Gram?Schmidt procedure

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1—10 of 23 matching pages

1: 17.18 Methods of Computation

… ►Method (1) can sometimes be improved by application of convergence acceleration procedures; see §3.9. …

2: 8.25 Methods of Computation

… ►A numerical inversion procedure is also given for calculating the value of

x

(with 10S accuracy), when

a

and

P\left(a,x\right)

are specified, based on Newton’s rule (§3.8(ii)). … ►An efficient procedure, based partly on the recurrence relations (8.8.5) and (8.8.6), is described in Gautschi (1979b, 1999). …

3: 27.18 Methods of Computation: Primes

… ►An alternative procedure is the binary quadratic sieve of Atkin and Bernstein (Crandall and Pomerance (2005, p. 170)). …

4: Bille C. Carlson

… ►In theoretical physics he is known for the “Carlson-Keller Orthogonalization”, published in 1957, Orthogonalization Procedures and the Localization of Wannier Functions, and the “Carlson-Keller Theorem”, published in 1961, Eigenvalues of Density Matrices. …

5: 27.16 Cryptography

… ►Procedures for finding such primes require very little computer time. …

6: 28.34 Methods of Computation

… ►

(d)

Solution of the systems of linear algebraic equations (28.4.5)–(28.4.8) and (28.14.4), with the conditions (28.4.9)–(28.4.12) and (28.14.5), by boundary-value methods (§3.6) to determine the Fourier coefficients. Subsequently, the Fourier series can be summed with the aid of Clenshaw’s algorithm (§3.11(ii)). See Meixner and Schäfke (1954, §2.87). This procedure can be combined with §28.34(ii)(d).

…

7: 6.18 Methods of Computation

… ►In addition, Acton (1974) developed a recurrence procedure, as follows. …

8: 30.7 Graphics

…

9: 3.11 Approximation Techniques

… ►A more efficient procedure is as follows. … ►A general procedure is to approximate

F

by a rational function

R

(vanishing at infinity) and then approximate

f

r={\mathscr{L}}^{-1}R

. … ►A set of functions

\phi_{0}(x),\phi_{1}(x),\dots,\phi_{n}(x)

that is linearly independent on the set

x_{1},x_{2},\dots,x_{J}

(compare (3.11.36)) can always be orthogonalized in the sense given in the preceding paragraph by the Gram–Schmidt procedure; see Gautschi (1997a). …

10: 19.36 Methods of Computation

… ►All cases of

R_{F}

R_{C}

R_{J}

, and

R_{D}

are computed by essentially the same procedure (after transforming Cauchy principal values by means of (19.20.14) and (19.2.20)). … ►If

t_{0}=a_{0}

and

\theta=-1

, so that

t_{n}=a_{n}

, then this procedure reduces to the AGM method for the complete integral. … ►A three-part computational procedure for

\Pi\left(\phi,\alpha^{2},k\right)

is described by Franke (1965) for

\alpha^{2}<1

. …