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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Rank deficiency in null space computation via SVD

Dear numerical algebra experts, I am trying to find $\alpha$ for which $A(\alpha)x=0$ has a non-trivial solution ($x\neq0$), i.e. I am looking for $\alpha_0$ for which the null space of $A(\alpha_0)$ is not empty. A (square, NxN, real) is the…
Michael Steinke
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Characterization of the behavior of the residuals in Conjugate Gradient

In conjugate gradient method for solving symmetric positive definite linear system $Ax=b$, which can also be regarded as a convex optimization problem $\dfrac{1}{2} x'Ax - x'b$, the $A$-norm of the error $\|x_k-x_\star\|_A$ at iterate $k$ is…
bernard
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Matrix in Matlab

I'd like to compute the centralizer of a subgroup $H$ of orthogonal group $O(8, R)$, so I need to solve the equation $AX=XA, BX=XB \mbox{ where } H=\langle A, B\rangle.$ The problem that I have is matrices A and B are symbolic, in fact their entries…
Jayq
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Gershgorin discs Theorem

Let $i_0\in \{1,\ldots ,n\},\alpha >0$. Then$$K_{i_0}:=\left \{z\in \mathbb{C}:|z-a_{i_0i_0}|\leq \alpha \sum \limits _{j=1 \\ j\neq i_0}^n|a_{i_0j}|\right \}$$has exactly one eigenvalue if and only if$$K_{i_0}\cap \left…
Uhmm
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A query about the reciprocals of condition numbers of nonsingular matrices

The condition number of a nonsingular matrix $A$ is defined as $$ \kappa(A) = \mbox{cond}(A) = \Vert A \Vert \Vert A^{-1} \Vert \tag{1} $$ and its value is dependent on the choice of the norm in (1). However, as all $p$-norms are equivalent in a…
Dr. Sundar
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Conjugate gradient convergence

Suppose a symmetric positive definite matrix $A$ has one thousand eigenvalues uniformly distributed between $1$ and $10$, and one eigenvalue of $10^4$. Suppose another symmetric positive definite matrix $B$ has an eigenvalue of $1$ and has one…
Tomer
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The sum of square of eigenvalues equals minimal Frobenius norm under similar transformation

I've been working out a question in Applied Numerical Linear Algebra by James W. Demmel in Chapter 4 but totally stuck. The question is: let $A$ be any n-by-n matrix with eigenvalues $\lambda_1, \ldots, \lambda_n$. Prove that $$ \sum_{i=1}^n…
Henry Davii
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Why do we use Jacobi method?

If Gauss-Seidel is more efficient than Jacobi method, then why do we need to learn it? Does it have any advantages over Gauss-Seidel?
sarach
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Short-cut to a group of long sums/differences

If I have data $a,b,c,d$, and want to calculate $x=a+b-c-d$, $y=a-b-c+d$ and $z=a+b+c-d$, I can save three adds by doing $e=a-c$, $f=b-d$, then $x=e+f$,$y=e-f$, $z=a+c+f$. If I have 100 data values $a_1..a_{100}$ and twenty sum/differences…
Empy2
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Interpret GMRes residual plateau

Forgive me if my question is simple and please feel free to refer to an old answer. I am experimenting with preconditioned GMRes with no restart. I wonder if there is an interpretation of this plateau in the GMRes residual plot? I mean that the…
lumberman
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why is $x(\cdot)(x(+)1)$ not backward stable for floating point input?

I am not able to understand this as i have also read an example $1(+) x$ is also not backward stable but still not able to figure it out
cannabis
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NLA Qualifying Exam Sample Question

I'm curious if anybody has some advice on a problem. Suppose $A$ is a real symmetric matrix with eigenvalues more or less uniformly distributed over $\left[2,18\right]$ together with an outlier at $\lambda=50$. How many steps of the conjugate…
Matt Tucker
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Algorithm to compute similarity computation

I have a similarity transformation of matrices from the type $B = P^{-1}AP$. It is known that $A$ and $P$ are invertible matrices, but not orthogonal. Given that I have the matrices $P$ and $A$ I need to compute $B$ in time $\frac{7}{3}n^{3} +…
oxy
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Singular Value Decomposition of $A^{T}A$

How to show that the singular values of $A^{T}A$ are $σ_{1}^{2},...,σ_{n}^{2}$, and thus $∥A^{T}A∥_{2}$ = $σ_{max}^{2}$. In order to do this I have to find a singular vale decomposition for $A^{T}A$ ?
user535425
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prove 1d back substitution is backward stable

I'm wondering why backward substitution is backward stable. For the 1D case $ax=b$, Let $x = f(a, b) = \frac{b}{a}$, and let $\bar{f}$ be an algorithm for $f$. Also let $\epsilon$ be a small error. If $a \neq b$: $\bar{f}(a, b) =…
Frank
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