absolute error

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11: 2.8 Differential Equations with a Parameter

… ►For error bounds, extensions to pure imaginary or complex $u$, an extension to inhomogeneous differential equations, and examples, see Olver (1997b, Chapter 10). … ►An alternative way of representing the error terms in (2.8.15) and (2.8.16) is as follows. … ►Again, an alternative way of representing the error terms in (2.8.29) and (2.8.30) is by means of envelope functions. … ►For error bounds, more delicate error estimates, extensions to complex $\xi$, $\nu$, and $u$, zeros, and examples see Olver (1997b, Chapter 12), Boyd (1990a), and Dunster (1990a). … ►For results, including error bounds, see Olver (1977c). …

12: 3.5 Quadrature

… ►Also, the error constant (3.5.20) is given by … ►Equation (3.5.36), without the error term, becomes … ►A frequent problem with contour integrals is heavy cancellation, which occurs especially when the value of the integral is exponentially small compared with the maximum absolute value of the integrand. … ►

Example

… ►where $\mathrm{erfc}z$ is the complementary error function, and from (7.12.1) it follows that …

13: Errata

… ►The Editors thank the users who have contributed to the accuracy of the DLMF Project by submitting reports of possible errors. For confirmed errors, the Editors have made the corrections listed here. … ►

In the line just below (18.15.4), it was previously stated “is less than twice the first neglected term in absolute value.” It now states “is less than twice the first neglected term in absolute value, in which one has to take $\cos {\theta }_{n,m,\mathrm{\ell }}=1$.”

Reported by Gergő Nemes on 2019-02-08

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Originally named as a complementary error function, $w\left(z\right)$ has been renamed as the Faddeeva (or Faddeyeva) function.

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In applying changes in Version 1.0.12 to (16.15.3), an editing error was made; it has been corrected.

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14: 3.2 Linear Algebra

… ►To avoid instability the rows are interchanged at each elimination step in such a way that the absolute value of the element that is used as a divisor, the pivot element, is not less than that of the other available elements in its column. … ►Because of rounding errors, the residual vector $𝐫=𝐛-𝐀𝐱$ is nonzero as a rule. … ►where $\rho (𝐀{𝐀}^{\mathrm{T}})$ is the largest of the absolute values of the eigenvalues of the matrix $𝐀{𝐀}^{\mathrm{T}}$; see §3.2(iv). … ►Then we have the a posteriori error bound …

15: 10.74 Methods of Computation

… ►The power-series expansions given in §§10.2 and 10.8, together with the connection formulas of §10.4, can be used to compute the Bessel and Hankel functions when the argument $x$ or $z$ is sufficiently small in absolute value. … ►And since there are no error terms they could, in theory, be used for all values of $z$; however, there may be severe cancellation when $|z|$ is not large compared with $n^2$. …

16: 10.75 Tables

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Abramowitz and Stegun (1964, p. 373) tabulates the three smallest zeros of $Y_0\left(z\right)$, $Y_1\left(z\right)$, $Y_1^{\prime }\left(z\right)$ in the sector , together with the corresponding values of $Y_1\left(z\right)$, $Y_0\left(z\right)$, $Y_1\left(z\right)$, respectively, to 9D. (There is an error in the value of $Y_0\left(z\right)$ at the 3rd zero of $Y_1\left(z\right)$: the last four digits should be 2533; see Amos (1985).)

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MacDonald (1989) tabulates the first 30 zeros, in ascending order of absolute value in the fourth quadrant, of the function $J_0\left(z\right)-\mathrm{i}{J}_1\left(z\right)$, 6D. (Other zeros of this function can be obtained by reflection in the imaginary axis).

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17: 4.15 Graphics

… ►In the graphics shown in this subsection height corresponds to the absolute value of the function and color to the phase. …