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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Does the convergence of conjugate gradient algorithm depend only on the condition number, or does the shape of the eigenspectrum matter?

I am working with a very large linear system, using a flexible pre-conditioned conjugate gradient solver. The system is (B + C)x = b, with B and C symmetric and positive definite, and I know the condition number of C, but not B. I have two…
Bill Campbell
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Very high condition number

I am solving a linear system of equations with a condition number of 10^18 with matlab with double precision. As I know as a rule of thumb every power of 10 in condition number reduces one significant digit of the answer and hence with double…
Soleiman
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What is the function of the pivot index vector in Gauss Jordan Elimination with full pivoting?

In numerical recipes, on page 39 (page 4 of the pdf) the following algorithm has been suggested for finding a pivot: void gaussj(float **a, int n, float **b, int m) /* Linear equation solution by Gauss-Jordan elimination, equation (2.1.1) above.…
igor
  • 473
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Matrix norm relation

I've been trying to solve this for 3 hours.. If $A$ is an $n \times n$ matrix with $\|A\|<1$ in any norm, then show that $\|(I-A)^{-1}\| \leq \frac {1}{1-\|A\|}$. My trying is: $$ \|(I-A)^{-1}\| = \|A^{-1} - I^{-1}\|=\| A^{-1} - I\| \leq…
AYARcom
  • 364
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Gauss Seidel Iteration for a specific matrix

We seek to solve $Au= f$ via iteration, where $$ A = \left ( \begin{array}{cc} I & S \\ -S^T & I \end{array} \right ) $$ Where $S$ is an arbitrary square matrix in $R^n$ and $I$ is the identity matrix for $R^n$. Now consider the relaxation…
InfiniteElementMethod
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Relationship between QR and LU factorization

Both algorithm return very similar results in terms of having a upper/right triangular matrix as one of the factors. What is the relationship between Q and L, and between R and U? What is the intuition behind this relationship?
mymindcastadrift
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Proof that Householder Triangularization for QR is backward stable

How do you prove that QR factorization via Householder Triangularization is backward stable? Theorem 16.1 (From Trefethen and Bau): Let the $QR$ factorization of a matrix $A$ be computed by Householder triangularization on a computer satisfying the…
mymindcastadrift
  • 80
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Linear algebra

Question. Let $a_1,...a_n\in\{0,1,-1\}^m$ and $\sum a_i=(1,...,1)$. Is there a permutation $\tau$ of $\{1,...,n\}$ Such that for each $k\in \{1,...,n\}$ the vector $\sum_{i=1}^k a_{\tau (i)}$ has entries greater or equal than -1?
Golab
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Determining Nullspace Basis such that only one column is deleted or added as row is added or deleted, and remaining columns of basis stay the same

I would like to compute, in MATLAB, the basis Z for the nullspace of an m by n matrix A, such that if one row of A is added (resulting in A_a), the basis for A_a is n-m-1 of the n-m columns of Z, i.e., is obtained by deleting one column from Z.…
Mark L. Stone
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QR Algorithm fails under certain conditions

First of all, i have to admit that i am really knew to this numeric stuff. I have to detect two complex Eigenvalues of a Matrix and therefor i implemented some easy QR-Algorithm with MatLab. I am using the Wilkinson-Shift - i think. My Code Looks…
Roland
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Hessenberg reduction

Given $A \in \mathbb{R}^{nxn}$ and $z \in \mathbb{R}^n$, find orthogonal $Q$ such that $Q^TAQ$ is upper Hessenberg and $Q^Tz = \beta e_1$. My attempt so far, Individually I can find the Householder matrix $H_1 z= \beta e_1$ and $Q_1^TAQ_1 = R$ such…
Srinivas Eswar
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solving given linear equation

So before you guys judge me, I honestly am so clueless with this so please bare with my dumb questions. I have been at this equation that I've been going for an hour now, $\frac{(3x-1)}{2} -2 = 5-\frac{4x}{3+1}$ and I'm trying to solve for $x$. So…
Ardine
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If normal linear regression finds $A$ such that $AX \sim Y$, then how do I solve $BAX \sim Y$?

If normal linear regression finds $A$ such that $$AX \sim Y$$ then how should I solve $$BAX \sim Y$$ where $B$, $X$ and $Y$ are given (non-invertible) matrices? I could of course derive the solution by hand, but I need a numerically stable…
Jumpersky
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Is it possible to optimize solution of this linear system?

I have a matrix of the form: A1 A2 A3 A4 A5 A6 ... An P | 1 0 0 0 0 0 ... 0 -1 | c * f1 f1 1 0 0 0 0 ... 0 -1 | c * f2 f2 f2 1 0 0 0 ... 0 -1 | c * f3 f3 f3 f3 1 0 0 ... 0 -1 | c * f4 f4 f4 f4 f4 1 0 ... 0 -1 | c * f5 f5 f5…
cpacheco
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If $A$ is Symmetric Positive Definite(SPD) matrix, is $A+E$ SPD?

Let $A$ be symmetric positive definite matrix and $E$ is symmetric with $||E||_{2} < ||A^{-1}||^{-1}_{2}$ then prove that $A+E$ is symmetric positive definite. -- \ Observation; Since $A$ is invertible and $A+E = A(I+A^{-1}E)$ since $||A^{-1}E||_{2}…
aranel
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