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perazzperazz
authoredApr 12, 2024··
Improve rotm, rotmg documentation (#795)
·
v0.7.0v0.6.0
2 parents c673e3b + 7fe9b3a commit c26253a

4 files changed

+93
-97
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‎src/stdlib_linalg_blas.fypp

+23-24
Original file line numberDiff line numberDiff line change
@@ -1019,17 +1019,17 @@ module stdlib_linalg_blas
10191019

10201020

10211021
interface rotm
1022-
!! APPLY THE MODIFIED GIVENS TRANSFORMATION, H, TO THE 2 BY N MATRIX
1023-
!! (DX**T) , WHERE **T INDICATES TRANSPOSE. THE ELEMENTS OF DX ARE IN
1024-
!! (DY**T)
1025-
!! DX(LX+I*INCX), I = 0 TO N-1, WHERE LX = 1 IF INCX >= 0, ELSE
1026-
!! LX = (-INCX)*N, AND SIMILARLY FOR SY USING LY AND INCY.
1027-
!! WITH DPARAM(1)=DFLAG, H HAS ONE OF THE FOLLOWING FORMS..
1028-
!! DFLAG=-1._dp DFLAG=0._dp DFLAG=1._dp DFLAG=-2.D0
1029-
!! (DH11 DH12) (1._dp DH12) (DH11 1._dp) (1._dp 0._dp)
1030-
!! H=( ) ( ) ( ) ( )
1031-
!! (DH21 DH22), (DH21 1._dp), (-1._dp DH22), (0._dp 1._dp).
1032-
!! SEE ROTMG FOR A DESCRIPTION OF DATA STORAGE IN DPARAM.
1022+
!! ROTM applies the modified Givens transformation, \(H\), to the 2-by-N matrix
1023+
!! $$ \left[ \begin{array}{c}DX^T\\DY^T\\ \end{array} \right], $$
1024+
!! where \(^T\) indicates transpose. The elements of \(DX\) are in
1025+
!! DX(LX+I*INCX), I = 0:N-1, where LX = 1 if INCX >= 0, else LX = (-INCX)*N,
1026+
!! and similarly for DY using LY and INCY.
1027+
!! With DPARAM(1)=DFLAG, \(H\) has one of the following forms:
1028+
!! $$ H=\underbrace{\begin{bmatrix}DH_{11} & DH_{12}\\DH_{21} & DH_{22}\end{bmatrix}}_{DFLAG=-1},
1029+
!! \underbrace{\begin{bmatrix}1 & DH_{12}\\DH_{21} & 1\end{bmatrix}}_{DFLAG=0},
1030+
!! \underbrace{\begin{bmatrix}DH_{11} & 1\\-1 & DH_{22}\end{bmatrix}}_{DFLAG=1},
1031+
!! \underbrace{\begin{bmatrix}1 & 0\\0 & 1\end{bmatrix}}_{DFLAG=-2}. $$
1032+
!! See ROTMG for a description of data storage in DPARAM.
10331033
#ifdef STDLIB_EXTERNAL_BLAS
10341034
pure subroutine drotm(n,dx,incx,dy,incy,dparam)
10351035
import sp,dp,qp,ilp,lk
@@ -1060,19 +1060,18 @@ module stdlib_linalg_blas
10601060

10611061

10621062
interface rotmg
1063-
!! CONSTRUCT THE MODIFIED GIVENS TRANSFORMATION MATRIX H WHICH ZEROS
1064-
!! THE SECOND COMPONENT OF THE 2-VECTOR (SQRT(DD1)*DX1,SQRT(DD2) DY2)**T.
1065-
!! WITH DPARAM(1)=DFLAG, H HAS ONE OF THE FOLLOWING FORMS..
1066-
!! DFLAG=-1._dp DFLAG=0._dp DFLAG=1._dp DFLAG=-2.D0
1067-
!! (DH11 DH12) (1._dp DH12) (DH11 1._dp) (1._dp 0._dp)
1068-
!! H=( ) ( ) ( ) ( )
1069-
!! (DH21 DH22), (DH21 1._dp), (-1._dp DH22), (0._dp 1._dp).
1070-
!! LOCATIONS 2-4 OF DPARAM CONTAIN DH11, DH21, DH12, AND DH22
1071-
!! RESPECTIVELY. (VALUES OF 1._dp, -1._dp, OR 0._dp IMPLIED BY THE
1072-
!! VALUE OF DPARAM(1) ARE NOT STORED IN DPARAM.)
1073-
!! THE VALUES OF GAMSQ AND RGAMSQ SET IN THE DATA STATEMENT MAY BE
1074-
!! INEXACT. THIS IS OK AS THEY ARE ONLY USED FOR TESTING THE SIZE
1075-
!! OF DD1 AND DD2. ALL ACTUAL SCALING OF DATA IS DONE USING GAM.
1063+
!! ROTMG Constructs the modified Givens transformation matrix \(H\) which zeros the
1064+
!! second component of the 2-vector
1065+
!! $$ \left[ {\sqrt{DD_1}\cdot DX_1,\sqrt{DD_2}\cdot DY_2} \right]^T. $$
1066+
!! With DPARAM(1)=DFLAG, \(H\) has one of the following forms:
1067+
!! $$ H=\underbrace{\begin{bmatrix}DH_{11} & DH_{12}\\DH_{21} & DH_{22}\end{bmatrix}}_{DFLAG=-1},
1068+
!! \underbrace{\begin{bmatrix}1 & DH_{12}\\DH_{21} & 1\end{bmatrix}}_{DFLAG=0},
1069+
!! \underbrace{\begin{bmatrix}DH_{11} & 1\\-1 & DH_{22}\end{bmatrix}}_{DFLAG=1},
1070+
!! \underbrace{\begin{bmatrix}1 & 0\\0 & 1\end{bmatrix}}_{DFLAG=-2}. $$
1071+
!! Locations 2-4 of DPARAM contain DH11, DH21, DH12 and DH22 respectively.
1072+
!! (Values of 1.0, -1.0, or 0.0 implied by the value of DPARAM(1) are not stored in DPARAM.)
1073+
!! The values of parameters GAMSQ and RGAMSQ may be inexact. This is OK as they are only
1074+
!! used for testing the size of DD1 and DD2. All actual scaling of data is done using GAM.
10761075
#ifdef STDLIB_EXTERNAL_BLAS
10771076
pure subroutine drotmg(dd1,dd2,dx1,dy1,dparam)
10781077
import sp,dp,qp,ilp,lk

‎src/stdlib_linalg_blas_d.fypp

+24-25
Original file line numberDiff line numberDiff line change
@@ -1028,18 +1028,18 @@ module stdlib_linalg_blas_d
10281028
end subroutine stdlib_drotg
10291029

10301030

1031-
pure subroutine stdlib_drotm(n,dx,incx,dy,incy,dparam)
1032-
!! APPLY THE MODIFIED GIVENS TRANSFORMATION, H, TO THE 2 BY N MATRIX
1033-
!! (DX**T) , WHERE **T INDICATES TRANSPOSE. THE ELEMENTS OF DX ARE IN
1034-
!! (DY**T)
1035-
!! DX(LX+I*INCX), I = 0 TO N-1, WHERE LX = 1 IF INCX >= 0, ELSE
1036-
!! LX = (-INCX)*N, AND SIMILARLY FOR SY USING LY AND INCY.
1037-
!! WITH DPARAM(1)=DFLAG, H HAS ONE OF THE FOLLOWING FORMS..
1038-
!! DFLAG=-1._dp DFLAG=0._dp DFLAG=1._dp DFLAG=-2.D0
1039-
!! (DH11 DH12) (1._dp DH12) (DH11 1._dp) (1._dp 0._dp)
1040-
!! H=( ) ( ) ( ) ( )
1041-
!! (DH21 DH22), (DH21 1._dp), (-1._dp DH22), (0._dp 1._dp).
1042-
!! SEE DROTMG FOR A DESCRIPTION OF DATA STORAGE IN DPARAM.
1031+
pure subroutine stdlib_drotm(n,dx,incx,dy,incy,dparam)
1032+
!! DROTM applies the modified Givens transformation, \(H\), to the 2-by-N matrix
1033+
!! $$ \left[ \begin{array}{c}DX^T\\DY^T\\ \end{array} \right], $$
1034+
!! where \(^T\) indicates transpose. The elements of \(DX\) are in
1035+
!! DX(LX+I*INCX), I = 0:N-1, where LX = 1 if INCX >= 0, else LX = (-INCX)*N,
1036+
!! and similarly for DY using LY and INCY.
1037+
!! With DPARAM(1)=DFLAG, \(H\) has one of the following forms:
1038+
!! $$ H=\underbrace{\begin{bmatrix}DH_{11} & DH_{12}\\DH_{21} & DH_{22}\end{bmatrix}}_{DFLAG=-1},
1039+
!! \underbrace{\begin{bmatrix}1 & DH_{12}\\DH_{21} & 1\end{bmatrix}}_{DFLAG=0},
1040+
!! \underbrace{\begin{bmatrix}DH_{11} & 1\\-1 & DH_{22}\end{bmatrix}}_{DFLAG=1},
1041+
!! \underbrace{\begin{bmatrix}1 & 0\\0 & 1\end{bmatrix}}_{DFLAG=-2}. $$
1042+
!! See DROTMG for a description of data storage in DPARAM.
10431043
! -- reference blas level1 routine --
10441044
! -- reference blas is a software package provided by univ. of tennessee, --
10451045
! -- univ. of california berkeley, univ. of colorado denver and nag ltd..--
@@ -1136,19 +1136,18 @@ module stdlib_linalg_blas_d
11361136

11371137

11381138
pure subroutine stdlib_drotmg(dd1,dd2,dx1,dy1,dparam)
1139-
!! CONSTRUCT THE MODIFIED GIVENS TRANSFORMATION MATRIX H WHICH ZEROS
1140-
!! THE SECOND COMPONENT OF THE 2-VECTOR (SQRT(DD1)*DX1,SQRT(DD2) DY2)**T.
1141-
!! WITH DPARAM(1)=DFLAG, H HAS ONE OF THE FOLLOWING FORMS..
1142-
!! DFLAG=-1._dp DFLAG=0._dp DFLAG=1._dp DFLAG=-2.D0
1143-
!! (DH11 DH12) (1._dp DH12) (DH11 1._dp) (1._dp 0._dp)
1144-
!! H=( ) ( ) ( ) ( )
1145-
!! (DH21 DH22), (DH21 1._dp), (-1._dp DH22), (0._dp 1._dp).
1146-
!! LOCATIONS 2-4 OF DPARAM CONTAIN DH11, DH21, DH12, AND DH22
1147-
!! RESPECTIVELY. (VALUES OF 1._dp, -1._dp, OR 0._dp IMPLIED BY THE
1148-
!! VALUE OF DPARAM(1) ARE NOT STORED IN DPARAM.)
1149-
!! THE VALUES OF GAMSQ AND RGAMSQ SET IN THE DATA STATEMENT MAY BE
1150-
!! INEXACT. THIS IS OK AS THEY ARE ONLY USED FOR TESTING THE SIZE
1151-
!! OF DD1 AND DD2. ALL ACTUAL SCALING OF DATA IS DONE USING GAM.
1139+
!! DROTMG Constructs the modified Givens transformation matrix \(H\) which zeros the
1140+
!! second component of the 2-vector
1141+
!! $$ \left[ {\sqrt{DD_1}\cdot DX_1,\sqrt{DD_2}\cdot DY_2} \right]^T. $$
1142+
!! With DPARAM(1)=DFLAG, \(H\) has one of the following forms:
1143+
!! $$ H=\underbrace{\begin{bmatrix}DH_{11} & DH_{12}\\DH_{21} & DH_{22}\end{bmatrix}}_{DFLAG=-1},
1144+
!! \underbrace{\begin{bmatrix}1 & DH_{12}\\DH_{21} & 1\end{bmatrix}}_{DFLAG=0},
1145+
!! \underbrace{\begin{bmatrix}DH_{11} & 1\\-1 & DH_{22}\end{bmatrix}}_{DFLAG=1},
1146+
!! \underbrace{\begin{bmatrix}1 & 0\\0 & 1\end{bmatrix}}_{DFLAG=-2}. $$
1147+
!! Locations 2-4 of DPARAM contain DH11, DH21, DH12 and DH22 respectively.
1148+
!! (Values of 1.0, -1.0, or 0.0 implied by the value of DPARAM(1) are not stored in DPARAM.)
1149+
!! The values of parameters GAMSQ and RGAMSQ may be inexact. This is OK as they are only
1150+
!! used for testing the size of DD1 and DD2. All actual scaling of data is done using GAM.
11521151
! -- reference blas level1 routine --
11531152
! -- reference blas is a software package provided by univ. of tennessee, --
11541153
! -- univ. of california berkeley, univ. of colorado denver and nag ltd..--

‎src/stdlib_linalg_blas_q.fypp

+23-24
Original file line numberDiff line numberDiff line change
@@ -1032,17 +1032,17 @@ module stdlib_linalg_blas_q
10321032

10331033

10341034
pure subroutine stdlib_qrotm(n,dx,incx,dy,incy,dparam)
1035-
!! APPLY THE MODIFIED GIVENS TRANSFORMATION, H, TO THE 2 BY N MATRIX
1036-
!! (DX**T) , WHERE **T INDICATES TRANSPOSE. THE ELEMENTS OF DX ARE IN
1037-
!! (DY**T)
1038-
!! DX(LX+I*INCX), I = 0 TO N-1, WHERE LX = 1 IF INCX >= 0, ELSE
1039-
!! LX = (-INCX)*N, AND SIMILARLY FOR SY USING LY AND INCY.
1040-
!! WITH DPARAM(1)=DFLAG, H HAS ONE OF THE FOLLOWING FORMS..
1041-
!! DFLAG=-1._qp DFLAG=0._qp DFLAG=1._qp DFLAG=-2.D0
1042-
!! (DH11 DH12) (1._qp DH12) (DH11 1._qp) (1._qp 0._qp)
1043-
!! H=( ) ( ) ( ) ( )
1044-
!! (DH21 DH22), (DH21 1._qp), (-1._qp DH22), (0._qp 1._qp).
1045-
!! SEE DROTMG FOR A DESCRIPTION OF DATA STORAGE IN DPARAM.
1035+
!! QROTM applies the modified Givens transformation, \(H\), to the 2-by-N matrix
1036+
!! $$ \left[ \begin{array}{c}DX^T\\DY^T\\ \end{array} \right], $$
1037+
!! where \(^T\) indicates transpose. The elements of \(DX\) are in
1038+
!! DX(LX+I*INCX), I = 0:N-1, where LX = 1 if INCX >= 0, else LX = (-INCX)*N,
1039+
!! and similarly for DY using LY and INCY.
1040+
!! With DPARAM(1)=DFLAG, \(H\) has one of the following forms:
1041+
!! $$ H=\underbrace{\begin{bmatrix}DH_{11} & DH_{12}\\DH_{21} & DH_{22}\end{bmatrix}}_{DFLAG=-1},
1042+
!! \underbrace{\begin{bmatrix}1 & DH_{12}\\DH_{21} & 1\end{bmatrix}}_{DFLAG=0},
1043+
!! \underbrace{\begin{bmatrix}DH_{11} & 1\\-1 & DH_{22}\end{bmatrix}}_{DFLAG=1},
1044+
!! \underbrace{\begin{bmatrix}1 & 0\\0 & 1\end{bmatrix}}_{DFLAG=-2}. $$
1045+
!! See QROTMG for a description of data storage in DPARAM.
10461046
! -- reference blas level1 routine --
10471047
! -- reference blas is a software package provided by univ. of tennessee, --
10481048
! -- univ. of california berkeley, univ. of colorado denver and nag ltd..--
@@ -1139,19 +1139,18 @@ module stdlib_linalg_blas_q
11391139

11401140

11411141
pure subroutine stdlib_qrotmg(dd1,dd2,dx1,dy1,dparam)
1142-
!! CONSTRUCT THE MODIFIED GIVENS TRANSFORMATION MATRIX H WHICH ZEROS
1143-
!! THE SECOND COMPONENT OF THE 2-VECTOR (SQRT(DD1)*DX1,SQRT(DD2) DY2)**T.
1144-
!! WITH DPARAM(1)=DFLAG, H HAS ONE OF THE FOLLOWING FORMS..
1145-
!! DFLAG=-1._qp DFLAG=0._qp DFLAG=1._qp DFLAG=-2.D0
1146-
!! (DH11 DH12) (1._qp DH12) (DH11 1._qp) (1._qp 0._qp)
1147-
!! H=( ) ( ) ( ) ( )
1148-
!! (DH21 DH22), (DH21 1._qp), (-1._qp DH22), (0._qp 1._qp).
1149-
!! LOCATIONS 2-4 OF DPARAM CONTAIN DH11, DH21, DH12, AND DH22
1150-
!! RESPECTIVELY. (VALUES OF 1._qp, -1._qp, OR 0._qp IMPLIED BY THE
1151-
!! VALUE OF DPARAM(1) ARE NOT STORED IN DPARAM.)
1152-
!! THE VALUES OF GAMSQ AND RGAMSQ SET IN THE DATA STATEMENT MAY BE
1153-
!! INEXACT. THIS IS OK AS THEY ARE ONLY USED FOR TESTING THE SIZE
1154-
!! OF DD1 AND DD2. ALL ACTUAL SCALING OF DATA IS DONE USING GAM.
1142+
!! QROTMG Constructs the modified Givens transformation matrix \(H\) which zeros the
1143+
!! second component of the 2-vector
1144+
!! $$ \left[ {\sqrt{DD_1}\cdot DX_1,\sqrt{DD_2}\cdot DY_2} \right]^T. $$
1145+
!! With DPARAM(1)=DFLAG, \(H\) has one of the following forms:
1146+
!! $$ H=\underbrace{\begin{bmatrix}DH_{11} & DH_{12}\\DH_{21} & DH_{22}\end{bmatrix}}_{DFLAG=-1},
1147+
!! \underbrace{\begin{bmatrix}1 & DH_{12}\\DH_{21} & 1\end{bmatrix}}_{DFLAG=0},
1148+
!! \underbrace{\begin{bmatrix}DH_{11} & 1\\-1 & DH_{22}\end{bmatrix}}_{DFLAG=1},
1149+
!! \underbrace{\begin{bmatrix}1 & 0\\0 & 1\end{bmatrix}}_{DFLAG=-2}. $$
1150+
!! Locations 2-4 of DPARAM contain DH11, DH21, DH12 and DH22 respectively.
1151+
!! (Values of 1.0, -1.0, or 0.0 implied by the value of DPARAM(1) are not stored in DPARAM.)
1152+
!! The values of parameters GAMSQ and RGAMSQ may be inexact. This is OK as they are only
1153+
!! used for testing the size of DD1 and DD2. All actual scaling of data is done using GAM.
11551154
! -- reference blas level1 routine --
11561155
! -- reference blas is a software package provided by univ. of tennessee, --
11571156
! -- univ. of california berkeley, univ. of colorado denver and nag ltd..--

‎src/stdlib_linalg_blas_s.fypp

+23-24
Original file line numberDiff line numberDiff line change
@@ -1209,17 +1209,17 @@ module stdlib_linalg_blas_s
12091209

12101210

12111211
pure subroutine stdlib_srotm(n,sx,incx,sy,incy,sparam)
1212-
!! APPLY THE MODIFIED GIVENS TRANSFORMATION, H, TO THE 2 BY N MATRIX
1213-
!! (SX**T) , WHERE **T INDICATES TRANSPOSE. THE ELEMENTS OF SX ARE IN
1214-
!! (SX**T)
1215-
!! SX(LX+I*INCX), I = 0 TO N-1, WHERE LX = 1 IF INCX >= 0, ELSE
1216-
!! LX = (-INCX)*N, AND SIMILARLY FOR SY USING USING LY AND INCY.
1217-
!! WITH SPARAM(1)=SFLAG, H HAS ONE OF THE FOLLOWING FORMS..
1218-
!! SFLAG=-1._sp SFLAG=0._sp SFLAG=1._sp SFLAG=-2.E0
1219-
!! (SH11 SH12) (1._sp SH12) (SH11 1._sp) (1._sp 0._sp)
1220-
!! H=( ) ( ) ( ) ( )
1221-
!! (SH21 SH22), (SH21 1._sp), (-1._sp SH22), (0._sp 1._sp).
1222-
!! SEE SROTMG FOR A DESCRIPTION OF DATA STORAGE IN SPARAM.
1212+
!! SROTM applies the modified Givens transformation, \(H\), to the 2-by-N matrix
1213+
!! $$ \left[ \begin{array}{c}SX^T\\SY^T\\ \end{array} \right], $$
1214+
!! where \(^T\) indicates transpose. The elements of \(SX\) are in
1215+
!! SX(LX+I*INCX), I = 0:N-1, where LX = 1 if INCX >= 0, else LX = (-INCX)*N,
1216+
!! and similarly for SY using LY and INCY.
1217+
!! With SPARAM(1)=SFLAG, \(H\) has one of the following forms:
1218+
!! $$ H=\underbrace{\begin{bmatrix}SH_{11} & SH_{12}\\SH_{21} & SH_{22}\end{bmatrix}}_{SFLAG=-1},
1219+
!! \underbrace{\begin{bmatrix}1 & SH_{12}\\SH_{21} & 1\end{bmatrix}}_{SFLAG=0},
1220+
!! \underbrace{\begin{bmatrix}SH_{11} & 1\\-1 & SH_{22}\end{bmatrix}}_{SFLAG=1},
1221+
!! \underbrace{\begin{bmatrix}1 & 0\\0 & 1\end{bmatrix}}_{SFLAG=-2}. $$
1222+
!! See SROTMG for a description of data storage in SPARAM.
12231223
! -- reference blas level1 routine --
12241224
! -- reference blas is a software package provided by univ. of tennessee, --
12251225
! -- univ. of california berkeley, univ. of colorado denver and nag ltd..--
@@ -1316,19 +1316,18 @@ module stdlib_linalg_blas_s
13161316

13171317

13181318
pure subroutine stdlib_srotmg(sd1,sd2,sx1,sy1,sparam)
1319-
!! CONSTRUCT THE MODIFIED GIVENS TRANSFORMATION MATRIX H WHICH ZEROS
1320-
!! THE SECOND COMPONENT OF THE 2-VECTOR (SQRT(SD1)*SX1,SQRT(SD2) SY2)**T.
1321-
!! WITH SPARAM(1)=SFLAG, H HAS ONE OF THE FOLLOWING FORMS..
1322-
!! SFLAG=-1._sp SFLAG=0._sp SFLAG=1._sp SFLAG=-2.E0
1323-
!! (SH11 SH12) (1._sp SH12) (SH11 1._sp) (1._sp 0._sp)
1324-
!! H=( ) ( ) ( ) ( )
1325-
!! (SH21 SH22), (SH21 1._sp), (-1._sp SH22), (0._sp 1._sp).
1326-
!! LOCATIONS 2-4 OF SPARAM CONTAIN SH11,SH21,SH12, AND SH22
1327-
!! RESPECTIVELY. (VALUES OF 1._sp, -1._sp, OR 0._sp IMPLIED BY THE
1328-
!! VALUE OF SPARAM(1) ARE NOT STORED IN SPARAM.)
1329-
!! THE VALUES OF GAMSQ AND RGAMSQ SET IN THE DATA STATEMENT MAY BE
1330-
!! INEXACT. THIS IS OK AS THEY ARE ONLY USED FOR TESTING THE SIZE
1331-
!! OF SD1 AND SD2. ALL ACTUAL SCALING OF DATA IS DONE USING GAM.
1319+
!! SROTMG Constructs the modified Givens transformation matrix \(H\) which zeros the
1320+
!! second component of the 2-vector
1321+
!! $$ \left[ {\sqrt{SD_1}\cdot SX_1,\sqrt{SD_2}\cdot SY_2} \right]^T. $$
1322+
!! With SPARAM(1)=SFLAG, \(H\) has one of the following forms:
1323+
!! $$ H=\underbrace{\begin{bmatrix}SH_{11} & SD_{12}\\SH_{21} & SH_{22}\end{bmatrix}}_{SFLAG=-1},
1324+
!! \underbrace{\begin{bmatrix}1 & SH_{12}\\SH_{21} & 1\end{bmatrix}}_{SFLAG=0},
1325+
!! \underbrace{\begin{bmatrix}SH_{11} & 1\\-1 & SH_{22}\end{bmatrix}}_{SFLAG=1},
1326+
!! \underbrace{\begin{bmatrix}1 & 0\\0 & 1\end{bmatrix}}_{SFLAG=2}. $$
1327+
!! Locations 2-4 of SPARAM contain SH11, SH21, SH12 and SH22 respectively.
1328+
!! (Values of 1.0, -1.0, or 0.0 implied by the value of SPARAM(1) are not stored in SPARAM.)
1329+
!! The values of parameters GAMSQ and RGAMSQ may be inexact. This is OK as they are only
1330+
!! used for testing the size of DD1 and DD2. All actual scaling of data is done using GAM.
13321331
! -- reference blas level1 routine --
13331332
! -- reference blas is a software package provided by univ. of tennessee, --
13341333
! -- univ. of california berkeley, univ. of colorado denver and nag ltd..--

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