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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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What math topic teaches you to optimize numerical computations?

I was recently reading Trefethen & Bau - Numerical Linear Algebra, when I came accross this paragraph: In fact, there is much more to the cost of an algorithm than operation counts. On a single-processor computer, the execution time is affected by…
Frank
  • 880
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Prove that for an induced matrix norm, the norm of Identity matrix is 1.

I am not sure how to even start this proof. I know that for any induced matrix norm, $||Ax||$ is less than or equal to $||Ax||$ $||x||$.
Vishnu P
  • 43
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Arrange the computation of $f(x)$ so that the loss of significant figures can be reduced using $4$-digit decimal arithmetic

For the function $f(x)=\sqrt{x^2+1}-x$, have computed $f(100)=0$ using 4-digit decimal arithmetic (rounding after every intermediate calculation). The value $f(100) = 0.0049998750$ (to 8sf) is given and I have computed a relative error of $1$ using…
confused
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$l^1$ norm estimate for inverse of Vandermonde matrix

As title, I would like to know the known upper bound for the $l^1$ norm for inverse of Vandermonde matrix. A quick search gives this paper by Gautschi 40 years ago, but it deals with the infinity norm instead. Thanks! Motivation: Part 2 of this…
user27126
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2 answers

Spectrum of Schur complement

Good afternoon! I have a question about the eigenvalues of a schur complement. Assume $\mu_1 \leq ... \leq \mu_n$ are eigenvalues of the schur-complement $S$ to a positive definite block-matrix $A$ and $0 \leq \lambda_1 \leq ...\leq \lambda_n$…
Donut
  • 235
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Shortest Distance between a Point and a Numerical 2D Curve

I have a 2D Curve. I have all the numerical values for the line within a certain range. I do not have an equation for this line. At several points in this 2D space I want to calculate the shortest distance from a point, P, to this non-linear line…
user1011182
  • 111
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Finding eigenvalues in a region

I have an eigenvalue problem (of a very large (n~1000000) but sparse complex system) wherein I need to determine all the eigenvalues in a certain region(a rectangle) in the positive half plane. I am currently using the traditional shift invert…
flying aboard
  • 21
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eigenvalues for symmetric and non-symmetric matrices

I know the Power methods and Jacobi methods are suitable to finding eigenvalues for symmetric matrices, please tell me other methods for this matrices. And what are the methods for the Non-symmetric eigenvalue problems? Thanks a lot.
user115927
  • 59
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1 answer

Computing the best-fit plane normal from n points

I've been working steadily through "3D Math Primer for Graphics and Game Development" and am stuck on how the authors derived their equation for the best-fit plane normal given n points. Please note, I'm not necessarily looking for someone to derive…
user3192018
  • 21
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Conditioning considerations in least square solutions via the normal equations

I'm a little unsure about how to classify conditioning issues with solving least squares equations via the normal equations approach. I'm hoping to get verification that what I say below is correct, and have it pointed out if it is wrong. There…
Fractal20
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$F(x)=Ax+b$ is a contraction mapping

I want to proof that $f(x)=Ax+b$ with \begin{align} A = \begin{bmatrix} 0 & -\frac{1}{8} & \frac{1}{4} \\ 0 & \frac{1}{3} & 0 \\ -\frac{1}{2} & -\frac{5}{22} & \frac{3}{4} \end{bmatrix} && b=[1,2,3]^{T} \end{align} is a contraction mapping, that…
TheWaveLad
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Numerical range of a matrix contains the convex hull of the eigenvalues.

I am stuck with the following question. Question: Let $A \in \mathbb{C}^{m \times m}$ be arbitrary. Let $W(A)$ be the numerical range i.e. the set of all Rayleigh quotients of $A$ corresponding to a all nonzero vectors $x \in \mathbb{C}^m$. Show…
Math Student 020
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Characterizing a matrix with identical eigenvalues

Suppose $n \times n$ hermitian and positive semi-definite matrix $A$ is given. We can rewrite $A$ using its eigen decomposition, $$ A = U_A \Lambda_A U_A^H. $$ Now suppose matrix $B$ is also $n \times n$, hermitian, and positive semi-definite.…
Mona Hajimomeni
  • 55
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Why use linear solve instead of inverse in inverse power iteration

Suppose $A$ is a matrix and that we want to apply inverse iteration with a shift of $\mu$. The inverse iteration method usually involves solving a linear system $(A-\mu I)w=v_{k-1}$ to obtain the $k$-th iterate, $v_k=w/\Vert w \Vert$. Now, I…
user2978125
  • 349
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Computing Singular Value question

Good afternoon. I'm studying for my finals of this year, currently studying for the exam "Numerical Linear Algebra". I'm trying to solve some of the questions the teacher asked the past years (for preparation), but I can't seem to figure out a few,…
Timmy Nelen
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