# Why was The Matrix 4 a flop?

What is LU decomposition in numerical method

In numerical analysis and linear algebra, lower–upper (LU) decomposition or factorization factors a matrix as the product of a lower triangular matrix and an upper triangular matrix (see matrix decomposition). The product sometimes includes a permutation matrix as well.

Is the time complexity of classical matrix multiplication

As of October 2022, the best announced bound on the asymptotic complexity of a matrix multiplication algorithm is O(n2.37188) time, given by Duan, Wu and Zhou announced in a preprint. This improves on the bound of O(n2.3728596) time, given by Josh Alman and Virginia Vassilevska Williams.

Why does LU decomposition fail

The LU decomposition can fail when the top-left entry in the matrix A is zero or very small compared to other entries. Pivoting is a strategy to mitigate this problem by rearranging the rows and/or columns of A to put a larger element in the top-left position.

What is the disadvantage of LU decomposition

It requires forward and backward substituion. Solving requires storing in memory the LU factors. It requires around n33 FLOPS. It requires (like most) pivoting to ensure numerical stability.

What is the fastest matrix multiplication time complexity

O(n2.37188)

As of October 2022, the best announced bound on the asymptotic complexity of a matrix multiplication algorithm is O(n2.37188) time, given by Duan, Wu and Zhou announced in a preprint.

What is the fastest matrix multiplication algorithm

the Strassen algorithm

In linear algebra, the Strassen algorithm, named after Volker Strassen, is an algorithm for matrix multiplication. It is faster than the standard matrix multiplication algorithm for large matrices, with a better asymptotic complexity, although the naive algorithm is often better for smaller matrices.

Why is Lu better than Gaussian elimination

Linear Equations and Eigensystems

Compared with Gaussian elimination, LU decomposition has a particular advantage when the equation system we wish to solve, Ax = b , has more than one right side or when the right sides are not known in advance.

Is LU decomposition faster than Gauss elimination

The LU Decomposition method is n/4 times more efficient in finding the inverse than Naïve Gaussian Elimination method.

Why is Matlab so fast in matrix multiplication

MATLAB uses highly optimized libraries for matrix multiplication which is why the plain MATLAB matrix multiplication is so fast. The gpuArray version uses MAGMA. This is why. MATLAB doesn't perform a naive matrix multiplication by looping over every single element the way you did in your C++ code.

Why is matrix multiplication faster than for loops

A straightforward for loop uses the Python interpreter, which is incredibly slow. It's more likely that a good matrix library you're using is just a Python wrapper for the much faster C or C++ math libraries.

Is matrix multiplication faster on CPU or GPU

The main advantage of using a GPU for sparse matrix multiplication is speed. A GPU can perform this task much faster than a CPU, especially when working with large, sparse matrices. This is because GPUs are optimized for parallel processing, which allows them to perform many calculations simultaneously.

What is faster than Karatsuba

The Schönhage–Strassen algorithm is faster than both Karatsuba and Toom-Cook for very large n n n ( \big( (on the order of n > 2 2 15 ) n > 2^{2^{15}}\big) n>2215) and runs in O ( n log ⁡ n log ⁡ log ⁡ n ) O(n \log n \log \log n) O(nlognloglogn).

Why does Gaussian elimination fail

Gaussian elimination, as described above, fails if any of the pivots is zero, it is worse yet if any pivot becomes close to zero. In this case, the method can be carried to completion, but the obtained results may be totally wrong.

Why is Lu better than Gaussian

However, LU-factorization has the following advantages: Gaussian elimination and Gauss–Jordan elimination both use the augmented matrix [A|b], so b must be known. In contrast, LU-decomposition uses only matrix A, so once that factorization is complete, it can be applied to any vector b.

Why GPU is faster for matrix multiplication

In your case of matrix multiplication. You can parallelize the computations, Because GPU have much more threads and in each thread you have multiple blocks. So a lot of computations are parallelized, resulting quick computations.

What is the fastest possible matrix multiplication

In linear algebra, the Strassen algorithm, named after Volker Strassen, is an algorithm for matrix multiplication. It is faster than the standard matrix multiplication algorithm for large matrices, with a better asymptotic complexity, although the naive algorithm is often better for smaller matrices.

What is the fastest known multiplication algorithm

The Karatsuba algorithm is a fast multiplication algorithm that uses a divide and conquer approach to multiply two numbers.

Why are GPUs good at matrix

On the contrary, if the application contains large-scale data to be processed and shows a large amount of data parallelism, the GPU would be a better choice because the GPU has a large number of programmable cores that can support large-scale multi-threaded operations and has a larger peak bandwidth than the CPU.

Are AMD GPU faster than Nvidia

Nvidia: What's the difference The most basic difference between AMD GPUs and Nvidia GPUs is that Nvidia chips tend to be more powerful, especially at the high-end, while AMD cards offer better value at lower price points and a more friendly user interface.

Do computers use Karatsuba

In Karatsuba algorithm, 2-digit multiplication operation can be performed with three multiplications (Dwivedi, 2013). Karatsuba algorithm is used by Intel and other companies to perform faster multiplication, because it requires less number of steps (Madke and Zafar, 2014).

What is the disadvantage of Gaussian elimination

Gaussian elimination, as described above, fails if any of the pivots is zero, it is worse yet if any pivot becomes close to zero. In this case, the method can be carried to completion, but the obtained results may be totally wrong.

Is Gaussian elimination stable

Gaussian elimination is numerically stable for diagonally dominant or positive-definite matrices. For general matrices, Gaussian elimination is usually considered to be stable, when using partial pivoting, even though there are examples of stable matrices for which it is unstable.

Why is LU factorization not unique

Let L be any unit lower triangular matrix and let U = 0 ∈ Rn×n. Then LU = 0 = A, implying that we have constructed an LU factorization of A. Since there is more than one unit lower triangular matrix in Rn×n with n ≥ 2, this proves the non-uniqueness of the factorization.

How much faster is CUDA than CPU

Besides, considering the operations in CUDA are asynchronous, I also need to add a synchronization statement to ensure printing the used time after all CUDA tasks are done. Almost 42x times faster than running in CPU. A model that needs several days' training in CPU may now take only a few hours in GPU.

What is world’s hardest multiplication

03/5​The hardest is 6X8!

The hardest multiplication, they found, was six times eight or 6X8 as 63% of the students could not solve it. This was followed by 8×6, 11×12, 12×8 and 8×12.