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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Gaussian Elimination and condition number

When we use Gaussian Elimination algorithm to linear system, do we care about the condition number of the matrix? Why or Why not?
Jianqiang Li
  • 3
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2 answers

Well-conditioned problem and a stable algorithm?

I am trying to figure out the difference between Well-conditioned problem and a Stable algorithm: A well-conditioned problem is one in which rounding errors play very little role. A stable algorithm is one in which rounding errors are guaranteed not…
dxdydz
  • 1,371
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calculating b splines with non-distinct knots

How can one calculate B splines with non distinct knots. I know about carl de boor algorithm, but that is to my knowledge only applied to non distinct knots. We can assume the knots are in increasing order. In particular im looking to find…
kayaya75
  • 3
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How to quickly determine whether the intersection of two linear subspaces contain nonzero vectors?

How to quickly determine whether the intersection of two linear subspaces contain nonzero vectors? Please give a algorithm. i.e., both $Ax=0$ and $Bx=0$ have nonzero solutions. where the vector x has the same dimension. We know that all the…
Wei Jiang
  • 77
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Approximate low-rank $U^\top U$ decomposition / Gaussian elimination.

Page 66 in this slideset discusses and presents an example for the following idea: given the idea that a (Laplacian, hence square) matrix can be exactly decomposed as $$ A = U^\top U $$ (which could be obtained by Gaussian elimination), define the…
user8948
  • 248
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Lanczos algorithm for complex symmetric matrices with partial reorthogonalization

My research is in electron spin resonance, where we deal with the time-independent stochastic Liouville equation and end up with the task of finding the eigenvectors and eigenvalues of complex symmetric matrices (size ~ 10000x10000, ~30% elements…
user3812405
  • 11
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When does GMRES method compute norm of residual incorrectly?

I'm trying to diagnose what I assume is a bug in a code that uses the GMRES method to solve a system of linear equations. At each GMRES iteration the norm of the residual is printed. When GMRES converges the norm of the residual is computed…
Andrew
  • 349
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why its paradoxical to try make $a$ the subject of $(a-b)/(b-a) = c?$

so one day i just made a random formula to try and make $a$ the subject and i made $a-b/b-a = c$ ok first you need to times both side by $b-a$ to get $a-b=cb-ca$ now you need to $b-ca$ on both sides to get $a+ca=b+cb$ now factorise to get…
Thomas Holloway
  • 25
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Finding the best 1-rank approximation

I am trying to solve a 5x5 matrix to find first SVD and then 1-rank approximation but I have a problem in finding the eigenvaluse of it
Ms Hamda
  • 1
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Searching for the uncertainty in QR decomposition

An invertible Matrix $A \in \Bbb R ^{2 \times2}$ and $b \in \Bbb R ^{2}$ are to $x \in \Bbb R^2$ the definite solution $Ax=b$. $x_N$ is the solution to the normalization equation $$A^TAx_N = A^Tb $$ and $x_Q \in \Bbb R^2$ the solution to $$QRx_Q = b…
Maxim
  • 161
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Question about the meaning of "exact" in backward stability

In Numerical Linear Algebra by Trefethen&Bau, backward stability It is mentioned that a backward stable algorithm gives "exactly" the right answer to nearly the right question. What I am thinking is, an algorithm has error. How can it be exactly…
Ronald Ku
  • 101
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Order of Convergence in the Simultaneous Power Method for finding Eigenvectors

Why the $C$ constant is defined like this in the Theorem: There is the proof of Theorem provided by the book, i underlined what i do not understood. The text where i found this Theorem was Numerical Linear Algebra , Trefethen and Bau.
Joan Guastalla
  • 75
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numerical linear algebra reflectors

This is an intro to numerical linear algebra course. I am not sure how to start with this proof. Let $u \in \mathbb{R}^n$ and suppose that $\| u \|_2 = 1$. If $P=uu^\top$, prove that a.)$p^2=p$ b.)$p^\top = p$
user123456
  • 11
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Efficiently updating a vector

What is the most efficient way to make this linear algebra computation? I am interested in computing a vector $y^{(k)}$ that updates as shown below. $$y^{(k)} = A^k B A^k x$$ where the matrices $A,B \in \mathbb{R}^{n \times n}$ are tridiagonal…
user17762
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Numerical linear algebra orthogonal

Need help on how to prove this numerical linear algebra problem. 1.)Let $Q \in M_n(\mathbb{R})$ and suppose that $\langle Qx, Qy \rangle = \langle x, y \rangle$ for every $x$, $y \in \mathbb{R}^n$. Prove that $Q$ is orthogonal.
user123456
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