Questions tagged [numerical-linear-algebra]

Questions on the various algorithms used in linear algebra computations (matrix computations).

2,241 questions
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Inverting a matrix with the same diagonal entries in a particular form

Hi I'm struggling with this inversion and any help would be greatly appreciated. I want to invert the following $\mathbb R^{m\times m}$ matrix \begin{bmatrix} 1 + m & m & \dots &...
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Simplify this expression with divided differences.

The divided differences are defined as follows $$f[x_i] := f(x_i), \quad f[x_0, \ldots, x_n] := \frac{f[x_1, \ldots, x_n] - f[x_0, \ldots, x_{n - 1}]}{x_n - x_0} \quad \text{for } n \ge 2$$ For ...
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How can I find the pivot matrix from LU-factorization?

I trying to solve LU-factrization with pivoting: $$PA=LU$$ By using the subroutine sgeft2 from Lapack. It's a Fortran 90 library for numerical linear algebra. I have found the $L$ and $U$ matrix, ...
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How to solve $Q$ matrix from Householder QR-factorization? - Lapack

I'm using the subroutine sgeqr2 from Lapack. This subroutine solves the QR-factorization $$A = QR$$ It's easy to find the $R$ matrix, because the in-out argument $A$ of subroutine sgeqr2 will return ...
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Schur complement condition number

How can I most simply show, without referring to advanced theorems, that the Schur complement is better-conditioned than the original SPD matrix itself?
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Why is the norm $1$ of matrix $A$ is equal to the maximum sum of column

first, I know that there exists a similar question to mine which is in here, and it is actually very well explained. However, there is just one part that I do not understand. That is the conclusion. ...
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LU decomposition implies Gaussian Elimination without pivoting

If Gaussian elimination can be carried out without pivoting for A, then A has an LU decomposition. Is the converse true: if A has an LU decomposition, then Gaussian elimination can be carried out (in ...
I want to solve Singular Valude Decomposition(SVD) $$A = USV^T$$ I have been using the QR-method before, but it takes lot of time and is very slow. Is there any easier method to use? Is it easy to ...