Mathematics Stack Exchange
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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condition number of rectangular matrix with epsilon

$ A = \begin{bmatrix} 1 & 1 & 1 \\ \epsilon & 0 & 0\\ 0 & \epsilon & 0 \\ 0 & 0 & \epsilon \end{bmatrix}$ Find $K_2(A)$ (where $K_2(A) $ represents the condition number of the matrix with respect to $\|.\|_2$ norm. ) I find $\|A\|_2 =…
Aritra Roy
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Given an M x N matrix, is there a way to produce an orthogonal set of N vectors of length M, where M < N?

Gram-Schmidt orthogonalization would only use the first M vectors to generate a basis of size M x M.
Jack
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Solving linear system corresponding to an upper triangular sparse matrix of exponential size

I have an algorithm for generating some upper triangular sparse matrix $A$ of size $2^n\times 2^n$, which is invertible. I want to solve the linear system $A\mathbf{x}=\mathbf{1}$ where $\mathbf{1}$ is all ones vector for $n=100$. I wonder if there…
kswim
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posteriori extimate of the relative error in solving $Ax = b$, $||b||_2 = 1$

Let $A = \begin{bmatrix} 1 & -1 \\ 1 & -1.00001\end{bmatrix}$, give a posteriori extimate of the relative error in solving $Ax = b$, $||b||_2 = 1$. To do this I was thinking of first finding the condition number of $A$ as follows. Letting $x =…
Math_Day
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How does this step occur in the proof?

I was looking at the proof for the 1-Norm of a matrix. $A$ is any $m x n$ matrix and $||A||_1$ is the maximum column sum of $A$. The proof goes like this: Write $A$ in terms of its columns. $A=[ a_1 | ... | a_n ]$, where each $a_j$ is an $m-$…
user733666
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How to calculate trend line

I have the following dataset 5 338.45 6 7 8 9 10 350.75 11 12 13 14 15 380.70 I need to calculate the missing values by plotting a linear trend line between the values I know (5, 10 and 15) I did that by doing (350.75-338.45)/6 = 2.05 then adding…
Ben
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Norm of inverted matrix

Let be $ \|\cdot \| $ an induced matrix norm and $A, B\in \mathbb{R}^{n,n}$ where that $A$ is invertible. Further more it is $ \|B\| < \|A^{-1}\|^{-1} $. Then $ A+B $ is invertible. My idea was to show this: If $ (A+B)\cdot x=0 $ then it follows $…
hallo007
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Rayleigh quotient of $q$ with respect to matrix A

my notes say (I'm self studying) that the Rayleigh quotient minimizes $Aq=pq$ I have no idea what that means. I did a quick search online and I found out that this quotient is a complex number $p$, $q^HAq = q^Hqp $ I have no idea how I shoud prove…
reiji sakamaki
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Calculate the rank of a circular matrix over GF(2)

How can we calculate the rank of a large circular matrix over GF(2) efficiently ? for instance ( 10^6* 10^6 matrix ) ( Best algorithm )
meh98
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LU factorization: which row should be chosen in partial pivoting?

I've seen partial pivoting described thus: during the $k$th step of LU factorization of $\mathbf{A}$, find the remaining element of $\mathbf{A}$ in column $k$ with the greatest absolute value, and swap its row with the $k$th row. However, I can…
Will
  • 505
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Finding pertubations that obtain condition numbers

I'm having trouble with number 5 of this question below. I managed do do the other 4 though. This question comes from Trefethen & Bau - Numerical Linear Algebra. $$ A = \begin{bmatrix} 1 & 1 \\ 1 & 1.0001\\ 1 & 1.0001\\ \end{bmatrix} \,\,\, B=…
Frank
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Cubic Curve critical point

a=-1.89;b=1; ContourPlot [y^2 ==x^3+a x+b, {x,-2,2},{y,-2,2},GridLines->Automatic] Clear[a];FindRoot [0==x^3+a x+b,{a,-1.9}] NSolve[ 0==x^3+a x+b,a] Trying to find constant $a\approx 1.89 $ more accurately for cross-over point. Please help.
Narasimham
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5.1.12, Matrix Analysis & Applied Linear Algebra - Carl D Meyer

$\alpha^\lambda .\beta^{(1-\lambda)}\le \lambda.\alpha+(1-\lambda).\beta\Rightarrow\sum_{i=1}^{n}|\hat x_i .\hat y_i|\le\frac{1}{p}.\sum_{i=1}^{n}|\hat x_i|^p+\frac{1}{q}.\sum_{i=1}^{n}|\hat y_i|^q$ $\hat x=\frac{x}{||x||_p}$ ; $\hat…
Rebeca Lie Yatsuzuka Silva
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power method and inverse iteration

True or False and please justify this answer. when the power method is applied to find the largest eigenvalue and the corresponding eigenvector of a matrix, the vectors are normalized at each step in order to increase the rate of convergence of the…
Tina
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Definition of a scalar, quadratic function of a vector

I've been studying Conjugate Gradient method. Before the method is explained, the book starts with the definition of a scalar, quadratic function of a vector like the below. $$f(x) = \frac{1}{2}x^TAx - b^Tx+c \quad \quad \text{where A : matrix, b:…
alryosha
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