Name

snorm2, dnorm2, scnorm2, dznorm2 − Square root of sum of the squares of the elements of a vector

FORMAT

{S,D}NORM2 (n, x, incx) SCNORM2 (n, x, incx) DZNORM2 (n, x, incx)

Function Value

sumreal∗4 | real∗8 | complex∗8 | complex∗16
The Euclidean norm of the vector x, that is, the square root of the sum of the squares of the elements of a real vector or the square root of the sum of the squares of the absolute value of the elements of the complex vector. If n<=0, (sum = 0.0.) 

Arguments

ninteger∗4
On entry, the number of elements of the vector x.
On exit, n is unchanged. 

xreal∗4 | real∗8 | complex∗8 | complex∗16
On entry, a one-dimensional array X of length at least (1+(n-1)∗|incx|), containing the elements of the vector x.
On exit, x is unchanged. 

incxinteger∗4
On entry, the increment for the array X.
If incx > 0, vector x is stored forward in the array, so that x(i) is stored in location X(1+(i-1)∗incx). 
If incx < 0, vector x is stored backward in the array, so that x(i) is stored in location X(1+(n-i)∗|incx|). 
If incx = 0, only the first element is accessed. 
On exit, incx is unchanged. 

Description

SNORM2 and DNORM2 compute the Euclidean norm of a real vector x.  The Euclidean norm is the square root of the sum of the squares of the elements of the vector: SUM(i=1...n,x(i)∗∗2)∗∗(1/2) SCNORM2 and DZNORM2 compute the square root of the sum of the squares of the absolute value of the elements of a complex vector x: SUM(i=1...n,|x(i)|∗∗2)∗∗(1/2)

For complex vectors, each element x(j) is a complex number.  In this subprogram, the absolute value of a complex number is defined as the square root of the sum of the squares of the real part and the imaginary part: |x(j)| = (a(j)∗∗2 + b(j)∗∗2) ∗∗(1/2) = ((real)∗∗2 + (imaginary)∗∗2) ∗∗(1/2)

If incx < 0, the result is identical to using |incx|.  If incx = 0, the computation is a time-consuming way of setting sum = (n∗x(1)∗∗2)∗∗(1/2) for real operations, and sum = |x(i)| (n)∗∗(1/2).  for complex operations. 

Because of efficient coding, rounding errors can cause the final result to differ from the result computed by a sequential evaluation of the Euclidean norm. 

Unlike the _NRM2 and __NRM2 subprograms in BLAS Level 1, the _NORM2 and __NORM2 subprograms do not perform any special scaling to ensure that intermediate results do not overflow or underflow.  Therefore, these routines must use an input vector x so that |(min)∗∗(1/2)|
 <= |x(i)|
 <= |(max)∗∗(1/2)|
The largest value of x must not overflow when it is squared; the smallest value must not underflow when it is squared.

Example

INTEGER∗4 N, INCX
REAL∗4 X(20), SUM
INCX = 1
N = 20
SUM = SNORM2(N,X,INCX)

This FORTRAN example shows how to compute the Euclidean norm of the vector x.