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Mathematics Stack Exchange

Questions tagged [numerical-linear-algebra]

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

Questions tagged with this tag can be about, but not limited to:

  1. Matrix decompositions like SVD, QR, Cholesky, etc.
  2. The solution of linear systems and least squares problems.
  3. Analysis of numerical linear algebra algorithms like condition numbers and stability analysis.
  4. Eigenvalue problems.
  5. The designs of direct or iterative methods to solve linear systems.
3541 questions
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Numerically stable determinant of matrix product

What is the best way in general to numerically calculate the determinant of a product of two matrices? Calculate the matrix product first and then calculate the determinant of the result or calculate the determinants of the two matrices and…
TheCat
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Approximate numerical diagonalization of a matrix

I need to calculate an approximate value of the Von Neumann entropy \begin{equation} S = - \text{Tr} ( \rho \log_2 \rho ) \end{equation} for a variable $N\times N$ matrix $\rho$ very often. My last approach was to diagonalize \begin{equation} \rho…
psicolor
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How to bound $\Delta x$ for having $|\Delta y|<10^{-4}$

I have an exercise and i would like to ask if someone has any idea what i have to use to solve it.( what i about the notes : absolute and relative error, general formula of error propagation). Any idea,is being accepted :) thanks Given the function…
Irene
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Reference that Explains Preconditioning

I would like to understand Preconditioning techniques and why they work. Could someone provide a good reference for this type of information?
NicNic8
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Proof with an Artificial Power Method

Suppose $A$ is $m\times m$ and has a complete set of orthonormal eigenvectors, $q_1, \ldots , q_m$, and with corresponding eigenvalues $\lambda_1,\ldots , \lambda_m$. Assume that the ordering is such that $\left|\lambda_j\right| \geq…
Razor7654
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Strategies for evaluating the action of the solution to a Sylvester equation on a vector

I have a Sylvester equation $AX+XB=C$ with a unique solution. I don't actually need $X$, but rather the matrix-vector product $Xv$ (for some known $v$). It seems most literature concerns itself with finding $X$, but I was hoping, much like…
Jason
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Prove if a square root of a Herimitian positive semi-definite matrix is also Herimitian positive semi-definite, then it is unique.

Prove if a square root of a Herimitian positive semi-definite matrix is also Herimitian positive semi-definite, then it is unique. Here is what I have done so far: By the spectrum theorem, suppose $A$ and $B$ are Herimitian positive semi-definite…
DDaren
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solve nonlinear system of equation numerically

solve the following system of equations numerically $$2x+2y - e^{xy} = 0$$ $$x^3 + y - xy^3 = 1$$ I'm also asked to solve analytically but I'm pretty sure the closed form solution doesn't exist because if yes, then the second equation would involve…
kensaii
  • 387
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Linear Relationship of two equations

if $0.46a = 120b$ and $2.68a = 60b$ The relationship is linear. what does $0b$ equal in terms of $a$? what does $1b$ equal in terms of $a$? A method to work this out would also be nice, I have tied myself up in knots. Many Thanks
Duncan
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How to link the eigenvalues to the components from PCA

I have a difference matrix from daily changes which I use to construct a covariance matrix. On this covariance matrix I use the power method to get the eigenvalues. The power method yields exactly the same eigenvalues as the eig function in matlab,…
Oamriotn
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Prove that I -xx* is singular if and only x*x = 1

(=>) Suppose I - xx* is singular if and only there is a y such that (I−xx*)y=0, i.e. xx* y=y. Now set λ=x* y. Then y=λx, i.e. xx* λx=λx Thus λx(x* x) = λx => x* x = 1 (<=) Suppose x*x = 1 Please help
shimura
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Normal Equations error bounds

$A^TAx = A^Tb$ $A^TA\hat{x} = A^Tb + f$ where $\lVert f\rVert \leq cu\lVert A\rVert\lVert b\rVert$ Show that $\frac{\lVert x-\hat{x}\rVert}{\lVert x\rVert} \leq cuK(A)^2\frac{\lVert A\rVert\lVert b\rVert}{\lVert A^Tb\rVert}$ My approach was…
Srinivas Eswar
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help to find SOR optimal parameter $w$

Please let following linear system as $Ax=b$: $$\begin{array}{l} 6a{x_1} + {x_2} + {x_3} = 1\\ {x_1} - 3a{x_2} + 4{x_3} = 2\\ {x_1} + {x_2} - 2a{x_3} = 3 \end{array}$$ Help me to prove that the optimal value $w$ in SOR method is $\dfrac{6a}{1+6a}$.
SKMohammadi
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Finding the minimum of Condition number for this matrix

Suppose $A=\left[ {\begin{array}{*{20}{c}} {0.1\alpha }&{0.1\alpha }\\ 1&{1.5} \end{array}} \right]$. How can we find minimum of condition number $k(A)=\Vert A\Vert \Vert A^{-1} \Vert$ (Assume $\infty$-Norm)? And for what value of $\alpha$?
SKMohammadi
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Jacobi Iteration with Shift

The question is to solve a linear system using Jacobi iterations with a shift of mu = 5. My code converges very quickly, but it does not yield the results that MATLAB gives with the backslash operator. Perhaps I'm implementing the shift part…
jb68321
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