Vector Processors and Data Level Parallelism
A complex number consists of a real and imaginary component and is usually written in the form where and are either integer or floating-point values and (the imaginary value) . Sometimes in engineering, the letter is used in place of because is used for other values.
Multiplying two complex numbers is done by applying the FOIL (Firsts, Outers, Inners and Lasts) method, similar to that of binomial multiplication. For example, multiplying (a + bi)(c + di) is accomplished as follows:
Firsts: a * c
Outers: a * di
Inners: bi * c
Lasts: bi * di
This produces (a+bi)(c+di) = ac + adi + bci + bdi2. The terms are combined to produce the product back in the form a + bi. Keep in mind that i2 = -1.
An example using actual values: (2.5 + 3i)(4.0 + 2i)
Firsts: 2.5 * 4.0
Outers: 2.5 * 2i
Inners: 3i * 4.0
Lasts: 3i * 2i
This produces 10 + 5i + 12i + 6i2 = 10 + 17i + 6(-1) = 4 + 17i.
Some contemporary programming languages natively support complex numbers (Python, MATLAB). Newer revisions of some older languages (C, FORTRAN) have added support for complex numbers. Some programming languages have no native support for complex numbers.
Consider the following high-level language code which multiplies two vectors that contain single-precision complex numbers:
Values a, b and c are vectors; _re is the real component element and _im is the imaginary component element in each vector.
Register s0 = loop counter & array index [i]
Vector registers: v0 – v31
MVL (maximum vector length) = 64
Instructions: vld (vector load)
vst (vector store)
vadd (vector add)
vsub (vector subtract)
vmul (vector multiply)
bne (branch if not equal)*
blt (branch if less than)*
j (unconditional jump)*
addi (integer add immediate)*
ori (logical or immediate)*
Note: instructions with an asterisk indicate the instructions are used only for setting initial index value and increments, and for loop control.
vld (vector load): vld vD, vec_ref
vst (vector store): vst vD, vec_ref
vadd (vector add): vadd vD, vS1, vS2
vsub (vector subtract): vsub vD, vS1, vS2
vmul (vector multiply): vmul vD, vS1, vS2
bne (branch if not equal): bne x1, x2, target_label
blt (branch if less than): blt x1, x2, target_label
j (unconditional jump): j target_label
addi (integer add immediate): addi xD, xS1, xS2
ori (logical or immediate): ori xD, xS1, const
vD = destination vector register
vS1 = first source vector register
vS2 = second source vector register
vec_ref = vector reference (name)
x1 = first general purpose register for comparison
x2 = second general purpose register for comparison
xS1 = first source general purpose register
xS2 = second source general purpose register
target_label = label of the target instruction for branch
const = an integer constant
The assignment is described in the attached document. The first page provides background information about complex numbers, the basis for the definition of the vectors you will work with in completing the assignment. This may provide insight as to why the original loop code is structured as it is. However, you will not be dealing with any data contained in any vectors; only references to the vectors for the purpose of completing the coding part of the assignment. The work of the assignment is described on the second page, consisting of four questions to be answered. The third page is reference information to assist in formulating the code needed to answer questions 1 and 2 of the assignment.
This assignment is similar to the example presented in the lecture materials, slides 8, 9 and 10.
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