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Here, c[i:i+3] represents the four array elements from c[i] to c[i+3] and the vector processor can perform four operations for a single vector instruction. Since the four vector operations complete in roughly the same time as one scalar instruction, the vector approach can run up to four times faster than the original code.
As another, more significant, example of compile-time loop unrolling, template metaprogramming can be used to create length-n vector classes (where n is known at compile time). The benefit over a more traditional length-n vector is that the loops can be unrolled, resulting in very optimized code. As an example, consider the addition operator.
Basic Linear Algebra Subprograms (BLAS) is a specification that prescribes a set of low-level routines for performing common linear algebra operations such as vector addition, scalar multiplication, dot products, linear combinations, and matrix multiplication.
Vectorial addition chains are well suited to perform multi-exponentiation: [1] Input: Elements x 0,...,x k-1 of an abelian group G and a vectorial addition chain of dimension k computing [n 0,...,n k-1] Output:The element x 0 n 0...x k-1 n r-1. for i =-k+1 to 0 do y i → x i+k-1; for i = 1 to s do y i →y j ×y r; return y s
In addition, the design had completely separate pipelines for different instructions, for example, addition/subtraction was implemented in different hardware than multiplication. This allowed a batch of vector instructions to be pipelined into each of the ALU subunits, a technique they called vector chaining. The Cray-1 normally had a ...
For example, to perform an element by element sum of two arrays, a and b to produce a third c, it is only necessary to write c = a + b In addition to support for vectorized arithmetic and relational operations, these languages also vectorize common mathematical functions such as sine. For example, if x is an array, then y = sin (x)
The cross product operation is an example of a vector rank function because it operates on vectors, not scalars. Matrix multiplication is an example of a 2-rank function, because it operates on 2-dimensional objects (matrices). Collapse operators reduce the dimensionality of an input data array by one or more dimensions. For example, summing ...
In Python NumPy arrays implement the flatten method, [note 1] while in R the desired effect can be achieved via the c() or as.vector() functions. In R , function vec() of package 'ks' allows vectorization and function vech() implemented in both packages 'ks' and 'sn' allows half-vectorization.