION SSQ( 2 ), COLSSQ( 2 )
* ..
* .. External Functions ..
LOGICAL LSAME, DISNAN
EXTERNAL LSAME, DISNAN
* ..
* .. External Subroutines ..
EXTERNAL DLASSQ, DCOMBSSQ
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN, SQRT
* ..
@ -311,46 +315,61 @@
ELSE IF( ( LSAME( NORM, 'F' ) ) .OR. ( LSAME( NORM, 'E' ) ) ) THEN
*
* Find normF(A).
* SSQ(1) is scale
* SSQ(2) is sum-of-squares
* For better accuracy, sum each column separately.
*
IF( LSAME( UPLO, 'U' ) ) THEN
IF( LSAME( DIAG, 'U' ) ) THEN
SCALE = ONE
SUM = N
SSQ( 1 ) = ONE
SSQ( 2 ) = N
IF( K.GT.0 ) THEN
DO 280 J = 2, N
COLSSQ( 1 ) = ZERO
COLSSQ( 2 ) = ONE
CALL DLASSQ( MIN( J-1, K ),
$ AB( MAX( K+2-J, 1 ), J ), 1, SCALE,
$ SUM )
$ AB( MAX( K+2-J, 1 ), J ), 1,
$ COLSSQ( 1 ), COLSSQ( 2 ) )
CALL DCOMBSSQ( SSQ, COLSSQ )
280 CONTINUE
END IF
ELSE
SCALE = ZERO
SUM = ONE
SSQ( 1 ) = ZERO
SSQ( 2 ) = ONE
DO 290 J = 1, N
COLSSQ( 1 ) = ZERO
COLSSQ( 2 ) = ONE
CALL DLASSQ( MIN( J, K+1 ), AB( MAX( K+2-J, 1 ), J ),
$ 1, SCALE, SUM )
$ 1, COLSSQ( 1 ), COLSSQ( 2 ) )
CALL DCOMBSSQ( SSQ, COLSSQ )
290 CONTINUE
END IF
ELSE
IF( LSAME( DIAG, 'U' ) ) THEN
SCALE = ONE
SUM = N
SSQ( 1 ) = ONE
SSQ( 2 ) = N
IF( K.GT.0 ) THEN
DO 300 J = 1, N - 1
CALL DLASSQ( MIN( N-J, K ), AB( 2, J ), 1, SCALE,
$ SUM )
COLSSQ( 1 ) = ZERO
COLSSQ( 2 ) = ONE
CALL DLASSQ( MIN( N-J, K ), AB( 2, J ), 1,
$ COLSSQ( 1 ), COLSSQ( 2 ) )
CALL DCOMBSSQ( SSQ, COLSSQ )
300 CONTINUE
END IF
ELSE
SCALE = ZERO
SUM = ONE
SSQ( 1 ) = ZERO
SSQ( 2 ) = ONE
DO 310 J = 1, N
CALL DLASSQ( MIN( N-J+1, K+1 ), AB( 1, J ), 1, SCALE,
$ SUM )
COLSSQ( 1 ) = ZERO
COLSSQ( 2 ) = ONE
CALL DLASSQ( MIN( N-J+1, K+1 ), AB( 1, J ), 1,
$ COLSSQ( 1 ), COLSSQ( 2 ) )
CALL DCOMBSSQ( SSQ, COLSSQ )
310 CONTINUE
END IF
END IF
VALUE = SCALE*SQRT( SUM )
VALUE = SSQ( 1 )*SQRT( SSQ( 2 ) )
END IF
*
DLANTB = VALUE

55
lapack-netlib/SRC/dlantp.f
View File

@ -129,6 +129,7 @@
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* December 2016
*
IMPLICIT NONE
* .. Scalar Arguments ..
CHARACTER DIAG, NORM, UPLO
INTEGER N
@ -146,15 +147,18 @@
* .. Local Scalars ..
LOGICAL UDIAG
INTEGER I, J, K
DOUBLE PRECISION SCALE, SUM, VALUE
DOUBLE PRECISION SUM, VALUE
* ..
* .. External Subroutines ..
EXTERNAL DLASSQ
* .. Local Arrays ..
DOUBLE PRECISION SSQ( 2 ), COLSSQ( 2 )
* ..
* .. External Functions ..
LOGICAL LSAME, DISNAN
EXTERNAL LSAME, DISNAN
* ..
* .. External Subroutines ..
EXTERNAL DLASSQ, DCOMBSSQ
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, SQRT
* ..
@ -306,45 +310,64 @@
ELSE IF( ( LSAME( NORM, 'F' ) ) .OR. ( LSAME( NORM, 'E' ) ) ) THEN
*
* Find normF(A).
* SSQ(1) is scale
* SSQ(2) is sum-of-squares
* For better accuracy, sum each column separately.
*
IF( LSAME( UPLO, 'U' ) ) THEN
IF( LSAME( DIAG, 'U' ) ) THEN
SCALE = ONE
SUM = N
SSQ( 1 ) = ONE
SSQ( 2 ) = N
K = 2
DO 280 J = 2, N
CALL DLASSQ( J-1, AP( K ), 1, SCALE, SUM )
COLSSQ( 1 ) = ZERO
COLSSQ( 2 ) = ONE
CALL DLASSQ( J-1, AP( K ), 1,
$ COLSSQ( 1 ), COLSSQ( 2 ) )
CALL DCOMBSSQ( SSQ, COLSSQ )
K = K + J
280 CONTINUE
ELSE
SCALE = ZERO
SUM = ONE
SSQ( 1 ) = ZERO
SSQ( 2 ) = ONE
K = 1
DO 290 J = 1, N
CALL DLASSQ( J, AP( K ), 1, SCALE, SUM )
COLSSQ( 1 ) = ZERO
COLSSQ( 2 ) = ONE
CALL DLASSQ( J, AP( K ), 1,
$ COLSSQ( 1 ), COLSSQ( 2 ) )
CALL DCOMBSSQ( SSQ, COLSSQ )
K = K + J
290 CONTINUE
END IF
ELSE
IF( LSAME( DIAG, 'U' ) ) THEN
SCALE = ONE
SUM = N
SSQ( 1 ) = ONE
SSQ( 2 ) = N
K = 2
DO 300 J = 1, N - 1
CALL DLASSQ( N-J, AP( K ), 1, SCALE, SUM )
COLSSQ( 1 ) = ZERO
COLSSQ( 2 ) = ONE
CALL DLASSQ( N-J, AP( K ), 1,
$ COLSSQ( 1 ), COLSSQ( 2 ) )
CALL DCOMBSSQ( SSQ, COLSSQ )
K = K + N - J + 1
300 CONTINUE
ELSE
SCALE = ZERO
SUM = ONE
SSQ( 1 ) = ZERO
SSQ( 2 ) = ONE
K = 1
DO 310 J = 1, N
CALL DLASSQ( N-J+1, AP( K ), 1, SCALE, SUM )
COLSSQ( 1 ) = ZERO
COLSSQ( 2 ) = ONE
CALL DLASSQ( N-J+1, AP( K ), 1,
$ COLSSQ( 1 ), COLSSQ( 2 ) )
CALL DCOMBSSQ( SSQ, COLSSQ )
K = K + N - J + 1
310 CONTINUE
END IF
END IF
VALUE = SCALE*SQRT( SUM )
VALUE = SSQ( 1 )*SQRT( SSQ( 2 ) )
END IF
*
DLANTP = VALUE

58
lapack-netlib/SRC/dlantr.f
View File

@ -146,6 +146,7 @@
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* December 2016
*
IMPLICIT NONE
* .. Scalar Arguments ..
CHARACTER DIAG, NORM, UPLO
INTEGER LDA, M, N
@ -163,15 +164,18 @@
* .. Local Scalars ..
LOGICAL UDIAG
INTEGER I, J
DOUBLE PRECISION SCALE, SUM, VALUE
DOUBLE PRECISION SUM, VALUE
* ..
* .. External Subroutines ..
EXTERNAL DLASSQ
* .. Local Arrays ..
DOUBLE PRECISION SSQ( 2 ), COLSSQ( 2 )
* ..
* .. External Functions ..
LOGICAL LSAME, DISNAN
EXTERNAL LSAME, DISNAN
* ..
* .. External Subroutines ..
EXTERNAL DLASSQ, DCOMBSSQ
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MIN, SQRT
* ..
@ -281,7 +285,7 @@
END IF
ELSE
IF( LSAME( DIAG, 'U' ) ) THEN
DO 210 I = 1, N
DO 210 I = 1, MIN( M, N )
WORK( I ) = ONE
210 CONTINUE
DO 220 I = N + 1, M
@ -311,38 +315,56 @@
ELSE IF( ( LSAME( NORM, 'F' ) ) .OR. ( LSAME( NORM, 'E' ) ) ) THEN
*
* Find normF(A).
* SSQ(1) is scale
* SSQ(2) is sum-of-squares
* For better accuracy, sum each column separately.
*
IF( LSAME( UPLO, 'U' ) ) THEN
IF( LSAME( DIAG, 'U' ) ) THEN
SCALE = ONE
SUM = MIN( M, N )
SSQ( 1 ) = ONE
SSQ( 2 ) = MIN( M, N )
DO 290 J = 2, N
CALL DLASSQ( MIN( M, J-1 ), A( 1, J ), 1, SCALE, SUM )
COLSSQ( 1 ) = ZERO
COLSSQ( 2 ) = ONE
CALL DLASSQ( MIN( M, J-1 ), A( 1, J ), 1,
$ COLSSQ( 1 ), COLSSQ( 2 ) )
CALL DCOMBSSQ( SSQ, COLSSQ )
290 CONTINUE
ELSE
SCALE = ZERO
SUM = ONE
SSQ( 1 ) = ZERO
SSQ( 2 ) = ONE
DO 300 J = 1, N
CALL DLASSQ( MIN( M, J ), A( 1, J ), 1, SCALE, SUM )
COLSSQ( 1 ) = ZERO
COLSSQ( 2 ) = ONE
CALL DLASSQ( MIN( M, J ), A( 1, J ), 1,
$ COLSSQ( 1 ), COLSSQ( 2 ) )
CALL DCOMBSSQ( SSQ, COLSSQ )
300 CONTINUE
END IF
ELSE
IF( LSAME( DIAG, 'U' ) ) THEN
SCALE = ONE
SUM = MIN( M, N )
SSQ( 1 ) = ONE
SSQ( 2 ) = MIN( M, N )
DO 310 J = 1, N
CALL DLASSQ( M-J, A( MIN( M, J+1 ), J ), 1, SCALE,
$ SUM )
COLSSQ( 1 ) = ZERO
COLSSQ( 2 ) = ONE
CALL DLASSQ( M-J, A( MIN( M, J+1 ), J ), 1,
$ COLSSQ( 1 ), COLSSQ( 2 ) )
CALL DCOMBSSQ( SSQ, COLSSQ )
310 CONTINUE
ELSE
SCALE = ZERO
SUM = ONE
SSQ( 1 ) = ZERO
SSQ( 2 ) = ONE
DO 320 J = 1, N
CALL DLASSQ( M-J+1, A( J, J ), 1, SCALE, SUM )
COLSSQ( 1 ) = ZERO
COLSSQ( 2 ) = ONE
CALL DLASSQ( M-J+1, A( J, J ), 1,
$ COLSSQ( 1 ), COLSSQ( 2 ) )
CALL DCOMBSSQ( SSQ, COLSSQ )
320 CONTINUE
END IF
END IF
VALUE = SCALE*SQRT( SUM )
VALUE = SSQ( 1 )*SQRT( SSQ( 2 ) )
END IF
*
DLANTR = VALUE

8
lapack-netlib/SRC/dlanv2.f
View File

@ -161,7 +161,6 @@
IF( C.EQ.ZERO ) THEN
CS = ONE
SN = ZERO
GO TO 10
*
ELSE IF( B.EQ.ZERO ) THEN
*
@ -174,12 +173,12 @@
A = TEMP
B = -C
C = ZERO
GO TO 10
*
ELSE IF( ( A-D ).EQ.ZERO .AND. SIGN( ONE, B ).NE.SIGN( ONE, C ) )
$ THEN
CS = ONE
SN = ZERO
GO TO 10
*
ELSE
*
TEMP = A - D
@ -207,6 +206,7 @@
SN = C / TAU
B = B - C
C = ZERO
*
ELSE
*
* Complex eigenvalues, or real (almost) equal eigenvalues.
@ -268,8 +268,6 @@
END IF
*
END IF
*
10 CONTINUE
*
* Store eigenvalues in (RT1R,RT1I) and (RT2R,RT2I).
*

248
lapack-netlib/SRC/dlaorhr_col_getrfnp.f Normal file
View File

@ -0,0 +1,248 @@
*> \brief \b DLAORHR_COL_GETRFNP
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAORHR_COL_GETRFNP + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaorhr_col_getrfnp.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaorhr_col_getrfnp.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaorhr_col_getrfnp.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAORHR_COL_GETRFNP( M, N, A, LDA, D, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), D( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAORHR_COL_GETRFNP computes the modified LU factorization without
*> pivoting of a real general M-by-N matrix A. The factorization has
*> the form:
*>
*> A - S = L * U,
*>
*> where:
*> S is a m-by-n diagonal sign matrix with the diagonal D, so that
*> D(i) = S(i,i), 1 <= i <= min(M,N). The diagonal D is constructed
*> as D(i)=-SIGN(A(i,i)), where A(i,i) is the value after performing
*> i-1 steps of Gaussian elimination. This means that the diagonal
*> element at each step of "modified" Gaussian elimination is
*> at least one in absolute value (so that division-by-zero not
*> not possible during the division by the diagonal element);
*>
*> L is a M-by-N lower triangular matrix with unit diagonal elements
*> (lower trapezoidal if M > N);
*>
*> and U is a M-by-N upper triangular matrix
*> (upper trapezoidal if M < N).
*>
*> This routine is an auxiliary routine used in the Householder
*> reconstruction routine DORHR_COL. In DORHR_COL, this routine is
*> applied to an M-by-N matrix A with orthonormal columns, where each
*> element is bounded by one in absolute value. With the choice of
*> the matrix S above, one can show that the diagonal element at each
*> step of Gaussian elimination is the largest (in absolute value) in
*> the column on or below the diagonal, so that no pivoting is required
*> for numerical stability [1].
*>
*> For more details on the Householder reconstruction algorithm,
*> including the modified LU factorization, see [1].
*>
*> This is the blocked right-looking version of the algorithm,
*> calling Level 3 BLAS to update the submatrix. To factorize a block,
*> this routine calls the recursive routine DLAORHR_COL_GETRFNP2.
*>
*> [1] "Reconstructing Householder vectors from tall-skinny QR",
*> G. Ballard, J. Demmel, L. Grigori, M. Jacquelin, H.D. Nguyen,
*> E. Solomonik, J. Parallel Distrib. Comput.,
*> vol. 85, pp. 3-31, 2015.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the M-by-N matrix to be factored.
*> On exit, the factors L and U from the factorization
*> A-S=L*U; the unit diagonal elements of L are not stored.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension min(M,N)
*> The diagonal elements of the diagonal M-by-N sign matrix S,
*> D(i) = S(i,i), where 1 <= i <= min(M,N). The elements can
*> be only plus or minus one.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*>
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2019
*
*> \ingroup doubleGEcomputational
*
*> \par Contributors:
* ==================
*>
*> \verbatim
*>
*> November 2019, Igor Kozachenko,
*> Computer Science Division,
*> University of California, Berkeley
*>
*> \endverbatim
*
* =====================================================================
SUBROUTINE DLAORHR_COL_GETRFNP( M, N, A, LDA, D, INFO )
IMPLICIT NONE
*
* -- LAPACK computational routine (version 3.9.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2019
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), D( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
INTEGER IINFO, J, JB, NB
* ..
* .. External Subroutines ..
EXTERNAL DGEMM, DLAORHR_COL_GETRFNP2, DTRSM, XERBLA
* ..
* .. External Functions ..
INTEGER ILAENV
EXTERNAL ILAENV
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -4
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DLAORHR_COL_GETRFNP', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( MIN( M, N ).EQ.0 )
$ RETURN
*
* Determine the block size for this environment.
*
NB = ILAENV( 1, 'DLAORHR_COL_GETRFNP', ' ', M, N, -1, -1 )
IF( NB.LE.1 .OR. NB.GE.MIN( M, N ) ) THEN
*
* Use unblocked code.
*
CALL DLAORHR_COL_GETRFNP2( M, N, A, LDA, D, INFO )
ELSE
*
* Use blocked code.
*
DO J = 1, MIN( M, N ), NB
JB = MIN( MIN( M, N )-J+1, NB )
*
* Factor diagonal and subdiagonal blocks.
*
CALL DLAORHR_COL_GETRFNP2( M-J+1, JB, A( J, J ), LDA,
$ D( J ), IINFO )
*
IF( J+JB.LE.N ) THEN
*
* Compute block row of U.
*
CALL DTRSM( 'Left', 'Lower', 'No transpose', 'Unit', JB,
$ N-J-JB+1, ONE, A( J, J ), LDA, A( J, J+JB ),
$ LDA )
IF( J+JB.LE.M ) THEN
*
* Update trailing submatrix.
*
CALL DGEMM( 'No transpose', 'No transpose', M-J-JB+1,
$ N-J-JB+1, JB, -ONE, A( J+JB, J ), LDA,
$ A( J, J+JB ), LDA, ONE, A( J+JB, J+JB ),
$ LDA )
END IF
END IF
END DO
END IF
RETURN
*
* End of DLAORHR_COL_GETRFNP
*
END

305
lapack-netlib/SRC/dlaorhr_col_getrfnp2.f Normal file
View File

@ -0,0 +1,305 @@
*> \brief \b DLAORHR_COL_GETRFNP2
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAORHR_GETRF2NP + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaorhr_col_getrfnp2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaorhr_col_getrfnp2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaorhr_col_getrfnp2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* RECURSIVE SUBROUTINE DLAORHR_COL_GETRFNP2( M, N, A, LDA, D, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), D( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAORHR_COL_GETRFNP2 computes the modified LU factorization without
*> pivoting of a real general M-by-N matrix A. The factorization has
*> the form:
*>
*> A - S = L * U,
*>
*> where:
*> S is a m-by-n diagonal sign matrix with the diagonal D, so that
*> D(i) = S(i,i), 1 <= i <= min(M,N). The diagonal D is constructed
*> as D(i)=-SIGN(A(i,i)), where A(i,i) is the value after performing
*> i-1 steps of Gaussian elimination. This means that the diagonal
*> element at each step of "modified" Gaussian elimination is at
*> least one in absolute value (so that division-by-zero not
*> possible during the division by the diagonal element);
*>
*> L is a M-by-N lower triangular matrix with unit diagonal elements
*> (lower trapezoidal if M > N);
*>
*> and U is a M-by-N upper triangular matrix
*> (upper trapezoidal if M < N).
*>
*> This routine is an auxiliary routine used in the Householder
*> reconstruction routine DORHR_COL. In DORHR_COL, this routine is
*> applied to an M-by-N matrix A with orthonormal columns, where each
*> element is bounded by one in absolute value. With the choice of
*> the matrix S above, one can show that the diagonal element at each
*> step of Gaussian elimination is the largest (in absolute value) in
*> the column on or below the diagonal, so that no pivoting is required
*> for numerical stability [1].
*>
*> For more details on the Householder reconstruction algorithm,
*> including the modified LU factorization, see [1].
*>
*> This is the recursive version of the LU factorization algorithm.
*> Denote A - S by B. The algorithm divides the matrix B into four
*> submatrices:
*>
*> [ B11 | B12 ] where B11 is n1 by n1,
*> B = [ -----|----- ] B21 is (m-n1) by n1,
*> [ B21 | B22 ] B12 is n1 by n2,
*> B22 is (m-n1) by n2,
*> with n1 = min(m,n)/2, n2 = n-n1.
*>
*>
*> The subroutine calls itself to factor B11, solves for B21,
*> solves for B12, updates B22, then calls itself to factor B22.
*>
*> For more details on the recursive LU algorithm, see [2].
*>
*> DLAORHR_COL_GETRFNP2 is called to factorize a block by the blocked
*> routine DLAORHR_COL_GETRFNP, which uses blocked code calling
*. Level 3 BLAS to update the submatrix. However, DLAORHR_COL_GETRFNP2
*> is self-sufficient and can be used without DLAORHR_COL_GETRFNP.
*>
*> [1] "Reconstructing Householder vectors from tall-skinny QR",
*> G. Ballard, J. Demmel, L. Grigori, M. Jacquelin, H.D. Nguyen,
*> E. Solomonik, J. Parallel Distrib. Comput.,
*> vol. 85, pp. 3-31, 2015.
*>
*> [2] "Recursion leads to automatic variable blocking for dense linear
*> algebra algorithms", F. Gustavson, IBM J. of Res. and Dev.,
*> vol. 41, no. 6, pp. 737-755, 1997.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the M-by-N matrix to be factored.
*> On exit, the factors L and U from the factorization
*> A-S=L*U; the unit diagonal elements of L are not stored.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension min(M,N)
*> The diagonal elements of the diagonal M-by-N sign matrix S,
*> D(i) = S(i,i), where 1 <= i <= min(M,N). The elements can
*> be only plus or minus one.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*>
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2019
*
*> \ingroup doubleGEcomputational
*
*> \par Contributors:
* ==================
*>
*> \verbatim
*>
*> November 2019, Igor Kozachenko,
*> Computer Science Division,
*> University of California, Berkeley
*>
*> \endverbatim
*
* =====================================================================
RECURSIVE SUBROUTINE DLAORHR_COL_GETRFNP2( M, N, A, LDA, D, INFO )
IMPLICIT NONE
*
* -- LAPACK computational routine (version 3.9.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2019
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), D( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
DOUBLE PRECISION SFMIN
INTEGER I, IINFO, N1, N2
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* ..
* .. External Subroutines ..
EXTERNAL DGEMM, DSCAL, DTRSM, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DSIGN, MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input parameters
*
INFO = 0
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -4
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DLAORHR_COL_GETRFNP2', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( MIN( M, N ).EQ.0 )
$ RETURN
IF ( M.EQ.1 ) THEN
*
* One row case, (also recursion termination case),
* use unblocked code
*
* Transfer the sign
*
D( 1 ) = -DSIGN( ONE, A( 1, 1 ) )
*
* Construct the row of U
*
A( 1, 1 ) = A( 1, 1 ) - D( 1 )
*
ELSE IF( N.EQ.1 ) THEN
*
* One column case, (also recursion termination case),
* use unblocked code
*
* Transfer the sign
*
D( 1 ) = -DSIGN( ONE, A( 1, 1 ) )
*
* Construct the row of U
*
A( 1, 1 ) = A( 1, 1 ) - D( 1 )
*
* Scale the elements 2:M of the column
*
* Determine machine safe minimum
*
SFMIN = DLAMCH('S')
*
* Construct the subdiagonal elements of L
*
IF( ABS( A( 1, 1 ) ) .GE. SFMIN ) THEN
CALL DSCAL( M-1, ONE / A( 1, 1 ), A( 2, 1 ), 1 )
ELSE
DO I = 2, M
A( I, 1 ) = A( I, 1 ) / A( 1, 1 )
END DO
END IF
*
ELSE
*
* Divide the matrix B into four submatrices
*
N1 = MIN( M, N ) / 2
N2 = N-N1
*
* Factor B11, recursive call
*
CALL DLAORHR_COL_GETRFNP2( N1, N1, A, LDA, D, IINFO )
*
* Solve for B21
*
CALL DTRSM( 'R', 'U', 'N', 'N', M-N1, N1, ONE, A, LDA,
$ A( N1+1, 1 ), LDA )
*
* Solve for B12
*
CALL DTRSM( 'L', 'L', 'N', 'U', N1, N2, ONE, A, LDA,
$ A( 1, N1+1 ), LDA )
*
* Update B22, i.e. compute the Schur complement
* B22 := B22 - B21*B12
*
CALL DGEMM( 'N', 'N', M-N1, N2, N1, -ONE, A( N1+1, 1 ), LDA,
$ A( 1, N1+1 ), LDA, ONE, A( N1+1, N1+1 ), LDA )
*
* Factor B22, recursive call
*
CALL DLAORHR_COL_GETRFNP2( M-N1, N2, A( N1+1, N1+1 ), LDA,
$ D( N1+1 ), IINFO )
*
END IF
RETURN
*
* End of DLAORHR_COL_GETRFNP2
*
END

2
lapack-netlib/SRC/dlaqps.f
View File

@ -127,7 +127,7 @@
*> \param[in,out] AUXV
*> \verbatim
*> AUXV is DOUBLE PRECISION array, dimension (NB)
*> Auxiliar vector.
*> Auxiliary vector.
*> \endverbatim
*>
*> \param[in,out] F

36
lapack-netlib/SRC/dlaqr0.f
View File

@ -67,7 +67,7 @@
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix H. N .GE. 0.
*> The order of the matrix H. N >= 0.
*> \endverbatim
*>
*> \param[in] ILO
@ -79,12 +79,12 @@
*> \verbatim
*> IHI is INTEGER
*> It is assumed that H is already upper triangular in rows
*> and columns 1:ILO-1 and IHI+1:N and, if ILO.GT.1,
*> and columns 1:ILO-1 and IHI+1:N and, if ILO > 1,
*> H(ILO,ILO-1) is zero. ILO and IHI are normally set by a
*> previous call to DGEBAL, and then passed to DGEHRD when the
*> matrix output by DGEBAL is reduced to Hessenberg form.
*> Otherwise, ILO and IHI should be set to 1 and N,
*> respectively. If N.GT.0, then 1.LE.ILO.LE.IHI.LE.N.
*> respectively. If N > 0, then 1 <= ILO <= IHI <= N.
*> If N = 0, then ILO = 1 and IHI = 0.
*> \endverbatim
*>
@ -97,19 +97,19 @@
*> decomposition (the Schur form); 2-by-2 diagonal blocks
*> (corresponding to complex conjugate pairs of eigenvalues)
*> are returned in standard form, with H(i,i) = H(i+1,i+1)
*> and H(i+1,i)*H(i,i+1).LT.0. If INFO = 0 and WANTT is
*> and H(i+1,i)*H(i,i+1) < 0. If INFO = 0 and WANTT is
*> .FALSE., then the contents of H are unspecified on exit.
*> (The output value of H when INFO.GT.0 is given under the
*> (The output value of H when INFO > 0 is given under the
*> description of INFO below.)
*>
*> This subroutine may explicitly set H(i,j) = 0 for i.GT.j and
*> This subroutine may explicitly set H(i,j) = 0 for i > j and
*> j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N.
*> \endverbatim
*>
*> \param[in] LDH
*> \verbatim
*> LDH is INTEGER
*> The leading dimension of the array H. LDH .GE. max(1,N).
*> The leading dimension of the array H. LDH >= max(1,N).
*> \endverbatim
*>
*> \param[out] WR
@ -125,7 +125,7 @@
*> and WI(ILO:IHI). If two eigenvalues are computed as a
*> complex conjugate pair, they are stored in consecutive
*> elements of WR and WI, say the i-th and (i+1)th, with
*> WI(i) .GT. 0 and WI(i+1) .LT. 0. If WANTT is .TRUE., then
*> WI(i) > 0 and WI(i+1) < 0. If WANTT is .TRUE., then
*> the eigenvalues are stored in the same order as on the
*> diagonal of the Schur form returned in H, with
*> WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2 diagonal
@ -143,7 +143,7 @@
*> IHIZ is INTEGER
*> Specify the rows of Z to which transformations must be
*> applied if WANTZ is .TRUE..
*> 1 .LE. ILOZ .LE. ILO; IHI .LE. IHIZ .LE. N.
*> 1 <= ILOZ <= ILO; IHI <= IHIZ <= N.
*> \endverbatim
*>
*> \param[in,out] Z
@ -153,7 +153,7 @@
*> If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is
*> replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the
*> orthogonal Schur factor of H(ILO:IHI,ILO:IHI).
*> (The output value of Z when INFO.GT.0 is given under
*> (The output value of Z when INFO > 0 is given under
*> the description of INFO below.)
*> \endverbatim
*>
@ -161,7 +161,7 @@
*> \verbatim
*> LDZ is INTEGER
*> The leading dimension of the array Z. if WANTZ is .TRUE.
*> then LDZ.GE.MAX(1,IHIZ). Otherwize, LDZ.GE.1.
*> then LDZ >= MAX(1,IHIZ). Otherwise, LDZ >= 1.
*> \endverbatim
*>
*> \param[out] WORK
@ -174,7 +174,7 @@
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK .GE. max(1,N)
*> The dimension of the array WORK. LWORK >= max(1,N)
*> is sufficient, but LWORK typically as large as 6*N may
*> be required for optimal performance. A workspace query
*> to determine the optimal workspace size is recommended.
@ -191,18 +191,18 @@
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> .GT. 0: if INFO = i, DLAQR0 failed to compute all of
*> > 0: if INFO = i, DLAQR0 failed to compute all of
*> the eigenvalues. Elements 1:ilo-1 and i+1:n of WR
*> and WI contain those eigenvalues which have been
*> successfully computed. (Failures are rare.)
*>
*> If INFO .GT. 0 and WANT is .FALSE., then on exit,
*> If INFO > 0 and WANT is .FALSE., then on exit,
*> the remaining unconverged eigenvalues are the eigen-
*> values of the upper Hessenberg matrix rows and
*> columns ILO through INFO of the final, output
*> value of H.
*>
*> If INFO .GT. 0 and WANTT is .TRUE., then on exit
*> If INFO > 0 and WANTT is .TRUE., then on exit
*>
*> (*) (initial value of H)*U = U*(final value of H)
*>
@ -210,7 +210,7 @@
*> value of H is upper Hessenberg and quasi-triangular
*> in rows and columns INFO+1 through IHI.
*>
*> If INFO .GT. 0 and WANTZ is .TRUE., then on exit
*> If INFO > 0 and WANTZ is .TRUE., then on exit
*>
*> (final value of Z(ILO:IHI,ILOZ:IHIZ)
*> = (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U
@ -218,7 +218,7 @@
*> where U is the orthogonal matrix in (*) (regard-
*> less of the value of WANTT.)
*>
*> If INFO .GT. 0 and WANTZ is .FALSE., then Z is not
*> If INFO > 0 and WANTZ is .FALSE., then Z is not
*> accessed.
*> \endverbatim
*
@ -678,7 +678,7 @@
END IF
END IF
*
* ==== Use up to NS of the the smallest magnatiude
* ==== Use up to NS of the the smallest magnitude
* . shifts. If there aren't NS shifts available,
* . then use them all, possibly dropping one to
* . make the number of shifts even. ====

2
lapack-netlib/SRC/dlaqr1.f
View File

@ -69,7 +69,7 @@
*> \verbatim
*> LDH is INTEGER
*> The leading dimension of H as declared in
*> the calling procedure. LDH.GE.N
*> the calling procedure. LDH >= N
*> \endverbatim
*>
*> \param[in] SR1

18
lapack-netlib/SRC/dlaqr2.f
View File

@ -103,7 +103,7 @@
*> \param[in] NW
*> \verbatim
*> NW is INTEGER
*> Deflation window size. 1 .LE. NW .LE. (KBOT-KTOP+1).
*> Deflation window size. 1 <= NW <= (KBOT-KTOP+1).
*> \endverbatim
*>
*> \param[in,out] H
@ -121,7 +121,7 @@
*> \verbatim
*> LDH is INTEGER
*> Leading dimension of H just as declared in the calling
*> subroutine. N .LE. LDH
*> subroutine. N <= LDH
*> \endverbatim
*>
*> \param[in] ILOZ
@ -133,7 +133,7 @@
*> \verbatim
*> IHIZ is INTEGER
*> Specify the rows of Z to which transformations must be
*> applied if WANTZ is .TRUE.. 1 .LE. ILOZ .LE. IHIZ .LE. N.
*> applied if WANTZ is .TRUE.. 1 <= ILOZ <= IHIZ <= N.
*> \endverbatim
*>
*> \param[in,out] Z
@ -149,7 +149,7 @@
*> \verbatim
*> LDZ is INTEGER
*> The leading dimension of Z just as declared in the
*> calling subroutine. 1 .LE. LDZ.
*> calling subroutine. 1 <= LDZ.
*> \endverbatim
*>
*> \param[out] NS
@ -194,13 +194,13 @@
*> \verbatim
*> LDV is INTEGER
*> The leading dimension of V just as declared in the
*> calling subroutine. NW .LE. LDV
*> calling subroutine. NW <= LDV
*> \endverbatim
*>
*> \param[in] NH
*> \verbatim
*> NH is INTEGER
*> The number of columns of T. NH.GE.NW.
*> The number of columns of T. NH >= NW.
*> \endverbatim
*>
*> \param[out] T
@ -212,14 +212,14 @@
*> \verbatim
*> LDT is INTEGER
*> The leading dimension of T just as declared in the
*> calling subroutine. NW .LE. LDT
*> calling subroutine. NW <= LDT
*> \endverbatim
*>
*> \param[in] NV
*> \verbatim
*> NV is INTEGER
*> The number of rows of work array WV available for
*> workspace. NV.GE.NW.
*> workspace. NV >= NW.
*> \endverbatim
*>
*> \param[out] WV
@ -231,7 +231,7 @@
*> \verbatim
*> LDWV is INTEGER
*> The leading dimension of W just as declared in the
*> calling subroutine. NW .LE. LDV
*> calling subroutine. NW <= LDV
*> \endverbatim
*>
*> \param[out] WORK

18
lapack-netlib/SRC/dlaqr3.f
View File

@ -100,7 +100,7 @@
*> \param[in] NW
*> \verbatim
*> NW is INTEGER
*> Deflation window size. 1 .LE. NW .LE. (KBOT-KTOP+1).
*> Deflation window size. 1 <= NW <= (KBOT-KTOP+1).
*> \endverbatim
*>
*> \param[in,out] H
@ -118,7 +118,7 @@
*> \verbatim
*> LDH is INTEGER
*> Leading dimension of H just as declared in the calling
*> subroutine. N .LE. LDH
*> subroutine. N <= LDH
*> \endverbatim
*>
*> \param[in] ILOZ
@ -130,7 +130,7 @@
*> \verbatim
*> IHIZ is INTEGER
*> Specify the rows of Z to which transformations must be
*> applied if WANTZ is .TRUE.. 1 .LE. ILOZ .LE. IHIZ .LE. N.
*> applied if WANTZ is .TRUE.. 1 <= ILOZ <= IHIZ <= N.
*> \endverbatim
*>
*> \param[in,out] Z
@ -146,7 +146,7 @@
*> \verbatim
*> LDZ is INTEGER
*> The leading dimension of Z just as declared in the
*> calling subroutine. 1 .LE. LDZ.
*> calling subroutine. 1 <= LDZ.
*> \endverbatim
*>
*> \param[out] NS
@ -191,13 +191,13 @@
*> \verbatim
*> LDV is INTEGER
*> The leading dimension of V just as declared in the
*> calling subroutine. NW .LE. LDV
*> calling subroutine. NW <= LDV
*> \endverbatim
*>
*> \param[in] NH
*> \verbatim
*> NH is INTEGER
*> The number of columns of T. NH.GE.NW.
*> The number of columns of T. NH >= NW.
*> \endverbatim
*>
*> \param[out] T
@ -209,14 +209,14 @@
*> \verbatim
*> LDT is INTEGER
*> The leading dimension of T just as declared in the
*> calling subroutine. NW .LE. LDT
*> calling subroutine. NW <= LDT
*> \endverbatim
*>
*> \param[in] NV
*> \verbatim
*> NV is INTEGER
*> The number of rows of work array WV available for
*> workspace. NV.GE.NW.
*> workspace. NV >= NW.
*> \endverbatim
*>
*> \param[out] WV
@ -228,7 +228,7 @@
*> \verbatim
*> LDWV is INTEGER
*> The leading dimension of W just as declared in the
*> calling subroutine. NW .LE. LDV
*> calling subroutine. NW <= LDV
*> \endverbatim
*>
*> \param[out] WORK

36
lapack-netlib/SRC/dlaqr4.f
View File

@ -74,7 +74,7 @@
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix H. N .GE. 0.
*> The order of the matrix H. N >= 0.
*> \endverbatim
*>
*> \param[in] ILO
@ -86,12 +86,12 @@
*> \verbatim
*> IHI is INTEGER
*> It is assumed that H is already upper triangular in rows
*> and columns 1:ILO-1 and IHI+1:N and, if ILO.GT.1,
*> and columns 1:ILO-1 and IHI+1:N and, if ILO > 1,
*> H(ILO,ILO-1) is zero. ILO and IHI are normally set by a
*> previous call to DGEBAL, and then passed to DGEHRD when the
*> matrix output by DGEBAL is reduced to Hessenberg form.
*> Otherwise, ILO and IHI should be set to 1 and N,
*> respectively. If N.GT.0, then 1.LE.ILO.LE.IHI.LE.N.
*> respectively. If N > 0, then 1 <= ILO <= IHI <= N.
*> If N = 0, then ILO = 1 and IHI = 0.
*> \endverbatim
*>
@ -104,19 +104,19 @@
*> decomposition (the Schur form); 2-by-2 diagonal blocks
*> (corresponding to complex conjugate pairs of eigenvalues)
*> are returned in standard form, with H(i,i) = H(i+1,i+1)
*> and H(i+1,i)*H(i,i+1).LT.0. If INFO = 0 and WANTT is
*> and H(i+1,i)*H(i,i+1) < 0. If INFO = 0 and WANTT is
*> .FALSE., then the contents of H are unspecified on exit.
*> (The output value of H when INFO.GT.0 is given under the
*> (The output value of H when INFO > 0 is given under the
*> description of INFO below.)
*>
*> This subroutine may explicitly set H(i,j) = 0 for i.GT.j and
*> This subroutine may explicitly set H(i,j) = 0 for i > j and
*> j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N.
*> \endverbatim
*>
*> \param[in] LDH
*> \verbatim
*> LDH is INTEGER
*> The leading dimension of the array H. LDH .GE. max(1,N).
*> The leading dimension of the array H. LDH >= max(1,N).
*> \endverbatim
*>
*> \param[out] WR
@ -132,7 +132,7 @@
*> and WI(ILO:IHI). If two eigenvalues are computed as a
*> complex conjugate pair, they are stored in consecutive
*> elements of WR and WI, say the i-th and (i+1)th, with
*> WI(i) .GT. 0 and WI(i+1) .LT. 0. If WANTT is .TRUE., then
*> WI(i) > 0 and WI(i+1) < 0. If WANTT is .TRUE., then
*> the eigenvalues are stored in the same order as on the
*> diagonal of the Schur form returned in H, with
*> WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2 diagonal
@ -150,7 +150,7 @@
*> IHIZ is INTEGER
*> Specify the rows of Z to which transformations must be
*> applied if WANTZ is .TRUE..
*> 1 .LE. ILOZ .LE. ILO; IHI .LE. IHIZ .LE. N.
*> 1 <= ILOZ <= ILO; IHI <= IHIZ <= N.
*> \endverbatim
*>
*> \param[in,out] Z
@ -160,7 +160,7 @@
*> If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is
*> replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the
*> orthogonal Schur factor of H(ILO:IHI,ILO:IHI).
*> (The output value of Z when INFO.GT.0 is given under
*> (The output value of Z when INFO > 0 is given under
*> the description of INFO below.)
*> \endverbatim
*>
@ -168,7 +168,7 @@
*> \verbatim
*> LDZ is INTEGER
*> The leading dimension of the array Z. if WANTZ is .TRUE.
*> then LDZ.GE.MAX(1,IHIZ). Otherwize, LDZ.GE.1.
*> then LDZ >= MAX(1,IHIZ). Otherwise, LDZ >= 1.
*> \endverbatim
*>
*> \param[out] WORK
@ -181,7 +181,7 @@
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK .GE. max(1,N)
*> The dimension of the array WORK. LWORK >= max(1,N)
*> is sufficient, but LWORK typically as large as 6*N may
*> be required for optimal performance. A workspace query
*> to determine the optimal workspace size is recommended.
@ -198,18 +198,18 @@
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> .GT. 0: if INFO = i, DLAQR4 failed to compute all of
*> > 0: if INFO = i, DLAQR4 failed to compute all of
*> the eigenvalues. Elements 1:ilo-1 and i+1:n of WR
*> and WI contain those eigenvalues which have been
*> successfully computed. (Failures are rare.)
*>
*> If INFO .GT. 0 and WANT is .FALSE., then on exit,
*> If INFO > 0 and WANT is .FALSE., then on exit,
*> the remaining unconverged eigenvalues are the eigen-
*> values of the upper Hessenberg matrix rows and
*> columns ILO through INFO of the final, output
*> value of H.
*>
*> If INFO .GT. 0 and WANTT is .TRUE., then on exit
*> If INFO > 0 and WANTT is .TRUE., then on exit
*>
*> (*) (initial value of H)*U = U*(final value of H)
*>
@ -217,7 +217,7 @@
*> value of H is upper Hessenberg and triangular in
*> rows and columns INFO+1 through IHI.
*>
*> If INFO .GT. 0 and WANTZ is .TRUE., then on exit
*> If INFO > 0 and WANTZ is .TRUE., then on exit
*>
*> (final value of Z(ILO:IHI,ILOZ:IHIZ)
*> = (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U
@ -225,7 +225,7 @@
*> where U is the orthogonal matrix in (*) (regard-
*> less of the value of WANTT.)
*>
*> If INFO .GT. 0 and WANTZ is .FALSE., then Z is not
*> If INFO > 0 and WANTZ is .FALSE., then Z is not
*> accessed.
*> \endverbatim
*
@ -677,7 +677,7 @@
END IF
END IF
*
* ==== Use up to NS of the the smallest magnatiude
* ==== Use up to NS of the the smallest magnitude
* . shifts. If there aren't NS shifts available,
* . then use them all, possibly dropping one to
* . make the number of shifts even. ====

52
lapack-netlib/SRC/dlaqr5.f
View File

@ -133,7 +133,7 @@
*> \verbatim
*> LDH is INTEGER
*> LDH is the leading dimension of H just as declared in the
*> calling procedure. LDH.GE.MAX(1,N).
*> calling procedure. LDH >= MAX(1,N).
*> \endverbatim
*>
*> \param[in] ILOZ
@ -145,7 +145,7 @@
*> \verbatim
*> IHIZ is INTEGER
*> Specify the rows of Z to which transformations must be
*> applied if WANTZ is .TRUE.. 1 .LE. ILOZ .LE. IHIZ .LE. N
*> applied if WANTZ is .TRUE.. 1 <= ILOZ <= IHIZ <= N
*> \endverbatim
*>
*> \param[in,out] Z
@ -161,7 +161,7 @@
*> \verbatim
*> LDZ is INTEGER
*> LDA is the leading dimension of Z just as declared in
*> the calling procedure. LDZ.GE.N.
*> the calling procedure. LDZ >= N.
*> \endverbatim
*>
*> \param[out] V
@ -173,7 +173,7 @@
*> \verbatim
*> LDV is INTEGER
*> LDV is the leading dimension of V as declared in the
*> calling procedure. LDV.GE.3.
*> calling procedure. LDV >= 3.
*> \endverbatim
*>
*> \param[out] U
@ -185,33 +185,14 @@
*> \verbatim
*> LDU is INTEGER
*> LDU is the leading dimension of U just as declared in the
*> in the calling subroutine. LDU.GE.3*NSHFTS-3.
*> \endverbatim
*>
*> \param[in] NH
*> \verbatim
*> NH is INTEGER
*> NH is the number of columns in array WH available for
*> workspace. NH.GE.1.
*> \endverbatim
*>
*> \param[out] WH
*> \verbatim
*> WH is DOUBLE PRECISION array, dimension (LDWH,NH)
*> \endverbatim
*>
*> \param[in] LDWH
*> \verbatim
*> LDWH is INTEGER
*> Leading dimension of WH just as declared in the
*> calling procedure. LDWH.GE.3*NSHFTS-3.
*> in the calling subroutine. LDU >= 3*NSHFTS-3.
*> \endverbatim
*>
*> \param[in] NV
*> \verbatim
*> NV is INTEGER
*> NV is the number of rows in WV agailable for workspace.
*> NV.GE.1.
*> NV >= 1.
*> \endverbatim
*>
*> \param[out] WV
@ -223,9 +204,28 @@
*> \verbatim
*> LDWV is INTEGER
*> LDWV is the leading dimension of WV as declared in the
*> in the calling subroutine. LDWV.GE.NV.
*> in the calling subroutine. LDWV >= NV.
*> \endverbatim
*
*> \param[in] NH
*> \verbatim
*> NH is INTEGER
*> NH is the number of columns in array WH available for
*> workspace. NH >= 1.
*> \endverbatim
*>
*> \param[out] WH
*> \verbatim
*> WH is DOUBLE PRECISION array, dimension (LDWH,NH)
*> \endverbatim
*>
*> \param[in] LDWH
*> \verbatim
*> LDWH is INTEGER
*> Leading dimension of WH just as declared in the
*> calling procedure. LDWH >= 3*NSHFTS-3.
*> \endverbatim
*>
* Authors:
* ========
*

2
lapack-netlib/SRC/dlarfb.f
View File

@ -92,6 +92,8 @@
*> K is INTEGER
*> The order of the matrix T (= the number of elementary
*> reflectors whose product defines the block reflector).
*> If SIDE = 'L', M >= K >= 0;
*> if SIDE = 'R', N >= K >= 0.
*> \endverbatim
*>
*> \param[in] V

2
lapack-netlib/SRC/dlarfx.f
View File

@ -94,7 +94,7 @@
*> \param[in] LDC
*> \verbatim
*> LDC is INTEGER
*> The leading dimension of the array C. LDA >= (1,M).
*> The leading dimension of the array C. LDC >= (1,M).
*> \endverbatim
*>
*> \param[out] WORK

2
lapack-netlib/SRC/dlarfy.f
View File

@ -103,7 +103,7 @@
*
*> \date December 2016
*
*> \ingroup double_eig
*> \ingroup doubleOTHERauxiliary
*
* =====================================================================
SUBROUTINE DLARFY( UPLO, N, V, INCV, TAU, C, LDC, WORK )

4
lapack-netlib/SRC/dlarrb.f
View File

@ -91,7 +91,7 @@
*> RTOL2 is DOUBLE PRECISION
*> Tolerance for the convergence of the bisection intervals.
*> An interval [LEFT,RIGHT] has converged if
*> RIGHT-LEFT.LT.MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) )
*> RIGHT-LEFT < MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) )
*> where GAP is the (estimated) distance to the nearest
*> eigenvalue.
*> \endverbatim
@ -117,7 +117,7 @@
*> WGAP is DOUBLE PRECISION array, dimension (N-1)
*> On input, the (estimated) gaps between consecutive
*> eigenvalues of L D L^T, i.e., WGAP(I-OFFSET) is the gap between
*> eigenvalues I and I+1. Note that if IFIRST.EQ.ILAST
*> eigenvalues I and I+1. Note that if IFIRST = ILAST
*> then WGAP(IFIRST-OFFSET) must be set to ZERO.
*> On output, these gaps are refined.
*> \endverbatim

2
lapack-netlib/SRC/dlarre.f
View File

@ -150,7 +150,7 @@
*> RTOL2 is DOUBLE PRECISION
*> Parameters for bisection.
*> An interval [LEFT,RIGHT] has converged if
*> RIGHT-LEFT.LT.MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) )
*> RIGHT-LEFT < MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) )
*> \endverbatim
*>
*> \param[in] SPLTOL

2
lapack-netlib/SRC/dlarrj.f
View File

@ -85,7 +85,7 @@
*> RTOL is DOUBLE PRECISION
*> Tolerance for the convergence of the bisection intervals.
*> An interval [LEFT,RIGHT] has converged if
*> RIGHT-LEFT.LT.RTOL*MAX(|LEFT|,|RIGHT|).
*> RIGHT-LEFT < RTOL*MAX(|LEFT|,|RIGHT|).
*> \endverbatim
*>
*> \param[in] OFFSET

2
lapack-netlib/SRC/dlarrv.f
View File

@ -149,7 +149,7 @@
*> RTOL2 is DOUBLE PRECISION
*> Parameters for bisection.
*> An interval [LEFT,RIGHT] has converged if
*> RIGHT-LEFT.LT.MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) )
*> RIGHT-LEFT < MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) )
*> \endverbatim
*>
*> \param[in,out] W

2
lapack-netlib/SRC/dlasd7.f
View File

@ -400,7 +400,7 @@
VL( I ) = VLW( IDXI )
50 CONTINUE
*
* Calculate the allowable deflation tolerence
* Calculate the allowable deflation tolerance
*
EPS = DLAMCH( 'Epsilon' )
TOL = MAX( ABS( ALPHA ), ABS( BETA ) )

2
lapack-netlib/SRC/dlasr.f
View File

@ -175,7 +175,7 @@
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> The M-by-N matrix A. On exit, A is overwritten by P*A if
*> SIDE = 'R' or by A*P**T if SIDE = 'L'.
*> SIDE = 'L' or by A*P**T if SIDE = 'R'.
*> \endverbatim
*>
*> \param[in] LDA

2
lapack-netlib/SRC/dlassq.f
View File

@ -60,7 +60,7 @@
*>
*> \param[in] X
*> \verbatim
*> X is DOUBLE PRECISION array, dimension (N)
*> X is DOUBLE PRECISION array, dimension (1+(N-1)*INCX)
*> The vector for which a scaled sum of squares is computed.
*> x( i ) = X( 1 + ( i - 1 )*INCX ), 1 <= i <= n.
*> \endverbatim

20
lapack-netlib/SRC/dlaswlq.f
View File

@ -1,3 +1,4 @@
*> \brief \b DLASWLQ
*
* Definition:
* ===========
@ -18,9 +19,20 @@
*>
*> \verbatim
*>
*> DLASWLQ computes a blocked Short-Wide LQ factorization of a
*> M-by-N matrix A, where N >= M:
*> A = L * Q
*> DLASWLQ computes a blocked Tall-Skinny LQ factorization of
*> a real M-by-N matrix A for M <= N:
*>
*> A = ( L 0 ) * Q,
*>
*> where:
*>
*> Q is a n-by-N orthogonal matrix, stored on exit in an implicit
*> form in the elements above the digonal of the array A and in
*> the elemenst of the array T;
*> L is an lower-triangular M-by-M matrix stored on exit in
*> the elements on and below the diagonal of the array A.
*> 0 is a M-by-(N-M) zero matrix, if M < N, and is not stored.
*>
*> \endverbatim
*
* Arguments:
@ -150,7 +162,7 @@
SUBROUTINE DLASWLQ( M, N, MB, NB, A, LDA, T, LDT, WORK, LWORK,
$ INFO)
*
* -- LAPACK computational routine (version 3.7.1) --
* -- LAPACK computational routine (version 3.9.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd. --
* June 2017

10
lapack-netlib/SRC/dlasyf_aa.f
View File

@ -284,7 +284,8 @@
*
* Swap A(I1, I2+1:M) with A(I2, I2+1:M)
*
CALL DSWAP( M-I2, A( J1+I1-1, I2+1 ), LDA,
IF( I2.LT.M )
$ CALL DSWAP( M-I2, A( J1+I1-1, I2+1 ), LDA,
$ A( J1+I2-1, I2+1 ), LDA )
*
* Swap A(I1, I1) with A(I2,I2)
@ -325,6 +326,7 @@
* Compute L(J+2, J+1) = WORK( 3:M ) / T(J, J+1),
* where A(J, J+1) = T(J, J+1) and A(J+2:M, J) = L(J+2:M, J+1)
*
IF( J.LT.(M-1) ) THEN
IF( A( K, J+1 ).NE.ZERO ) THEN
ALPHA = ONE / A( K, J+1 )
CALL DCOPY( M-J-1, WORK( 3 ), 1, A( K, J+2 ), LDA )
@ -334,6 +336,7 @@
$ A( K, J+2 ), LDA)
END IF
END IF
END IF
J = J + 1
GO TO 10
20 CONTINUE
@ -432,7 +435,8 @@
*
* Swap A(I2+1:M, I1) with A(I2+1:M, I2)
*
CALL DSWAP( M-I2, A( I2+1, J1+I1-1 ), 1,
IF( I2.LT.M )
$ CALL DSWAP( M-I2, A( I2+1, J1+I1-1 ), 1,
$ A( I2+1, J1+I2-1 ), 1 )
*
* Swap A(I1, I1) with A(I2, I2)
@ -473,6 +477,7 @@
* Compute L(J+2, J+1) = WORK( 3:M ) / T(J, J+1),
* where A(J, J+1) = T(J, J+1) and A(J+2:M, J) = L(J+2:M, J+1)
*
IF( J.LT.(M-1) ) THEN
IF( A( J+1, K ).NE.ZERO ) THEN
ALPHA = ONE / A( J+1, K )
CALL DCOPY( M-J-1, WORK( 3 ), 1, A( J+2, K ), 1 )
@ -482,6 +487,7 @@
$ A( J+2, K ), LDA )
END IF
END IF
END IF
J = J + 1
GO TO 30
40 CONTINUE

4
lapack-netlib/SRC/dlasyf_rk.f
View File

@ -321,7 +321,7 @@
* of A and working backwards, and compute the matrix W = U12*D
* for use in updating A11
*
* Initilize the first entry of array E, where superdiagonal
* Initialize the first entry of array E, where superdiagonal
* elements of D are stored
*
E( 1 ) = ZERO
@ -649,7 +649,7 @@
* of A and working forwards, and compute the matrix W = L21*D
* for use in updating A22
*
* Initilize the unused last entry of the subdiagonal array E.
* Initialize the unused last entry of the subdiagonal array E.
*
E( N ) = ZERO
*

2
lapack-netlib/SRC/dlasyf_rook.f
View File

@ -21,7 +21,7 @@
* SUBROUTINE DLASYF_ROOK( UPLO, N, NB, KB, A, LDA, IPIV, W, LDW, INFO )
*
* .. Scalar Arguments ..
* CHARADLATER UPLO
* CHARACTER UPLO
* INTEGER INFO, KB, LDA, LDW, N, NB
* ..
* .. Array Arguments ..

4
lapack-netlib/SRC/dlatdf.f
View File

@ -85,7 +85,7 @@
*> RHS is DOUBLE PRECISION array, dimension (N)
*> On entry, RHS contains contributions from other subsystems.
*> On exit, RHS contains the solution of the subsystem with
*> entries acoording to the value of IJOB (see above).
*> entries according to the value of IJOB (see above).
*> \endverbatim
*>
*> \param[in,out] RDSUM
@ -260,7 +260,7 @@
*
* Solve for U-part, look-ahead for RHS(N) = +-1. This is not done
* in BSOLVE and will hopefully give us a better estimate because
* any ill-conditioning of the original matrix is transfered to U
* any ill-conditioning of the original matrix is transferred to U
* and not to L. U(N, N) is an approximation to sigma_min(LU).
*
CALL DCOPY( N-1, RHS, 1, XP, 1 )

23
lapack-netlib/SRC/dlatsqr.f
View File

@ -1,3 +1,4 @@
*> \brief \b DLATSQR
*
* Definition:
* ===========
@ -19,8 +20,22 @@
*> \verbatim
*>
*> DLATSQR computes a blocked Tall-Skinny QR factorization of
*> an M-by-N matrix A, where M >= N:
*> A = Q * R .
*> a real M-by-N matrix A for M >= N:
*>
*> A = Q * ( R ),
*> ( 0 )
*>
*> where:
*>
*> Q is a M-by-M orthogonal matrix, stored on exit in an implicit
*> form in the elements below the digonal of the array A and in
*> the elemenst of the array T;
*>
*> R is an upper-triangular N-by-N matrix, stored on exit in
*> the elements on and above the diagonal of the array A.
*>
*> 0 is a (M-N)-by-N zero matrix, and is not stored.
*>
*> \endverbatim
*
* Arguments:
@ -149,10 +164,10 @@
SUBROUTINE DLATSQR( M, N, MB, NB, A, LDA, T, LDT, WORK,
$ LWORK, INFO)
*
* -- LAPACK computational routine (version 3.7.0) --
* -- LAPACK computational routine (version 3.9.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd. --
* December 2016
* November 2019
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, M, N, MB, NB, LDT, LWORK

306
lapack-netlib/SRC/dorgtsqr.f Normal file
View File

@ -0,0 +1,306 @@
*> \brief \b DORGTSQR
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORGTSQR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dorgtsqr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dorgtsqr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dorgtsqr.f">
*> [TXT]</a>
*>
* Definition:
* ===========
*
* SUBROUTINE DORGTSQR( M, N, MB, NB, A, LDA, T, LDT, WORK, LWORK,
* $ INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, LDT, LWORK, M, N, MB, NB
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), T( LDT, * ), WORK( * )
* ..
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DORGTSQR generates an M-by-N real matrix Q_out with orthonormal columns,
*> which are the first N columns of a product of real orthogonal
*> matrices of order M which are returned by DLATSQR
*>
*> Q_out = first_N_columns_of( Q(1)_in * Q(2)_in * ... * Q(k)_in ).
*>
*> See the documentation for DLATSQR.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. M >= N >= 0.
*> \endverbatim
*>
*> \param[in] MB
*> \verbatim
*> MB is INTEGER
*> The row block size used by DLATSQR to return
*> arrays A and T. MB > N.
*> (Note that if MB > M, then M is used instead of MB
*> as the row block size).
*> \endverbatim
*>
*> \param[in] NB
*> \verbatim
*> NB is INTEGER
*> The column block size used by DLATSQR to return
*> arrays A and T. NB >= 1.
*> (Note that if NB > N, then N is used instead of NB
*> as the column block size).
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*>
*> On entry:
*>
*> The elements on and above the diagonal are not accessed.
*> The elements below the diagonal represent the unit
*> lower-trapezoidal blocked matrix V computed by DLATSQR
*> that defines the input matrices Q_in(k) (ones on the
*> diagonal are not stored) (same format as the output A
*> below the diagonal in DLATSQR).
*>
*> On exit:
*>
*> The array A contains an M-by-N orthonormal matrix Q_out,
*> i.e the columns of A are orthogonal unit vectors.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in] T
*> \verbatim
*> T is DOUBLE PRECISION array,
*> dimension (LDT, N * NIRB)
*> where NIRB = Number_of_input_row_blocks
*> = MAX( 1, CEIL((M-N)/(MB-N)) )
*> Let NICB = Number_of_input_col_blocks
*> = CEIL(N/NB)
*>
*> The upper-triangular block reflectors used to define the
*> input matrices Q_in(k), k=(1:NIRB*NICB). The block
*> reflectors are stored in compact form in NIRB block
*> reflector sequences. Each of NIRB block reflector sequences
*> is stored in a larger NB-by-N column block of T and consists
*> of NICB smaller NB-by-NB upper-triangular column blocks.
*> (same format as the output T in DLATSQR).
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*> LDT is INTEGER
*> The leading dimension of the array T.
*> LDT >= max(1,min(NB1,N)).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> (workspace) DOUBLE PRECISION array, dimension (MAX(2,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> The dimension of the array WORK. LWORK >= (M+NB)*N.
*> If LWORK = -1, then a workspace query is assumed.
*> The routine only calculates the optimal size of the WORK
*> array, returns this value as the first entry of the WORK
*> array, and no error message related to LWORK is issued
*> by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*>
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2019
*
*> \ingroup doubleOTHERcomputational
*
*> \par Contributors:
* ==================
*>
*> \verbatim
*>
*> November 2019, Igor Kozachenko,
*> Computer Science Division,
*> University of California, Berkeley
*>
*> \endverbatim
*
* =====================================================================
SUBROUTINE DORGTSQR( M, N, MB, NB, A, LDA, T, LDT, WORK, LWORK,
$ INFO )
IMPLICIT NONE
*
* -- LAPACK computational routine (version 3.9.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2019
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, LDT, LWORK, M, N, MB, NB
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), T( LDT, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY
INTEGER IINFO, LDC, LWORKOPT, LC, LW, NBLOCAL, J
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DLAMTSQR, DLASET, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC DBLE, MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input parameters
*
LQUERY = LWORK.EQ.-1
INFO = 0
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 .OR. M.LT.N ) THEN
INFO = -2
ELSE IF( MB.LE.N ) THEN
INFO = -3
ELSE IF( NB.LT.1 ) THEN
INFO = -4
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -6
ELSE IF( LDT.LT.MAX( 1, MIN( NB, N ) ) ) THEN
INFO = -8
ELSE
*
* Test the input LWORK for the dimension of the array WORK.
* This workspace is used to store array C(LDC, N) and WORK(LWORK)
* in the call to DLAMTSQR. See the documentation for DLAMTSQR.
*
IF( LWORK.LT.2 .AND. (.NOT.LQUERY) ) THEN
INFO = -10
ELSE
*
* Set block size for column blocks
*
NBLOCAL = MIN( NB, N )
*
* LWORK = -1, then set the size for the array C(LDC,N)
* in DLAMTSQR call and set the optimal size of the work array
* WORK(LWORK) in DLAMTSQR call.
*
LDC = M
LC = LDC*N
LW = N * NBLOCAL
*
LWORKOPT = LC+LW
*
IF( ( LWORK.LT.MAX( 1, LWORKOPT ) ).AND.(.NOT.LQUERY) ) THEN
INFO = -10
END IF
END IF
*
END IF
*
* Handle error in the input parameters and return workspace query.
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DORGTSQR', -INFO )
RETURN
ELSE IF ( LQUERY ) THEN
WORK( 1 ) = DBLE( LWORKOPT )
RETURN
END IF
*
* Quick return if possible
*
IF( MIN( M, N ).EQ.0 ) THEN
WORK( 1 ) = DBLE( LWORKOPT )
RETURN
END IF
*
* (1) Form explicitly the tall-skinny M-by-N left submatrix Q1_in
* of M-by-M orthogonal matrix Q_in, which is implicitly stored in
* the subdiagonal part of input array A and in the input array T.
* Perform by the following operation using the routine DLAMTSQR.
*
* Q1_in = Q_in * ( I ), where I is a N-by-N identity matrix,
* ( 0 ) 0 is a (M-N)-by-N zero matrix.
*
* (1a) Form M-by-N matrix in the array WORK(1:LDC*N) with ones
* on the diagonal and zeros elsewhere.
*
CALL DLASET( 'F', M, N, ZERO, ONE, WORK, LDC )
*
* (1b) On input, WORK(1:LDC*N) stores ( I );
* ( 0 )
*
* On output, WORK(1:LDC*N) stores Q1_in.
*
CALL DLAMTSQR( 'L', 'N', M, N, N, MB, NBLOCAL, A, LDA, T, LDT,
$ WORK, LDC, WORK( LC+1 ), LW, IINFO )
*
* (2) Copy the result from the part of the work array (1:M,1:N)
* with the leading dimension LDC that starts at WORK(1) into
* the output array A(1:M,1:N) column-by-column.
*
DO J = 1, N
CALL DCOPY( M, WORK( (J-1)*LDC + 1 ), 1, A( 1, J ), 1 )
END DO
*
WORK( 1 ) = DBLE( LWORKOPT )
RETURN
*
* End of DORGTSQR
*
END

440
lapack-netlib/SRC/dorhr_col.f Normal file
View File

@ -0,0 +1,440 @@
*> \brief \b DORHR_COL
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORHR_COL + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dorhr_col.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dorhr_col.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dorhr_col.f">
*> [TXT]</a>
*>
* Definition:
* ===========
*
* SUBROUTINE DORHR_COL( M, N, NB, A, LDA, T, LDT, D, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, LDT, M, N, NB
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), D( * ), T( LDT, * )
* ..
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DORHR_COL takes an M-by-N real matrix Q_in with orthonormal columns
*> as input, stored in A, and performs Householder Reconstruction (HR),
*> i.e. reconstructs Householder vectors V(i) implicitly representing
*> another M-by-N matrix Q_out, with the property that Q_in = Q_out*S,
*> where S is an N-by-N diagonal matrix with diagonal entries
*> equal to +1 or -1. The Householder vectors (columns V(i) of V) are
*> stored in A on output, and the diagonal entries of S are stored in D.
*> Block reflectors are also returned in T
*> (same output format as DGEQRT).
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. M >= N >= 0.
*> \endverbatim
*>
*> \param[in] NB
*> \verbatim
*> NB is INTEGER
*> The column block size to be used in the reconstruction
*> of Householder column vector blocks in the array A and
*> corresponding block reflectors in the array T. NB >= 1.
*> (Note that if NB > N, then N is used instead of NB
*> as the column block size.)
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*>
*> On entry:
*>
*> The array A contains an M-by-N orthonormal matrix Q_in,
*> i.e the columns of A are orthogonal unit vectors.
*>
*> On exit:
*>
*> The elements below the diagonal of A represent the unit
*> lower-trapezoidal matrix V of Householder column vectors
*> V(i). The unit diagonal entries of V are not stored
*> (same format as the output below the diagonal in A from
*> DGEQRT). The matrix T and the matrix V stored on output
*> in A implicitly define Q_out.
*>
*> The elements above the diagonal contain the factor U
*> of the "modified" LU-decomposition:
*> Q_in - ( S ) = V * U
*> ( 0 )
*> where 0 is a (M-N)-by-(M-N) zero matrix.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] T
*> \verbatim
*> T is DOUBLE PRECISION array,
*> dimension (LDT, N)
*>
*> Let NOCB = Number_of_output_col_blocks
*> = CEIL(N/NB)
*>
*> On exit, T(1:NB, 1:N) contains NOCB upper-triangular
*> block reflectors used to define Q_out stored in compact
*> form as a sequence of upper-triangular NB-by-NB column
*> blocks (same format as the output T in DGEQRT).
*> The matrix T and the matrix V stored on output in A
*> implicitly define Q_out. NOTE: The lower triangles
*> below the upper-triangular blcoks will be filled with
*> zeros. See Further Details.
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*> LDT is INTEGER
*> The leading dimension of the array T.
*> LDT >= max(1,min(NB,N)).
*> \endverbatim
*>
*> \param[out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension min(M,N).
*> The elements can be only plus or minus one.
*>
*> D(i) is constructed as D(i) = -SIGN(Q_in_i(i,i)), where
*> 1 <= i <= min(M,N), and Q_in_i is Q_in after performing
*> i-1 steps of “modified” Gaussian elimination.
*> See Further Details.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*>
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The computed M-by-M orthogonal factor Q_out is defined implicitly as
*> a product of orthogonal matrices Q_out(i). Each Q_out(i) is stored in
*> the compact WY-representation format in the corresponding blocks of
*> matrices V (stored in A) and T.
*>
*> The M-by-N unit lower-trapezoidal matrix V stored in the M-by-N
*> matrix A contains the column vectors V(i) in NB-size column
*> blocks VB(j). For example, VB(1) contains the columns
*> V(1), V(2), ... V(NB). NOTE: The unit entries on
*> the diagonal of Y are not stored in A.
*>
*> The number of column blocks is
*>
*> NOCB = Number_of_output_col_blocks = CEIL(N/NB)
*>
*> where each block is of order NB except for the last block, which
*> is of order LAST_NB = N - (NOCB-1)*NB.
*>
*> For example, if M=6, N=5 and NB=2, the matrix V is
*>
*>
*> V = ( VB(1), VB(2), VB(3) ) =
*>
*> = ( 1 )
*> ( v21 1 )
*> ( v31 v32 1 )
*> ( v41 v42 v43 1 )
*> ( v51 v52 v53 v54 1 )
*> ( v61 v62 v63 v54 v65 )
*>
*>
*> For each of the column blocks VB(i), an upper-triangular block
*> reflector TB(i) is computed. These blocks are stored as
*> a sequence of upper-triangular column blocks in the NB-by-N
*> matrix T. The size of each TB(i) block is NB-by-NB, except
*> for the last block, whose size is LAST_NB-by-LAST_NB.
*>
*> For example, if M=6, N=5 and NB=2, the matrix T is
*>
*> T = ( TB(1), TB(2), TB(3) ) =
*>
*> = ( t11 t12 t13 t14 t15 )
*> ( t22 t24 )
*>
*>
*> The M-by-M factor Q_out is given as a product of NOCB
*> orthogonal M-by-M matrices Q_out(i).
*>
*> Q_out = Q_out(1) * Q_out(2) * ... * Q_out(NOCB),
*>
*> where each matrix Q_out(i) is given by the WY-representation
*> using corresponding blocks from the matrices V and T:
*>
*> Q_out(i) = I - VB(i) * TB(i) * (VB(i))**T,
*>
*> where I is the identity matrix. Here is the formula with matrix
*> dimensions:
*>
*> Q(i){M-by-M} = I{M-by-M} -
*> VB(i){M-by-INB} * TB(i){INB-by-INB} * (VB(i))**T {INB-by-M},
*>
*> where INB = NB, except for the last block NOCB
*> for which INB=LAST_NB.
*>
*> =====
*> NOTE:
*> =====
*>
*> If Q_in is the result of doing a QR factorization
*> B = Q_in * R_in, then:
*>
*> B = (Q_out*S) * R_in = Q_out * (S * R_in) = O_out * R_out.
*>
*> So if one wants to interpret Q_out as the result
*> of the QR factorization of B, then corresponding R_out
*> should be obtained by R_out = S * R_in, i.e. some rows of R_in
*> should be multiplied by -1.
*>
*> For the details of the algorithm, see [1].
*>
*> [1] "Reconstructing Householder vectors from tall-skinny QR",
*> G. Ballard, J. Demmel, L. Grigori, M. Jacquelin, H.D. Nguyen,
*> E. Solomonik, J. Parallel Distrib. Comput.,
*> vol. 85, pp. 3-31, 2015.
*> \endverbatim
*>
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2019
*
*> \ingroup doubleOTHERcomputational
*
*> \par Contributors:
* ==================
*>
*> \verbatim
*>
*> November 2019, Igor Kozachenko,
*> Computer Science Division,
*> University of California, Berkeley
*>
*> \endverbatim
*
* =====================================================================
SUBROUTINE DORHR_COL( M, N, NB, A, LDA, T, LDT, D, INFO )
IMPLICIT NONE
*
* -- LAPACK computational routine (version 3.9.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2019
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, LDT, M, N, NB
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), D( * ), T( LDT, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, IINFO, J, JB, JBTEMP1, JBTEMP2, JNB,
$ NPLUSONE
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DLAORHR_COL_GETRFNP, DSCAL, DTRSM,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input parameters
*
INFO = 0
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 .OR. N.GT.M ) THEN
INFO = -2
ELSE IF( NB.LT.1 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -5
ELSE IF( LDT.LT.MAX( 1, MIN( NB, N ) ) ) THEN
INFO = -7
END IF
*
* Handle error in the input parameters.
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DORHR_COL', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( MIN( M, N ).EQ.0 ) THEN
RETURN
END IF
*
* On input, the M-by-N matrix A contains the orthogonal
* M-by-N matrix Q_in.
*
* (1) Compute the unit lower-trapezoidal V (ones on the diagonal
* are not stored) by performing the "modified" LU-decomposition.
*
* Q_in - ( S ) = V * U = ( V1 ) * U,
* ( 0 ) ( V2 )
*
* where 0 is an (M-N)-by-N zero matrix.
*
* (1-1) Factor V1 and U.
CALL DLAORHR_COL_GETRFNP( N, N, A, LDA, D, IINFO )
*
* (1-2) Solve for V2.
*
IF( M.GT.N ) THEN
CALL DTRSM( 'R', 'U', 'N', 'N', M-N, N, ONE, A, LDA,
$ A( N+1, 1 ), LDA )
END IF
*
* (2) Reconstruct the block reflector T stored in T(1:NB, 1:N)
* as a sequence of upper-triangular blocks with NB-size column
* blocking.
*
* Loop over the column blocks of size NB of the array A(1:M,1:N)
* and the array T(1:NB,1:N), JB is the column index of a column
* block, JNB is the column block size at each step JB.
*
NPLUSONE = N + 1
DO JB = 1, N, NB
*
* (2-0) Determine the column block size JNB.
*
JNB = MIN( NPLUSONE-JB, NB )
*
* (2-1) Copy the upper-triangular part of the current JNB-by-JNB
* diagonal block U(JB) (of the N-by-N matrix U) stored
* in A(JB:JB+JNB-1,JB:JB+JNB-1) into the upper-triangular part
* of the current JNB-by-JNB block T(1:JNB,JB:JB+JNB-1)
* column-by-column, total JNB*(JNB+1)/2 elements.
*
JBTEMP1 = JB - 1
DO J = JB, JB+JNB-1
CALL DCOPY( J-JBTEMP1, A( JB, J ), 1, T( 1, J ), 1 )
END DO
*
* (2-2) Perform on the upper-triangular part of the current
* JNB-by-JNB diagonal block U(JB) (of the N-by-N matrix U) stored
* in T(1:JNB,JB:JB+JNB-1) the following operation in place:
* (-1)*U(JB)*S(JB), i.e the result will be stored in the upper-
* triangular part of T(1:JNB,JB:JB+JNB-1). This multiplication
* of the JNB-by-JNB diagonal block U(JB) by the JNB-by-JNB
* diagonal block S(JB) of the N-by-N sign matrix S from the
* right means changing the sign of each J-th column of the block
* U(JB) according to the sign of the diagonal element of the block
* S(JB), i.e. S(J,J) that is stored in the array element D(J).
*
DO J = JB, JB+JNB-1
IF( D( J ).EQ.ONE ) THEN
CALL DSCAL( J-JBTEMP1, -ONE, T( 1, J ), 1 )
END IF
END DO
*
* (2-3) Perform the triangular solve for the current block
* matrix X(JB):
*
* X(JB) * (A(JB)**T) = B(JB), where:
*
* A(JB)**T is a JNB-by-JNB unit upper-triangular
* coefficient block, and A(JB)=V1(JB), which
* is a JNB-by-JNB unit lower-triangular block
* stored in A(JB:JB+JNB-1,JB:JB+JNB-1).
* The N-by-N matrix V1 is the upper part
* of the M-by-N lower-trapezoidal matrix V
* stored in A(1:M,1:N);
*
* B(JB) is a JNB-by-JNB upper-triangular right-hand
* side block, B(JB) = (-1)*U(JB)*S(JB), and
* B(JB) is stored in T(1:JNB,JB:JB+JNB-1);
*
* X(JB) is a JNB-by-JNB upper-triangular solution
* block, X(JB) is the upper-triangular block
* reflector T(JB), and X(JB) is stored
* in T(1:JNB,JB:JB+JNB-1).
*
* In other words, we perform the triangular solve for the
* upper-triangular block T(JB):
*
* T(JB) * (V1(JB)**T) = (-1)*U(JB)*S(JB).
*
* Even though the blocks X(JB) and B(JB) are upper-
* triangular, the routine DTRSM will access all JNB**2
* elements of the square T(1:JNB,JB:JB+JNB-1). Therefore,
* we need to set to zero the elements of the block
* T(1:JNB,JB:JB+JNB-1) below the diagonal before the call
* to DTRSM.
*
* (2-3a) Set the elements to zero.
*
JBTEMP2 = JB - 2
DO J = JB, JB+JNB-2
DO I = J-JBTEMP2, NB
T( I, J ) = ZERO
END DO
END DO
*
* (2-3b) Perform the triangular solve.
*
CALL DTRSM( 'R', 'L', 'T', 'U', JNB, JNB, ONE,
$ A( JB, JB ), LDA, T( 1, JB ), LDT )
*
END DO
*
RETURN
*
* End of DORHR_COL
*
END