initial values

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1: 5.21 Methods of Computation

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

2: 28.33 Physical Applications

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Initial-value problems, in which only one equation (28.2.1) or (28.20.1) is involved. See §28.33(iii).

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§28.33(iii) Stability and Initial-Value Problems

… ►References for other initial-value problems include: …

3: 3.7 Ordinary Differential Equations

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§3.7(ii) Taylor-Series Method: Initial-Value Problems

… ►If the solution $w(z)$ that we are seeking grows in magnitude at least as fast as all other solutions of (3.7.1) as we pass along $𝒫$ from $a$ to $b$, then $w(z)$ and $w^{\prime }(z)$ may be computed in a stable manner for $z=z_0,z_1,\mathrm{\dots },z_P$ by successive application of (3.7.5) for $j=0,1,\mathrm{\dots },P-1$, beginning with initial values $w(a)$ and $w^{\prime }(a)$. … ►Similarly, if $w(z)$ is decaying at least as fast as all other solutions along $𝒫$, then we may reverse the labeling of the $z_j$ along $𝒫$ and begin with initial values $w(b)$ and $w^{\prime }(b)$. … ►It will be observed that the present formulation of the Taylor-series method permits considerable parallelism in the computation, both for initial-value and boundary-value problems. …

4: 10.74 Methods of Computation

… ►In the interval , $J_{\nu }\left(x\right)$ needs to be integrated in the forward direction and $Y_{\nu }\left(x\right)$ in the backward direction, with initial values for the former obtained from the power-series expansion (10.2.2) and for the latter from asymptotic expansions (§§10.17(i) and 10.20(i)). … ►Similarly, to maintain stability in the interval the integration direction has to be forwards in the case of $I_{\nu }\left(x\right)$ and backwards in the case of $K_{\nu }\left(x\right)$, with initial values obtained in an analogous manner to those for $J_{\nu }\left(x\right)$ and $Y_{\nu }\left(x\right)$. … ►In the case of $J_n\left(x\right)$, the need for initial values can be avoided by application of Olver’s algorithm (§3.6(v)) in conjunction with Equation (10.12.4) used as a normalizing condition, or in the case of noninteger orders, (10.23.15). …

5: 9.2 Differential Equation

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§9.2(ii) Initial Values

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6: 9.17 Methods of Computation

… ►In the case of $\mathrm{Ai}\left(z\right)$, for example, this means that in the sectors we may integrate along outward rays from the origin with initial values obtained from §9.2(ii). … ►Zeros of the Airy functions, and their derivatives, can be computed to high precision via Newton’s rule (§3.8(ii)) or Halley’s rule (§3.8(v)), using values supplied by the asymptotic expansions of §9.9(iv) as initial approximations. …

7: 28.34 Methods of Computation

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Direct numerical integration of the differential equation (28.2.1), with initial values given by (28.2.5) (§§3.7(ii), 3.7(v)).

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8: 12.4 Power-Series Expansions

… ►where the initial values are given by (12.2.6)–(12.2.9), and $u_1(a,z)$ and $u_2(a,z)$ are the even and odd solutions of (12.2.2) given by …

9: 13.29 Methods of Computation

… ►For $M(a,b,z)$ and $M_{\kappa ,\mu }\left(z\right)$ this means that in the sector $|\mathrm{ph}z|\le \pi$ we may integrate along outward rays from the origin with initial values obtained from (13.2.2) and (13.14.2). ►For $U(a,b,z)$ and $W_{\kappa ,\mu }\left(z\right)$ we may integrate along outward rays from the origin in the sectors , with initial values obtained from connection formulas in §13.2(vii), §13.14(vii). …

10: 6.18 Methods of Computation

… ► $A_0$, $B_0$, and $C_0$ can be computed by Miller’s algorithm (§3.6(iii)), starting with initial values $(A_N,B_N,C_N)=(1,0,0)$, say, where $N$ is an arbitrary large integer, and normalizing via $C_0=1/z$. … ►Zeros of $\mathrm{Ci}\left(x\right)$ and $\mathrm{si}\left(x\right)$ can be computed to high precision by Newton’s rule (§3.8(ii)), using values supplied by the asymptotic expansion (6.13.2) as initial approximations. …