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Questions tagged [numerical-analysis]

99 questions
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2 answers

Computing inverse matrix when an element changes

Given an $n \times n$ matrix $\mathbf{A}$. Let the inverse matrix of $\mathbf{A}$ be $\mathbf{A}^{-1}$ (that is, $\mathbf{A}\mathbf{A}^{-1} = \mathbf{I}$). Assume that one element in $\mathbf{A}$ is changed (let's say $a _{ij}$ to $a' _{ij}$). The…
AJed
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Alternatives to SVD for rank factorization

I have rank-deficient matrix $M \in \mathbb{R}^{n\times m}$ with $\text{rank}(M) = k$ and I want to find a rank factorization $M = PQ$ with $P \in \mathbb{R}^{n \times k}$ and $Q \in \mathbb{R}^{k \times m}$. A popular approach is to compute the…
Artem Kaznatcheev
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14
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Floating point rounding

Can an IEEE-754 floating point number < 1 (i.e. generated with a random number generator which generates a number >= 0.0 and < 1.0) ever be multiplied by some integer (in floating point form) to get a number equal to or larger than that integer due…
Cade Roux
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Overflow safe summation

Suppose I am given $n$ fixed width integers (i.e. they fit in a register of width $w$), $a_1, a_2, \dots a_n$ such that their sum $a_1 + a_2 + \dots + a_n = S$ also fits in a register of width $w$. It seems to me that we can always permute the…
Aryabhata
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Is 2**x faster to compute than exp(x)?

Forgive the naïveté that will be obvious in the way I ask this question as well as the fact that I'm asking it. Mathematicians typically use $\exp$ as it's the simplest/nicest base in theory (due to calculus). But computers seem to do everything in…
isomorphismes
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Does Automatic Differentiation handle conditional branches, if yes how?

I'm trying to understand how Automatic Differentiation (AD) works. For simple algebraic operation, I get the chain rule thing. But, when the code contains conditional statement like 1: test_sign = x*y < 0 2: if test_sign : 3: biggest =…
user10018
8
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2 answers

Is "ternary search" an appropriate term for the algorithm that optimizes a unimodal function on a real interval?

Suppose that I want to optimize a unimodal function defined on some real interval. I can use the well-known algorithm as described in Wikipedia under the name of ternary search. In case of the algorithm that repeatedly halving intervals, it is…
Pteromys
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Detecting overflow in summation

Suppose I am given an array of $n$ fixed width integers (i.e. they fit in a register of width $w$), $a_1, a_2, \dots a_n$. I want to compute the sum $S = a_1 + \ldots + a_n$ on a machine with 2's complement arithmetic, which performs additions…
Gilles 'SO- stop being evil'
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numerical integral vs counting roots

I have a problem that can be viewed in two different ways: Compute an $n$-dimensional integral, numerical context. The domain of integration is an $n$-dimensional hyper-cube of side $L$. Count (just count) the roots of an $n$-dimensional function…
labotsirc
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Reception of numerical infinities

A group of computer scientists associated with numerical analyst Yaroslav Sergeyev have published numerous publications recently on a scheme proposed by Sergeyev that uses terms like Infinity Computer, numerical infinitesimal and numerical infinity,…
Mikhail Katz
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Numerical methods: why doesn't this python code return 1.0?

I typed the following into the python console: >>>from numpy import float64 >>>x=float64(1.98682855148322934369) >>>x np.float64(1.9868285514832293) >>>y=float64(1)/x >>>x*y np.float64(0.9999999999999999) My argument for why the above code should…
Beatnik Dopa
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Big O notation: removing big O from denominator

In A First Course in the Numerical Analysis of Differential Equations (page 26) Arieh Iserles gives the following…
Simon Dirckx
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3 answers

Why is computation of this function numerically unstable?

Why is computing the function at certain ranges of h $$f(x) = \dfrac{\sqrt{-x+a} - \sqrt{2x+a}}{ 4a}$$ unstable? How would I rewrite this function so that it is stable?
John
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Arc-Length parameterization of a cubic bezier curve

I like to implement an arc-length Parameterization of a cubic bezier curve. So far I have implemented the method of calculating the arc length of the curve and now I'm stuck at calculating the times to divide the original curve into equal arc length…
user39558
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Testing whether an analytic function vanishes identically

I have an application that basically reduces to testing whether a given function vanishes identically. The function is given symbolically, using unary and binary operators on complex numbers. For example, we might want to test the function…
user20095
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