Questions tagged [matrix-multiplication]
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34 questions
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Classical matrix multiplication via Min-Plus matrix multiplication
Just a thought I had in mind. I can use classical matrix multiplication to compute min-plus matrix multiplication. Generally speaking, considering $(n+1)^{a_{i,j}}$ for each entry and then taking the ...
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Usage of matrix multiplication for distance products
This is more of a validation question, for the current best known results.
On one hand, we have classical matrix multiplication. Its running time is denoted as $n^\omega$.
On the other, we have ...
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Algorithm for solving linear equations if interested only in the first component
If I want to solve $\mathbf A \mathbf x = \mathbf b$, but I am only interested in the value of $x_1$, what algorithm should I use, and will it always be strictly more efficient than solving for all of ...
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Floating point operations in a zero padded Strassen multiplication
So I've seen other posts here that do discuss this, but I'm not quite sure how the time complexity (I think?) relates to the actual number of floating point operations done per second when you're ...
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Compute matrix inversion / multiplication using a black box
Suppose you're given a black box $A$, and you're told $A$ can invert a matrix (assuming the matrix is invertible) $M$ in $O(T_A)$. You're also given a black box $B$, and you're told $B$ can multiply ...
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Complexity of multiplying 3 matrices
There are algorithms that speed up matrix multiplication over the naive $n^3$ algorithm. But supposing you have 3 matrices $A$, $B$ and $C$, is there a way to compute $ABC$ that is asymptotically ...
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Matrix multiplication of natural numbers
I know matrix multiplication of matrices with real numbers is bounded by $ \Omega (n^2 log(n))$, but what about if all numbers are natural? Can we use the same methods to get a lower bound for this ...
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Has Triangle Finding ever been faster than Matrix Multiplication?
The Triangle Finding problem (TF) in Graph Theory was shown by Itai and Rodeh in 1977 [1] to be solvable as fast$^1$ as Boolean Matrix Multiplication (BMM, Matrix Multiplication over $\{0, 1\}$ with ...
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What's the fastest known non-galactic algorithm for matrix multiplication of large matrices
"A galactic algorithm is one that outperforms any other algorithm for problems that are sufficiently large, but where "sufficiently large" is so big that the algorithm is never used in ...
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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 ...
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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 two 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(...
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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 ...
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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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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-...
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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 ...
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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 ...
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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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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 ...
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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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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 ...
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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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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 ...
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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 ...
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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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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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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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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 ...
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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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