Haar method

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11: 1 Algebraic and Analytic Methods

Chapter 1 Algebraic and Analytic Methods

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12: 3 Numerical Methods

Chapter 3 Numerical Methods

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13: Amparo Gil

… ► Temme) of the book Numerical Methods for Special Functions, published by SIAM in 2007. … ►

3 Numerical Methods

14: Javier Segura

… ►Temme) of the book Numerical Methods for Special Functions, published by SIAM in 2007. … ►

3 Numerical Methods

15: 14.32 Methods of Computation

§14.32 Methods of Computation

… ►In other cases recurrence relations (§14.10) provide a powerful method when applied in a stable direction (§3.6); see Olver and Smith (1983) and Gautschi (1967). ►Other methods include: …

16: 35.4 Partitions and Zonal Polynomials

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35.4.3 $$Z_{\kappa }\left(\mathbf{T}\right)=Z_{\kappa }\left(\mathbf{I}\right){\left|\mathbf{T}\right|}^{k_m}\underset{\mathbf{O}(m)}{\int }\prod _{j=1}^{m-1}{|{(\mathbf{H}\mathbf{T}{\mathbf{H}}^{-1})}_j|}^{k_j-k_{j+1}}\mathrm{d}\mathbf{H},$$ $\mathbf{T}\in \mathbf{𝒮}$.

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Symbols:

$\in$: element of, $\int$: integral, $Z_{\kappa }\left(\mathbf{T}\right)$: zonal polynomial, $\mathbf{I}$: $m\times m$ identity matrix, $\mathbf{𝒮}$: space of real symmetric $m\times m$ matrices, $\mathbf{T}$: real symmetric $m\times m$ matrix, $\left|\mathbf{X}\right|$: determinant of $\mathbf{X}$ , $|{(\mathbf{X})}_j|$: $j$th principal minor of $\mathbf{X}$ , $\mathbf{O}(m)$: space of $m\times m$ orthogonal matrices, $\mathbf{H}$: orthogonal $m\times m$ matrix, $\mathrm{d}\mathbf{H}$: normalized Haar measure on $\mathbf{O}(m)$ , $j$: nonnegative integer, $k$: nonnegative integer, $m$: positive integer and $\kappa$: vector of nonnegative integers

Referenced by:

§35.4(i)

Permalink:

http://dlmf.nist.gov/35.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §35.4(i), §35.4 and Ch.35

►See Muirhead (1982, pp. 68–72) for the definition and properties of the Haar measure $\mathrm{d}\mathbf{H}$. … ►

35.4.7 $${\int }_{\mathbf{O}(m)}{Z}_{\kappa }\left(\mathbf{S}\mathbf{H}\mathbf{T}{\mathbf{H}}^{-1}\right)\mathrm{d}\mathbf{H}=\frac{Z_{\kappa }\left(\mathbf{S}\right)Z_{\kappa }\left(\mathbf{T}\right)}{Z_{\kappa }\left(\mathbf{I}\right)}.$$

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Symbols:

$\int$: integral, $Z_{\kappa }\left(\mathbf{T}\right)$: zonal polynomial, $\mathbf{I}$: $m\times m$ identity matrix, $\mathbf{S}$: real symmetric $m\times m$ matrix, $\mathbf{T}$: real symmetric $m\times m$ matrix, $\mathbf{O}(m)$: space of $m\times m$ orthogonal matrices, $\mathbf{H}$: orthogonal $m\times m$ matrix, $\mathrm{d}\mathbf{H}$: normalized Haar measure on $\mathbf{O}(m)$ , $m$: positive integer and $\kappa$: vector of nonnegative integers

Referenced by:

§35.9

Permalink:

http://dlmf.nist.gov/35.4.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §35.4(ii), §35.4(ii), §35.4 and Ch.35

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17: 9.17 Methods of Computation

§9.17 Methods of Computation

… ►The former reference includes a parallelized version of the method. … ►In these cases boundary-value methods need to be used instead; see §3.7(iii). … ►The second method is to apply generalized Gauss–Laguerre quadrature (§3.5(v)) to the integral (9.5.8). For the second method see also Gautschi (2002a). …

18: 35.1 Special Notation

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$a,b$	complex variables.
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$\mathrm{d}\mathbf{H}$	normalized Haar measure on $\mathbf{O}(m)$.
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19: 27.19 Methods of Computation: Factorization

§27.19 Methods of Computation: Factorization

… ►Type I probabilistic algorithms include the Brent–Pollard rho algorithm (also called Monte Carlo method), the Pollard $p\mathrm{-}\mathrm{1}$ algorithm, and the Elliptic Curve Method (ecm). …As of January 2009 the largest prime factors found by these methods are a 19-digit prime for Brent–Pollard rho, a 58-digit prime for Pollard $p-1$, and a 67-digit prime for ecm. …

20: 35.10 Methods of Computation

§35.10 Methods of Computation

… ►Other methods include numerical quadrature applied to double and multiple integral representations. …