Questions tagged [matrix-multiplication]

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Fast compute of F*P*FT matrix product

Let $P$ be a symmetric (positive definite, if that helps...) matrix of size $n$. Let $F$ be a matrix of size $n$. Is there an existing efficient algorithm implementation to calculate $FPF^T$ ? Is ...
1 vote
2 answers
227 views

Can dot producting the result of vector-matrix multiplication speed up the runtime

Suppose we have a matrix $A$ of dimension $n \times n$ and 2 vectors $\vec{u}$ and $\vec{v}$ of dimension $n$. Then we have $A\vec{v} = \vec{x}$ with time complexity $O(n^2)$ and space complexity $O(n)...
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What will be the Computational Complexity in terms of order O of the operations shown in the following figure

Suppose I have L bits. First, I want to multiply the L bits with L orthogonal codes of length N, and then I want to add all the vectors. So, first, I have to do a scalar multiplication with a vector ...
3 votes
3 answers
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Are there absolute reasons to prefer row/column-major memory ordering?

I've heard it said that "fortran uses column-major ordering because it's faster" but I'm not sure that's true. Certainly, matching column-major data to a column-major implementation will ...
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3 answers
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Super-linear parallelism or speedup in parallel matrix multiplication algorithms

I'm reading this slides from a MIT course on parallel software performance. They introduced the concepts of Work $T_1$, Span $T_\infty$ and Parallelism (ratio $T_1/T_\infty$). What is called ''...
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runtime of solving matrix differential equation wrt dimensions of matrix

Suppose a computer solves a coupled differential equations (with given boundary conditions) of which each equation deals with $2^n \times 2^n$ size of matrices as solutions. My question is Does time-...
3 votes
0 answers
54 views

fast multiplication of power of a matrix by a vector

I'm interested in computing of the product $M^n v$, where $M$ is an $m\times m$ matrix (over a semiring) and $v$ is column-vector, with the smallest number of multiplications in the underlying ...
4 votes
2 answers
85 views

Best-known complexity for $l \times m$ by $m \times n$ matrix multiplication?

I know that the fastest known algorithm for multiplying two $m \times m$ matrices runs in time $m^{\omega}$, where currently we have $\omega = 2.3728596$ due to Virginia Williams's latest result (see ...
3 votes
0 answers
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Computing a series of matrix power - matrix products

Assuming we have two dense matrices $A \in \mathbb{R}^{m\times m}, B \in \mathbb{R}^{m\times n}$, is there a smart way to compute all entries of the series $A^1 B, A^2 B, A^3 B, \dots, A^k B$ up to ...
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1 vote
1 answer
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Matrix Multiplication Verification

This question relates to the 2nd edition of Probability and Computation by Mitzenmacher and Upfal. The authors then use the Law of Total Probability My question concerns the line that begins with $\...
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Merging the submatrices' time complexity in matrix multiplication

This is a problem of CLRS: What is the largest $k$ such that if you can multiply $3 \times 3$ matrices using $k$ multiplications (not assuming commutativity of multiplication), then you can multiply $...
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Matrix-vector multiplication using only lower triangular of matrix

Suppose one has a large sparse symmetric positive definite matrix $A$ and wants to multiply it by a vector $x$. Only the lower triangular part of matrix A is stored/known. The multiplication $Ax$ ...
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Why do researchers only count the number of multiplications when analyse the time complexity of Matrix Multiplication?

In this article about the recent breakthough in Matrix Multiplication, it quotes Chris Umans's words: Multiplications are everything. The exponent on the eventual running time is fully dependent only ...
1 vote
1 answer
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Iterated multiplication of permutation matrices

Given $m$ matrices of size $n\times n$ each of which is promised to be a permutation is it in $\mathit{quasiAC}^0$ or $\mathit{AC}^0$ to multiply the permutations where $m=\mathit{poly}(n)$ $m=\...
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matrix multiplication speedup when the matrix elements are 0, 1 and -1

I would like to compute matrix multiplication A * B where A is Nx3 and B is 3x3. We also ...
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2 votes
1 answer
514 views

In Strassen's algorithm, why does padding the matrices with zeros not affect the asymptopic complexity?

In Strassen's algorithm, why does padding the matrices with zeros, in order to multiply matrices that are not powers of 2, not affect the asymptopic complexity? Hi, I was reading this question but I ...
5 votes
1 answer
90 views

Is there a polynomial sized arithmetic formula for iterated matrix multiplication?

I found an article on Catalytic space which describes how additional memory (which must be returned to it's arbitrary, initial state) can be useful for computation. There's also an expository follow ...
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1 vote
1 answer
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Why is the weight matrix diagonal in weighted least squares regression?

I was going through the theory for weighted least-squares fitting and I understood its basic underlying concepts, but I couldn't understand why exactly do we keep the weights as a diagonal matrix ...
22 votes
2 answers
1k views

What is the intuition behind Strassen's Algorithm?

I came across Strassen's algorithm for matrix multiplication, which has time complexity $O(n^{2.81})$, significantly better than the naive $O(n^3)$. Of course, there have been several other ...
1 vote
0 answers
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Lower bounds for orthogonal matrix multiplication

Is it possible, according to the current state of knowledge, that orthogonal matrices can be multiplied faster than arbitrary matrices? More precisely, let $T(N)$ denote the worst-case time of the ...
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1 answer
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Is this benchmark sufficient to consider my algorithm as an efficient matrix multiplication algorithm?

I built a matrix multiplication algorithm and now I need some thoughts about following benchmark. C++ chrono:: high resolution clock Time(micro second) (Dim)256--> (Naive algo ) 296807, (My algo) ...
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1 answer
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How can follow this this guide to construct a graph with matrix/reachability

Let's we have k matrices. For example we have 3 now, where first one is 8x5 ($a_1$ x $b_1$), second one is 5 x 6 ($a_2$ x $b_2$) and last one is 6 x 8 ($a_3$ x $b_3$). And our goal is to figure out ...
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In a LP problem Ax = b, how to solve for A instead of x?

I have a multi-objective linear programming problem of the form Ax = b, where A is a matrix and x and b are vectors. x is known, and I'm looking to minimise each row of b by solving for A. Constraints ...
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1 answer
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Is matrix multiplication cheaper than inverse?

In wiki, it is shown that the time complexity of matrix multiplication and matrix inverse is similar. But people always to argue it is easier to do matrix multiplication rather than inverse. Is this ...
3 votes
1 answer
305 views

Calculate boolean matrix multiplication (BMM) using transitive closure

Let us say I am given an algorithm that calculates the transitive closure of a given graph $G = \{ V, E \}$. How can I use this algorithm in order to perform the Boolean Matrix Multiplication of two ...
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