# Tag Info

Accepted

### Relating condition number of hessian to the rate of convergence

The classic zig-zag picture is misleading since it suggests that the slow convergence is due to overshooting. But for ill-conditioned problems, convergence is still slow even on a purely quadratic ...
• 19k
Accepted

• 19k
Accepted

### Matrix condition number and loss of accuracy

It is a rule of thumb, but there are some estimates backing that rule. Given a matrix $A∈ℝ^{n×n}$ and a vector $b∈ℝ^n$ the linear equation system ought to be solved is $$Ax=b.$$ Now we have ...
• 3,672
Accepted

### Condition number of random matrix gets worse as dimension grows

This is correct, and is a (very famous) result of Alan Edelman's from his PhD thesis: Edelman, Alan, Eigenvalues and condition numbers of random matrices, SIAM J. Matrix Anal. Appl. 9, No. 4, 543-560 (...
• 26.1k
Accepted

### Condition number of matrix is $1$ iff $A^TA=\alpha I$

Note that if $\kappa$ is the condition number (with respect to the Euclidean norm), then $$\kappa(A) = \frac{\sigma_1(A)}{\sigma_n(A)}$$ Where $\sigma_1$ and $\sigma_n$ denote the highest and lowest ...
• 227k

### How is $f(x)=x+1$ not backwards stable if I consider the error propagated in the addition?

The fundamental problem is that the domain is not clearly stated. We have two functions which are relevant in this context. Let $\mathcal F$ denote our set of floating point numbers. Then the relevant ...
• 12.9k
Accepted

### Is Schur complement better conditioned than the original matrix?

Lemma. If $P$ and $Q$ are two $n\times n$ Hermitian matrices and $\operatorname{nullity}(Q)=k>0$, the minimum eigenvalue of $P+Q$ is bounded above by the $k$-th largest eigenvalue of $P$. Proof of ...
• 141k

### Why is the condition number of a matrix given by these eigenvalues?

$$\operatorname{cond}(A)=\dfrac{\max_{|w|=1}|Aw|}{\min_{|w|=1}|Aw|}$$using SVD we have$$A=UDV$$where $U$ and $V$ are unitary and $D$ is diagonal whose main diagonal entries are eigenvalues of $A$. ...
• 32.5k
Accepted

• 10.4k