forward elimination

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1: 3.2 Linear Algebra

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§3.2(i) Gaussian Elimination

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Iterative Refinement

… ►Forward elimination for solving $𝐀𝐱=𝐟$ then becomes $y_1=f_1$, …

2: 3.6 Linear Difference Equations

… ►where $\mathrm{\Delta }{w}_{n-1}=w_n-w_{n-1}$, ${\mathrm{\Delta }}^2{w}_{n-1}=\mathrm{\Delta }{w}_n-\mathrm{\Delta }{w}_{n-1}$, and $n\in \mathbb{Z}$. … ►If, as $n\to \mathrm{\infty }$, the wanted solution $w_n$ grows (decays) in magnitude at least as fast as any solution of the corresponding homogeneous equation, then forward (backward) recursion is stable. … ►Then computation of $w_n$ by forward recursion is unstable. … ►(This part of the process is equivalent to forward elimination.) … ►Within this framework forward and backward recursion may be regarded as the special cases $\mathrm{\ell }=0$ and $\mathrm{\ell }=k$, respectively. …

3: 16.25 Methods of Computation

… ►In these cases integration, or recurrence, in either a forward or a backward direction is unstable. …

4: 5.21 Methods of Computation

… ►Or we can use forward recurrence, with an initial value obtained e. …

5: 11.13 Methods of Computation

… ►Then from the limiting forms for small argument (§§11.2(i), 10.7(i), 10.30(i)), limiting forms for large argument (§§11.6(i), 10.7(ii), 10.30(ii)), and the connection formulas (11.2.5) and (11.2.6), it is seen that ${𝐇}_{\nu }\left(x\right)$ and ${𝐋}_{\nu }\left(x\right)$ can be computed in a stable manner by integrating forwards, that is, from the origin toward infinity. The solution ${𝐊}_{\nu }\left(x\right)$ needs to be integrated backwards for small $x$, and either forwards or backwards for large $x$ depending whether or not $\nu$ exceeds $\frac{1}{2}$. For ${𝐌}_{\nu }\left(x\right)$ both forward and backward integration are unstable, and boundary-value methods are required (§3.7(iii)). … ►In consequence forward recurrence, backward recurrence, or boundary-value methods may be necessary. …

6: 3.9 Acceleration of Convergence

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3.9.2 $$S=\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k{2}^{-k-1}{\mathrm{\Delta }}^k{a}_0,$$

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Symbols:

$\mathrm{\Delta }$: forward difference operator and $S$: a convergent series

Permalink:

http://dlmf.nist.gov/3.9.E2

Encodings:

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See also:

Annotations for §3.9(ii), §3.9 and Ch.3

… ►Here $\mathrm{\Delta }$ is the forward difference operator: ►

3.9.3 $${\mathrm{\Delta }}^k{a}_0={\mathrm{\Delta }}^{k-1}{a}_1-{\mathrm{\Delta }}^{k-1}{a}_0,$$ $k=1,2,\mathrm{\dots }$.

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Symbols:

$\mathrm{\Delta }$: forward difference operator

A&S Ref:

3.6.27

Permalink:

http://dlmf.nist.gov/3.9.E3

Encodings:

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Annotations for §3.9(ii), §3.9 and Ch.3

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3.9.4 $${\mathrm{\Delta }}^k{a}_0=\sum _{m=0}^k{(-1)}^m\left(\genfrac{}{}{0.0pt}{}{k}{m}\right)a_{k-m}.$$

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Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient and $\mathrm{\Delta }$: forward difference operator

A&S Ref:

3.6.27

Permalink:

http://dlmf.nist.gov/3.9.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §3.9(ii), §3.9 and Ch.3

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7: Bille C. Carlson

… ►The main theme of Carlson’s mathematical research has been to expose previously hidden permutation symmetries that can eliminate a set of transformations and thereby replace many formulas by a few. …

8: 3.10 Continued Fractions

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Forward Recurrence Algorithm

… ►In general this algorithm is more stable than the forward algorithm; see Jones and Thron (1974). ►

Forward Series Recurrence Algorithm

… ►In Gautschi (1979c) the forward series algorithm is used for the evaluation of a continued fraction of an incomplete gamma function (see §8.9). … ►This forward algorithm achieves efficiency and stability in the computation of the convergents $C_n=A_n/B_n$, and is related to the forward series recurrence algorithm. …

9: 7.22 Methods of Computation

… ►See Gautschi (1977a), where forward and backward recursions are used; see also Gautschi (1961). …

10: 18.1 Notation

… ►Forward differences: ►

$${\mathrm{\Delta }}_x\left(f(x)\right)=f(x+1)-f(x),$$

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$${\mathrm{\Delta }}_x^{n+1}\left(f(x)\right)={\mathrm{\Delta }}_x\left({\mathrm{\Delta }}_x^n(f(x))\right).$$

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