
OpenVMS VAX RTL Mathematics (MTH$) Manual
The conjugated versions of this routine, BLAS1$VCDOTC, BLAS1$VZDOTC,
and BLAS1$VWDOTC return the dot product of the conjugate of the first
nelement vector with a second nelement vector, as
follows:
Vectors x and y contain n elements that are accessed
from arrays x and y by stepping
incx and incy elements at a time. The
vectors x and y can be rows or columns of a matrix. Both forward and
backward indexing are permitted.
The routine name determines the data type you should specify for
arguments x and y. Specify the same
data type for these arguments.
Rounding in BLAS1$VxDOTx occurs in a different order than in a
sequential evaluation of the dot product. The final result may differ
from the result of a sequential evaluation.
Example

C
C To compute the dot product of two vectors, x and y,
C and return the result in DOTPR:
C
INTEGER INCX,INCY
REAL X(20),Y(20),DOTPR
INCX = 1
INCY = 1
N = 20
DOTPR = BLAS1$VSDOT(N,X,INCX,Y,INCY)

BLAS1$VxNRM2
The Obtain the Euclidean Norm of a Vector routine obtains the Euclidean
norm of an nelement vector x , expressed as follows:
Format
BLAS1$VSNRM2 n ,x ,incx
BLAS1$VDNRM2 n ,x ,incx
BLAS1$VGNRM2 n ,x ,incx
BLAS1$VSCNRM2 n ,x ,incx
BLAS1$VDZNRM2 n ,x ,incx
BLAS1$VGWNRM2 n ,x ,incx
Use BLAS1$VSNRM2 for singleprecision real operations. Use
BLAS1$VDNRM2 for doubleprecision real (Dfloating) operations.
Use BLAS1$VGNRM2 for doubleprecision real (Gfloating) operations. Use
BLAS1$VSCNRM2 for singleprecision complex operations. Use
BLAS1$VDZNRM2 for doubleprecision complex (Dfloating) operations.
Use BLAS1$VGWNRM2 for doubleprecision complex (Gfloating)
operations.
RETURNS
OpenVMS usage: 
floating_point 
type: 
F_floating, D_floating, or G_floating real 
access: 
write only 
mechanism: 
by value 
The function value, called e_norm, is the Euclidean
norm of the vector x . The data type of the function value is a real
number; for the BLAS1$VSCNRM2, BLAS1$VDZNRM2, and BLAS1$VGWNRM2
routines, the data type of the function value is the real data type
corresponding to the complex argument data type.
Arguments
n
OpenVMS usage: 
longword_signed 
type: 
longword integer (signed) 
access: 
read only 
mechanism: 
by reference 
Number of elements in vector x to be processed. The n
argument is the address of a signed longword integer containing the
number of elements.
x
OpenVMS usage: 
floating_point or complex_number 
type: 
F_floating, D_floating, G_floating real or F_floating,
D_floating, G_floating complex 
access: 
read only 
mechanism: 
by reference, array reference 
Array containing the elements to be accessed. All elements of array
x are accessed only if the increment argument of
x, called incx, is 1. The
x argument is the address of a floatingpoint or
floatingpoint complex number that is this array. This argument is an
array of length at least:
where:
n

=

number of vector elements specified in
n

incx

=

increment argument for the array
x specified in
incx

Specify the data type as follows:
Routine 
Data Type for x 
BLAS1$VSNRM2

Ffloating real

BLAS1$VDNRM2

Dfloating real

BLAS1$VGNRM2

Gfloating real

BLAS1$VSCNRM2

Ffloating complex

BLAS1$VDZNRM2

Dfloating complex

BLAS1$VGWNRM2

Gfloating complex

If n is less than or equal to 0, then
e_norm is 0.0.
incx
OpenVMS usage: 
longword_signed 
type: 
longword integer (signed) 
access: 
read only 
mechanism: 
by reference 
Increment argument for the array x. The
incx argument is the address of a signed longword
integer containing the increment argument. If incx is
greater than or equal to 0, then x is referenced forward in array
x; that is, x[i] is referenced in:
where:
x

=

array specified in
x

i

=

element of the vector x

incx

=

increment argument for the array
x specified in
incx

If you specify a negative value for incx, it is
interpreted as the absolute value of incx.
Description
BLAS1$VxNRM2 obtains the Euclidean norm of an nelement vector
x , expressed as follows:
Vector x contains n elements that are accessed from
array x by stepping incx elements at
a time. The vector x is a real or complex singleprecision or
doubleprecision (D and G) nelement vector. The vector can be
a row or a column of a matrix. Both forward and backward indexing are
permitted.
The publicdomain BLAS Level 1 xNRM2 routines require a positive value
for incx. The RunTime Library BLAS Level 1 routines
interpret a negative value for incx as the absolute
value of incx.
The algorithm does not provide a special case for incx
= 0. Therefore, specifying 0 for incx has the effect
of using vector operations to set e_norm as follows:
e_norm = n^{0.5}*x[1]
For BLAS1$VDNRM2, BLAS1$VGNRM2, BLAS1$VDZNRM2, and BLAS1$VGWNRM2 (the
doubleprecision routines), the elements of the vector x are scaled to
avoid intermediate overflow or underflow. BLAS1$VSNRM2 and
BLAS1$VSCNRM2 (the singleprecision routines) use a backup data type to
avoid intermediate overflow or underflow.
Rounding in BLAS1$VxNRM2 occurs in a different order than in a
sequential evaluation of the Euclidean norm. The final result may
differ from the result of a sequential evaluation.
Example

C
C To obtain the Euclidean norm of the vector x:
C
INTEGER INCX,N
REAL X(20),E_NORM
INCX = 1
N = 20
E_NORM = BLAS1$VSNRM2(N,X,INCX)

BLAS1$VxROT
The Apply a Givens Plane Rotation routine applies a Givens plane
rotation to a pair of nelement vectors x and y .
Format
BLAS1$VSROT n ,x ,incx ,y ,incy ,c ,s
BLAS1$VDROT n ,x ,incx ,y ,incy ,c ,s
BLAS1$VGROT n ,x ,incx ,y ,incy ,c ,s
BLAS1$VCSROT n ,x ,incx ,y ,incy ,c ,s
BLAS1$VZDROT n ,x ,incx ,y ,incy ,c ,s
BLAS1$VWGROT n ,x ,incx ,y ,incy ,c ,s
Use BLAS1$VSROT for singleprecision real operations. Use
BLAS1$VDROT for doubleprecision real (Dfloating) operations. Use
BLAS1$VGROT for doubleprecision real (Gfloating) operations. Use
BLAS1$VCSROT for singleprecision complex operations. Use
BLAS1$VZDROT for doubleprecision complex (Dfloating) operations.
Use BLAS1$VWGROT for doubleprecision complex (Gfloating) operations.
BLAS1$VCSROT, BLAS1$VZDROT, and BLAS1$VWGROT are real rotations
applied to a complex vector.
RETURNS
None.
Arguments
n
OpenVMS usage: 
longword_signed 
type: 
longword integer (signed) 
access: 
read only 
mechanism: 
by reference 
Number of elements in vectors x and y to be rotated. The
n argument is the address of a signed longword integer
containing the number of elements to be rotated. If n
is less than or equal to 0, then x and
y are unchanged.
x
OpenVMS usage: 
floating_point or complex_number 
type: 
F_floating, D_floating, G_floating real or F_floating,
D_floating, G_floating complex 
access: 
modify 
mechanism: 
by reference, array reference 
Array containing the elements to be accessed. All elements of array
x are accessed only if the increment argument of
x, called incx, is 1. The
x argument is the address of a floatingpoint or
floatingpoint complex number that is this array. On entry, this
argument is an array of length at least:
where:
n

=

number of vector elements specified in
n

incx

=

increment argument for the array
x specified in
incx

Specify the data type as follows:
Routine 
Data Type for x 
BLAS1$VSROT

Ffloating real

BLAS1$VDROT

Dfloating real

BLAS1$VGROT

Gfloating real

BLAS1$VCSROT

Ffloating complex

BLAS1$VZDROT

Dfloating complex

BLAS1$VWGROT

Gfloating complex

If n is less than or equal to 0, then
x and y are unchanged. If
c equals 1.0 and s equals 0, then
x and y are unchanged. If any element
of x shares a memory location with an element of y , then the results
are unpredictable.
On exit, x contains the rotated vector x , as follows:
where:
x

=

array
x specified in
x

y

=

array
y specified in
y

i

=

i = 1,2,...,n

c

=

rotation element generated by the BLAS1$VxROTG routines

s

=

rotation element generated by the BLAS1$VxROTG routines

incx
OpenVMS usage: 
longword_signed 
type: 
longword integer (signed) 
access: 
read only 
mechanism: 
by reference 
Increment argument for the array x. The
incx argument is the address of a signed longword
integer containing the increment argument. If incx is
greater than or equal to 0, then x is referenced forward in array
x; that is, x[i] is referenced in:
where:
x

=

array specified in
x

i

=

element of the vector x

incx

=

increment argument for the array
x specified in
incx

If incx is less than 0, then x is referenced backward
in array x; that is, x[i] is referenced in:
where:
x

=

array specified in
x

n

=

number of vector elements specified in
n

i

=

element of the vector x

incx

=

increment argument for the array
x specified in
incx

y
OpenVMS usage: 
floating_point or complex_number 
type: 
F_floating, D_floating, G_floating real or F_floating,
D_floating, G_floating complex 
access: 
modify 
mechanism: 
by reference, array reference 
Array containing the elements to be accessed. All elements of array
y are accessed only if the increment argument of
y, called incy, is 1. The
y argument is the address of a floatingpoint or
floatingpoint complex number that is this array. On entry, this
argument is an array of length at least:
where:
n

=

number of vector elements specified in
n

incx

=

increment argument for the array
x specified in
incx

Specify the data type as follows:
Routine 
Data Type for y 
BLAS1$VSROT

Ffloating real

BLAS1$VDROT

Dfloating real

BLAS1$VGROT

Gfloating real

BLAS1$VCSROT

Ffloating complex

BLAS1$VZDROT

Dfloating complex

BLAS1$VWGROT

Gfloating complex

If n is less than or equal to 0, then
x and y are unchanged. If
c equals 1.0 and s equals 0, then
x and y are unchanged. If any element
of x shares a memory location with an element of y , then the results
are unpredictable.
On exit, y contains the rotated vector y , as follows:
where:
x

=

array
x specified in
x

y

=

array
y specified in
y

i

=

i = 1,2,...,n

c

=

real rotation element (can be generated by the BLAS1$VxROTG routines)

s

=

complex rotation element (can be generated by the BLAS1$VxROTG routines)

incy
OpenVMS usage: 
longword_signed 
type: 
longword integer (signed) 
access: 
read only 
mechanism: 
by reference 
Increment argument for the array y. The
incy argument is the address of a signed longword
integer containing the increment argument. If incy is
greater than or equal to 0, then y is referenced forward in array
y; that is, y[i] is referenced in:
where:
y

=

array specified in
y

i

=

element of the vector y

incy

=

increment argument for the array
y specified in
incy

If incy is less than 0, then y is referenced backward
in array y; that is, y[i] is referenced in:
where:
y

=

array specified in
y

n

=

number of vector elements specified in
n

i

=

element of the vector y

incy

=

increment argument for the array
y specified in
incy

c
OpenVMS usage: 
floating_point 
type: 
F_floating, D_floating, or G_floating real 
access: 
read only 
mechanism: 
by reference 
First rotation element, which can be interpreted as the cosine of the
angle of rotation. The c argument is the address of a
floatingpoint or floatingpoint complex number that is this vector
element. The c argument is the first rotation element
generated by the BLAS1$VxROTG routines.
Specify the data type (which is always real) as follows:
Routine 
Data Type for c 
BLAS1$VSROT and BLAS1$VCSROT

Ffloating real

BLAS1$VDROT and BLAS1$VZDROT

Dfloating real

BLAS1$VGROT and BLAS1$VWGROT

Gfloating real

s
OpenVMS usage: 
floating_point or complex_number 
type: 
F_floating, D_floating, G_floating real or F_floating,
D_floating, G_floating complex 
access: 
read only 
mechanism: 
by reference 
Second rotation element, which can be interpreted as the sine of the
angle of rotation. The s argument is the address of a
floatingpoint or floatingpoint complex number that is this vector
element. The s argument is the second rotation element
generated by the BLAS1$VxROTG routines.
Specify the data type (which can be either real or complex) as follows:
Routine 
Data Type for s 
BLAS1$VSROT and BLAS1$VCSROT

Ffloating real or Ffloating complex

BLAS1$VDROT and BLAS1$VZDROT

Dfloating real or Dfloating complex

BLAS1$VGROT and BLAS1$VWGROT

Gfloating real or Gfloating complex

Description
BLAS1$VSROT, BLAS1$VDROT, and BLAS1$VGROT apply a real Givens plane
rotation to a pair of real vectors. BLAS1$VCSROT, BLAS1$VZDROT, and
BLAS1$VWGROT apply a real Givens plane rotation to a pair of complex
vectors. The vectors x and y are real or complex singleprecision or
doubleprecision (D and G) vectors. The vectors can be rows or columns
of a matrix. Both forward and backward indexing are permitted. The
routine name determines the data type you should specify for arguments
x and y. Specify the same data type
for each of these arguments.
The Givens plane rotation is applied to n elements,
where the elements to be rotated are contained in vectors x and y
(i equals 1,2,...,n). These elements are accessed
from arrays x and y by stepping
incx and incy elements at a time. The
cosine and sine of the angle of rotation are c and
s, respectively. The arguments c and
s are usually generated by the BLAS Level 1 routine
BLAS1$VxROTG, using a=x and b=y :
The BLAS1$VxROT routines can be used to introduce zeros selectively
into a matrix.
Example

C
C To rotate the first two rows of a matrix and zero
C out the element in the first column of the second row:
C
INTEGER INCX,N
REAL X(20,20),A,B,C,S
INCX = 20
N = 20
A = X(1,1)
B = X(2,1)
CALL BLAS1$VSROTG(A,B,C,S)
CALL BLAS1$VSROT(N,X,INCX,X(2,1),INCX,C,S)

