# Gram?Schmidt procedure

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## 1—10 of 23 matching pages

##### 1: 17.18 Methods of Computation

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►Method (1) can sometimes be improved by application of convergence acceleration procedures; see §3.9.
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##### 2: 8.25 Methods of Computation

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►A numerical inversion procedure is also given for calculating the value of $x$ (with 10S accuracy), when $a$ and $P(a,x)$ are specified, based on Newton’s rule (§3.8(ii)).
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►An efficient procedure, based partly on the recurrence relations (8.8.5) and (8.8.6), is described in Gautschi (1979b, 1999).
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##### 3: 27.18 Methods of Computation: Primes

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►An alternative procedure is the

*binary quadratic sieve*of Atkin and Bernstein (Crandall and Pomerance (2005, p. 170)). …##### 4: Bille C. Carlson

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►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.
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##### 5: 27.16 Cryptography

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►Procedures for finding such primes require very little computer time.
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##### 6: 28.34 Methods of Computation

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►
(d)
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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

##### 8: 30.7 Graphics

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##### 9: 3.11 Approximation Techniques

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►A more efficient procedure is as follows.
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►A general procedure is to approximate $F$ by a rational function $R$ (vanishing at infinity) and then approximate $f$ by $r={\mathcal{L}}^{-1}R$.
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►A set of functions ${\varphi}_{0}(x),{\varphi}_{1}(x),\mathrm{\dots},{\varphi}_{n}(x)$ that is linearly independent on the set ${x}_{1},{x}_{2},\mathrm{\dots},{x}_{J}$ (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

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►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)).
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►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.
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►A three-part computational procedure for $\mathrm{\Pi}(\varphi ,{\alpha}^{2},k)$ is described by Franke (1965) for $$.
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