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Dividing/Multiplying Numbers Stored in two memory locations

I have two numbers x and y. The upper bits of x are stored at location m, while the lower bits of x are stored at location n. The upper bits of y are stored at location i, while the lower bits of y ...
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1answer
44 views

In fast multiplier circuits, what is the difference between a Counter and a Compressor?

When working on fast parallel multiplier circuit designs, like Wallace tree multipliers or Dada tree multipliers I found many papers and books refer to different components used in the tree to reduce ...
2
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2answers
39 views

Controlling overflow and loss of precision during floating point multiplication

I have a large number of floating point numbers (~10,000 numbers) , each having 6 digits after decimal. Now, the multiplication of all these numbers would yield about 60,000 digits. But the double ...
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0answers
95 views

Algorithm for multiplying multivariate polynomials

Let $R$ be a commutative ring. Let $f(x_1, \dots, x_n), g(x_1, \dots, x_n)$ be two multidimensional polynomials in $R$ with maximal total degree $\delta$. How fast can we compute the product of $f$ ...
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2answers
37 views

Efficient (sublinear) approximation algorithms for matrix-vector multiplication?

Given a matrix $A \in \mathbb{R}^{n \times p}$ and a vector $x \in \mathbb{R}^p$, I am interested in computing the value of the mean matrix-vector product: $$v = \frac{1}{n} Ax$$ If I did this using ...
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1answer
23 views

Scalar by N component vector multiplication faster than O(N)?

Is there a way to multiply scalar by vector faster than just multiplying each element of the vector by that scalar? It feels to me that there should be some exploit to do that. After all we will ...
3
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1answer
178 views

Fixed base exponentiation with precomputations

I'm trying to compute $g^m$ mod $n$ where $m,n$ are 1024-bit numbers. The method I want to use is fixed base exponentiation with precomputations, also known as fixed-base windowing.The paper I'm ...
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2answers
230 views

How can I compute an exponential modulo a large integer?

Does anyone have the computational power to check whether or not $F(m)^d \equiv m \pmod n$, where the values of the variables are found below. According to Wolfram Alpha, I found the result of the ...
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0answers
183 views

Booth's Algorithm Multiplication

When multiplying signed integers by Booth's algorithm, does the multiplicand always have to be negative? What happens if multiplier and multiplicand are both negative? Does the algorithm still work? ...
2
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2answers
161 views

Where does the lg(lg(N)) factor come from in Schönhage–Strassen's run time?

According to page 53 of Modern Computer Arithmetic (pdf), all of the steps in the Schönhage–Strassen Algorithm cost $O(N \cdot lg(N))$ except for the recursion step which ends up costing $O(N\cdot ...
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2answers
187 views

Why is the transform in Schönhage–Strassen's multiplication algorithm cheap?

The Schönhage–Strassen multiplication algorithm works by turning multiplications of size $N$ into many multiplications of size $lg(N)$ with a number-theoretic transform, and recursing. At least I ...
8
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1answer
975 views

Why doesn't Knuth's linear-time multiplication algorithm “count”?

The wikipedia page on multiplication algorithms mentions an interesting one by Donald Knuth. Basically, it involves combining fourier-transform multiplication with a precomputed table of ...
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1answer
396 views

Fast algorithm for matrix chain multiplication in special case

An exercise from the book Foundations of Algorithms Using Java Pseudocode: Write an efficient algorithm that will find an optimal order for multiplying $n$ matrices $A_1 \times A_2 \times \ldots ...
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2answers
126 views

Understanding Intel's algorithm for reducing a polynomial modulo an irreducible polynomial

I'm reading this Intel white paper on carry-less multiplication. It describes multiplication of polynomials in $\text{GF}(2^n)$. On a high level, this is performed in two steps: (1) multiplication of ...
1
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1answer
100 views

NFA for right left multiplication

Given the following multiplication table how could one construct an NFA such that it accepts all strings that have a certain product (say a) ? The string "abcb" would be evaluated as (a(b(cb))) = a ...
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3answers
145 views

Does the performance of matrix multiplication depend on the storage of the array?

Two matrices can be stored in either row major or column major order in contiguous memory. Does the time complexity of computing their multiplication vary depending on the storage scheme? That is, I ...
3
votes
1answer
464 views

Shift-and-or multiplication operation

Continuing in the same vein as Carry-free multiplication operation, a followup question is as follows (differences in bold): Let $r = p \oplus q$ be an operation similar to multiplication, but ...
3
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2answers
260 views

Carry-free multiplication operation

In long-multiplication, you shift and add, once for each $1$ bit in the lower number. Let $r = p \otimes q$ be an operation similar to multiplication, but slightly simpler: when expressed via ...