bluuebunny/crossref_metadata_2025_split huggingface dataset:
A uniform distribution question related to numerical analysis
Authors: Niederreiter Harald , Osgood Charles F. 
Published: June 2010

Using the theory of uniform distribution modulo one, it is shown that under certain conditions on the real-valued functions

α
(
x
)

\alpha (x)

and

g
(
x
)

g(x)

on [0,1 ],
\[

h

∑

γ
=
1

[

h

−
1

]

{

h

−
1

α
(
γ
h
)
}

m

g
(
γ
h
)
=

(
m
+
1
)

−
1

∫
0
1

g
(
x
)

d
x
+
o

(

h

1

/

3

log
⁡

1
h

)

as

h
→
0
+
,

h\;\sum \limits _{\gamma = 1}^{[{h^{ - 1}}]} {{{\{ {h^{ - 1}}\alpha (\gamma h)\} }^m}g(\gamma h) = {{(m + 1)}^{ - 1}}\int _0^1 {g(x)\;dx + o\left ( {{h^{1/3}}\log \frac {1}{h}} \right )} \quad {\text {as}}\;h \to 0 + ,}

\]
where

m
>
0

m > 0

and x denotes the fractional part of x. The conditions are as follows:

α
(
x
)

\alpha (x)

exists for all but finitely many points in [0, 1], changes sign at most finitely often, and is bounded away in absolute value from both 0 and

∞
\infty

, whereas

g
(
x
)

g(x)

is of bounded variation on [0,1]. Also, under these conditions on

α
(
x
)

\alpha (x)

,
\[

h

∑

γ
=
1

[

h

−
1

]

{

h

−
1

α
(
γ
h
)
}

m

=

(
m
+
1
)

−
1

+
o
(

h

1

/

3

)

as

h
→
0
+
.

h\;\sum \limits _{\gamma = 1}^{[{h^{ - 1}}]} {{{\{ {h^{ - 1}}\alpha (\gamma h)\} }^m} = {{(m + 1)}^{ - 1}} + o({h^{1/3}})\quad {\text {as}}\;h \to 0 + .}

\]
These results, which are, in fact, deduced from somewhat more general propositions, answer questions of Feldstein connected with discretization methods for differential equations.
An error functional expansion for 𝑁-dimensional quadrature with an integrand function singular at a point
Authors: Lyness J. N. 
Published: June 2010

Let If be the integral of

f
(

x
→

)

f(\vec x)

over an N-dimensional hypercube and

Q

(
m
)

f

{Q^{(m)}}f

be the approximation to If obtained by subdividing the hypercube into

m
N

{m^N}

equal subhypercubes and applying the same quadrature rule Q to each. In order to extrapolate efficiently for If on the basis of several different approximations

Q

(

m
i

)

f

{Q^{({m_i})}}f

, it is necessary to know the form of the error functional

Q

(
m
)

f
−
I
f

{Q^{(m)}}f - If

as an expansion in m. When

f
(

x
→

)

f(\vec x)

has a singularity, the conventional form (with inverse even powers of m) is not usually valid. In this paper we derive the expansion in the case in which

f
(

x
→

)

f(\vec x)

has the form
\[

f
(

x
→

)
=

r
α

φ
(

θ
→

)
h
(
r
)
g
(

x
→

)
,

α
>
−
N
,

f(\vec x) = {r^\alpha }\varphi (\vec \theta )h(r)g(\vec x),\quad \alpha > - N,

\]
the only singularity being at the origin, a vertex of the unit hypercube of integration. Here

(
r
,

θ
→

)

(r,\vec \theta )

represents the hyperspherical coordinates of

(

x
→

)

(\vec x)

. It is shown that for this integrand the error function expansion includes only terms

A

α
+
N
+
t

/

m

α
+
N
+
t

,

B
t

/

m
t

,

C

α
+
N
+
t

ln
⁡
m

/

m

α
+
N
+
t

,
t
=
1
,
2
,
…

{A_{\alpha + N + t}}/{m^{\alpha + N + t}},{B_t}/{m^t},{C_{\alpha + N + t}}\ln m/{m^{\alpha + N + t}},t = 1,2, \ldots

. The coefficients depend only on the integrand function

f
(

x
→

)

f(\vec x)

and the quadrature rule Q. For several easily recognizable classes of integrand function and for most familiar quadrature rules some of these coefficients are zero. An analogous expansion for the error functional with integrand function

F
(

x
→

)
=
ln
⁡
r
f
(

x
→

)

F(\vec x) = \ln rf(\vec x)

is also derived.
Tridiagonal fourth order approximations to general two-point nonlinear boundary value problems with mixed boundary conditions
Authors: Stepleman Robert S. 
Published: June 2010

This paper develops fourth order discretizations to the two-point boundary value problem
\[

y

(
2
)

(
t
)
=
f
(
t
,
y
(
t
)
,

y

(
1
)

(
t
)
)
,

α
0

y
(
0
)
−

β
0

y

(
1
)

(
0
)
=

δ
0

,

α
1

y
(
1
)
+

β
1

y

(
1
)

(
1
)
=

δ
1

.

\begin {array}{*{20}{c}} {{y^{(2)}}(t) = f(t,y(t),{y^{(1)}}(t)),} \\ {{\alpha _0}y(0) - {\beta _0}{y^{(1)}}(0) = {\delta _0},\quad {\alpha _1}y(1) + {\beta _1}{y^{(1)}}(1) = {\delta _1}.} \\ \end {array}

\]
These discretizations have the desirable properties that they are tridiagonal and of "positive type".
Roots of two transcendental equations involving spherical Bessel functions
Authors: Pexton Robert L. , Steiger Arno D. 
Published: June 2010

Roots of the transcendental equations

j
l

(
α
λ
)

y
l

(
λ
)
=

j
l

(
λ
)

y
l

(
α
λ
)

{j_l}(\alpha \lambda ){y_l}(\lambda ) = {j_l}(\lambda ){y_l}(\alpha \lambda )

and
\[

[
x

j
l

(
x
)

]

x
=
α
η

′

[
x

y
l

(
x
)

]

x
=
η

′

=

[
x

j
l

(
x
)

]

x
=
η

′

[
x

y
l

(
x
)

]

x
=
α
η

′

{[x{j_l}(x)]’_{x = \alpha \eta }}{[x{y_l}(x)]’_{x = \eta }} = {[x{j_l}(x)]’_{x = \eta }}{[x{y_l}(x)]’_{x = \alpha \eta }}

\]
for the spherical Bessel functions of the first and second kind,

j
l

(
z
)

{j_l}(z)

and

y
l

(
z
)

{y_l}(z)

, have been computed. The ranges for the parameter

α
\alpha

, the order l and the root index n are:

α
=
0.1
(
0.1
)
0.7

\alpha = 0.1(0.1)0.7

,

l
=
1
(
1
)
15

l = 1(1)15

,

n
=
1
(
1
)
30

n = 1(1)30

.
Growth of partial sums of divergent series
Authors: Boas R. P. 
Published: June 2010

Let

Σ
f
(
n
)

\Sigma f(n)

be a divergent series of decreasing positive terms, with partial sums

s
n

{s_n}

, where f decreases sufficiently smoothly; let

φ
(
x
)
=

∫
1
x

f
(
t
)
d
t

\varphi (x) = \smallint _1^xf(t)dt

and let

ψ
\psi

be the inverse of

φ
\varphi

. Let

n
A

{n_A}

be the smallest integer n such that

s
n

⩾
A

{s_n} \geqslant A

but

s

n
−
1

>
A
(
A
=
2
,
3
,
…
)

{s_{n - 1}} > A(A = 2,3, \ldots )

; let

γ
=
lim
{

Σ
1
n

f
(
k
)
−
φ
(
n
)
}

\gamma = \lim \{ \Sigma _1^nf(k) - \varphi (n)\}

be the analog of Euler’s constant; let

m
=
[
ψ
(
A
−
γ
)
]

m = [\psi (A - \gamma )]

. Call

ω
\omega

a Comtet function for

Σ
f
(
n
)

\Sigma f(n)

if

n
A

=
m

{n_A} = m

when the fractional part of

ψ
(
A
−
γ
)

\psi (A - \gamma )

is less than

ω
(
A
)

\omega (A)

and

n
A

=
m
+
1

{n_A} = m + 1

when the fractional part of

ψ
(
A
−
γ
)

\psi (A - \gamma )

is greater than

ω
(
A
)

\omega (A)

. It has been conjectured that

ω
(
A
)
=
1

/

2

\omega (A) = 1/2

is a Comtet function for

Σ
1

/

n

\Sigma 1/n

. It is shown that in general there is a Comtet function of the form
\[

ω
(
A
)
=

1
2

+

1
24

{

|

f
′
(
m
)

|

/

f
(
m
)
}

(
1
+
o
(
1
)
)
.

\omega (A) = \frac {1}{2} + \frac {1}{24} \left \{ |f\prime (m)|/f(m) \right \} (1 + o(1)).

\]
For

Σ
1

/

n

\Sigma 1/n

there is a Comtet function of the form

1

/

2
+
1

/

(
24
)

{
1

/

(
48

m
2

)
}

(
1
+
o
(
1
)
)

1/2 + 1/(24) \left \{ 1/(48m^2) \right \} (1 + o(1))

. Some numerical results are presented.
Roots of two transcendental equations determining the frequency spectra of standing spherical electromagnetic waves
Authors: Pexton Robert L. , Steiger Arno D. 
Published: June 2010

Roots of the transcendental equations
\[

j
l

(
λ
)

y
l

(
λ
)

=

j
l

(
α
λ
)

i

l
−
1

(
α
λ

|

ε

|

)

i
l

(
α
λ

|

ε

|

−

1

|

ε

|

j

l
−
1

(
α
λ
)

y
l

(
α
λ
)

i

l
−
1

(
α
λ

|

ε

|

)

i
l

(
α
λ

|

ε

|

−

1

|

ε

|

y

l
−
1

(
α
λ
)

\frac {{{j_l}(\lambda )}}{{{y_l}(\lambda )}} = \frac {{{j_l}(\alpha \lambda )\frac {{{i_{l - 1}}(\alpha \lambda \sqrt {|\varepsilon |} )}}{{{i_l}(\alpha \lambda \sqrt {|\varepsilon |} }} - \frac {1}{{\sqrt {|\varepsilon |} }}{j_{l - 1}}(\alpha \lambda )}}{{{y_l}(\alpha \lambda )\frac {{{i_{l - 1}}(\alpha \lambda \sqrt {|\varepsilon |} )}}{{{i_l}(\alpha \lambda \sqrt {|\varepsilon |} }} - \frac {1}{{\sqrt {|\varepsilon |} }}{y_{l - 1}}(\alpha \lambda )}}

\]
and
\[

η

j

l
−
1

(
η
)
−
l

j
l

(
η
)

η

y

l
−
1

(
η
)
−
l

y
l

(
η
)

=

|

ε

|

1
+

|

ε

|

α
η

j

l
−
1

(
α
η
)
−
l

j
l

(
α
η
)
+

|

ε

|

1
+

|

ε

|

α
η

j
l

(
α
η
)

i

l
−
1

(
α
η

|

ε

|

)

i
l

(
α
η

|

ε

|

)

|

ε

|

1
+

|

ε

|

α
η

y

l
−
1

(
α
η
)
−
l

y
l

(
α
η
)
+

|

ε

|

1
+

|

ε

|

α
η

y
l

(
α
η
)

i

l
−
1

(
α
η

|

ε

|

)

i
l

(
α
η

|

ε

|

)

\frac {{\eta {j_{l - 1}}(\eta ) - l{j_l}(\eta )}}{{\eta {y_{l - 1}}(\eta ) - l{y_l}(\eta )}} = \frac {{\frac {{|\varepsilon |}}{{1 + |\varepsilon |}}\alpha \eta {j_{l - 1}}(\alpha \eta ) - l{j_l}(\alpha \eta ) + \frac {{\sqrt {|\varepsilon |} }}{{1 + |\varepsilon |}}\alpha \eta {j_l}(\alpha \eta )\frac {{{i_{l - 1}}(\alpha \eta \sqrt {|\varepsilon |)} }}{{{i_l}(\alpha \eta \sqrt {|\varepsilon |)} }}}}{{\frac {{|\varepsilon |}}{{1 + |\varepsilon |}}\alpha \eta {y_{l - 1}}(\alpha \eta ) - l{y_l}(\alpha \eta ) + \frac {{\sqrt {|\varepsilon |} }}{{1 + |\varepsilon |}}\alpha \eta {y_l}(\alpha \eta )\frac {{{i_{l - 1}}(\alpha \eta \sqrt {|\varepsilon |)} }}{{{i_l}(\alpha \eta \sqrt {|\varepsilon |)} }}}}

\]
for the spherical Bessel functions of the first and second kind,

j
l

(
x
)

{j_l}(x)

and

y
l

(
x
)

{y_l}(x)

, and for the modified spherical Bessel functions of the first kind,

i
l

(
x
)

{i_l}(x)

, have been computed. The ranges for the parameters

|

ε

|

\sqrt {|\varepsilon |}

and

α
\alpha

, the order l and the root index n are:
\[

|

ε

|

=
1.0
,
10.0
,
100.0
,
500.0
;

α
=
0.1
(
0.1
)
0.7
;

l
=
1
(
1
)
15
;

n
=
1
(
1
)
30.

\sqrt {|\varepsilon |} = 1.0,10.0,100.0,500.0;\quad \alpha = 0.1(0.1)0.7;\quad l = 1(1)15;\quad n = 1(1)30.

\]
Improved error estimates for numerical solutions of symmetric integral equations
Authors: Rakotch E. 
Published: June 2010

The most widely employed method for a numerical solution of a symmetric integral equation with kernel

K
(
x
,
t
)

K(x,t)

in interval

I
≡
[
a
,
b
]

I \equiv [a,b]

is the replacement of the original problem by the sequence of eigenproblems
\[

K

(
n
)

y
i

(
n
)

=

μ

i
n

y
i

(
n
)

,

K

(
n
)

≡
{

w

j
n

K
(

x

i
n

,

x

j
n

)
}
,

i
=
1
,
…
,
n
,

{K^{(n)}}y_i^{(n)} = {\mu _{in}}y_i^{(n)},\quad {K^{(n)}} \equiv \{ {w_{jn}}K({x_{in}},{x_{jn}})\} ,\quad i = 1, \ldots ,n,

\]
with

w

j
n

>
0

{w_{jn}} > 0

and

x

j
n

∈
I
,
j
=
1
,
…
,
n

{x_{jn}} \in I,j = 1, \ldots ,n

. The eigenvectors

y
i

(
n
)

y_i^{(n)}

are further used to obtain an approximation, with improved error estimates, of the numerical eigensolution for some

N
>
n

N > n

, with no necessity of computing

μ

i
N

{\mu _{iN}}

and

y
i

(
N
)

,
i
=
1
,
…
,
N

y_i^{(N)},i = 1, \ldots ,N

, and of constructing another matrix.
On the convergence of difference approximations to nonlinear contraction semigroups in Hilbert spaces
Authors: Nevanlinna Olavi 
Published: June 2010

Convergence properties of the difference schemes (S)
\[

h

−
1

∑

j
=
0

k

α
j

u

n
+
j

+

∑

j
=
0

k

β
j

A

u

n
+
j

=
0
,

n
⩾
0
,

{h^{ - 1}}\sum \limits _{j = 0}^k {{\alpha _j}{u_{n + j}}} + \sum \limits _{j = 0}^k {{\beta _j}A{u_{n + j}}} = 0,\quad n \geqslant 0,

\]
, for evolution equations (E)
\[

d
u
(
t
)

d
t

+
A
u
(
t
)
=
0
,

t
⩾
0
;

u
(
0
)
=

u
0

∈

D
(
A
)

¯

\frac {{du(t)}}{{dt}} + Au(t) = 0,\quad t \geqslant 0;\quad u(0) = {u_0} \in \overline {D(A)}

\]
are studied. Here A is a nonlinear, maximally monotone operator in a real Hilbert space. It is shown, in particular, that if the scheme (S) is consistent and stable for the test equation

x
′
=
λ
x

x\prime = \lambda x

for

λ
∈

C

−
K

\lambda \in {\text {C}} - K

, where K is a compact subset of the right half-plane, then (S) is convergent as

h
↓
0

h \downarrow 0

, with suitable initial values, for (E), on compact intervals [0, T]. Moreover, the convergence is uniform on the half-axis

t
⩾
0

t \geqslant 0

, if the solution

u
(
t
)

u(t)

tends strongly to a constant as

t
→
∞

t \to \infty

. We also show that under weaker stability conditions one can construct conditionally convergent methods.
The effective choice of the smoothing norm in regularization
Authors: Cullum Jane 
Published: June 2010

We consider ill-posed problems of the form
\[

g
(
t
)
=

∫
0
1

K
(
t
,
s
)
f
(
s
)
d
s
,

0
⩽
t
⩽
1
,

g(t) = \int _0^1 {K(t,s)f(s)ds,\quad 0 \leqslant t \leqslant 1,}

\]
where g and K are given, and we must compute f. The Tikhonov regularization procedure replaces (1) by a one-parameter family of minimization problems-Minimize

(

‖

K
f
−
g

‖

2

+
α
Ω
(
f
)
)

({\left \| {Kf - g} \right \|^2} + \alpha \Omega (f))

-where

Ω
\Omega

is a smoothing norm chosen by the user. We demonstrate by example that the choice of

Ω
\Omega

is not simply a matter of convenience. We then show how this choice affects the convergence rate, and the condition of the problems generated by the regularization. An appropriate choice for

Ω
\Omega

depends upon the character of the compactness of K and upon the smoothness of the desired solution.
A Legendre polynomial integral
Authors: Blue James L. 
Published: June 2010

Let

{

P
n

(
x
)
}

\{ {P_n}(x)\}

be the usual Legendre polynomials. The following integral is apparently new.
\[

∫
0
1

P
n

(
2
x
−
1
)
log
⁡

1
x

d
x
=

(
−
1
)

n

n
(
n
+
1
)

for

n
⩾
1.

\int _0^1{P_n}(2x - 1)\log \frac {1}{x}dx = \frac {{{{( - 1)}^n}}}{{n(n + 1)}}\quad {\text {for}}\;n \geqslant 1.

\]
It has an application in the construction of Gauss quadrature formulas on (0, 1) with weight function

log
⁡
(
1

/

x
)

\log (1/x)

.
A note on the integral ∫^{∞}₀ 𝑡^{2𝛼-1}(1+𝑡²) ^{1-𝛼-𝛽}𝐽_{𝜈}(𝑥√(1+𝑡²))𝑑𝑡
Authors: Glasser M. L. 
Published: June 2010

The integral
\[

I
v

α
,
β

(
x
)
=

∫
0
∞

t

2
α
−
1

(
1
+

t
2

)

1
−
α
−
β

J
v

(
x

1
+

t
2

)

d
t

I_v^{\alpha ,\beta }(x) = \int _0^\infty {{t^{2\alpha - 1}}{{(1 + {t^2})}^{1 - \alpha - \beta }}{J_v}(x} \sqrt {1 + {t^2})} \;dt

\]
is expressed in terms of Bessel and related functions for various values of the parameters by summing the hypergeometric series representation given by Schmidt.
Degenerate roots of three transcendental equations involving spherical Bessel functions
Authors: Pexton Robert L. , Steiger Arno D. 
Published: June 2010

Roots of the three transcendental equations
\[

j
l

(
α
λ
)

y
l

(
λ
)
=

j
l

(
λ
)

y
l

(
α
λ
)
,

[
x

j
l

(
x
)

]

x
=
α
η

′

[
x

y
l

(
x
)

]

x
=
η

′

=
[
x

j
l

(
x
)

]

x
=
η

′

[
x

y
l

(
x
)

]

x
=
α
η

′

,

\begin {array}{*{20}{c}} {{j_l}(\alpha \lambda ){y_l}(\lambda ) = {j_l}(\lambda ){y_l}(\alpha \lambda ),} \\ {[x{j_l}(x)]_{x = \alpha \eta }^\prime [x{y_l}(x)]_{x = \eta }^\prime = [x{j_l}(x)]_{x = \eta }^\prime [x{y_l}(x)]_{x = \alpha \eta }^\prime ,} \\ \end {array}

\]
and
\[

[

j
l

(
x
)

]

x
=
α
μ

′

[

y
l

(
x
)

]

x
=
μ

′

=
[

j
l

(
x
)

]

x
=
μ

′

[

y
l

(
x
)

]

x
=
α
μ

′

[{j_l}(x)]_{x = \alpha \mu }^\prime [{y_l}(x)]_{x = \mu }^\prime = [{j_l}(x)]_{x = \mu }^\prime [{y_l}(x)]_{x = \alpha \mu }^\prime

\]
that are degenerate for certain values of the parameter

α
∈
(
0
,
1
)

\alpha \in (0,1)

are presented. The symbols

j
l

{j_l}

and

y
l

{y_l}

denote the spherical Bessel functions of the first and second kind. Root degeneracies are discussed for each equation individually as well as for pairs of equations. Only positive roots are considered, since the equations are invariant under the transformations

λ
→
−
λ

\lambda \to - \lambda

,

η
→
−
η

\eta \to - \eta

, and

μ
→
−
μ

\mu \to - \mu

. When

l
=
0

l = 0

, only the third equation has nontrivial roots. These roots are identical with the roots of the first equation for

l
=
1

l = 1

, i.e.

μ

0
n

=

λ

1
n

(
n
=
1
,
2
,
…
)

{\mu _{0n}} = {\lambda _{1n}}(n = 1,2, \ldots )

. Various graphs of

λ

l
n

{\lambda _{ln}}

,

η

l
n

{\eta _{ln}}

, and

μ

l
n

{\mu _{ln}}

display root-degeneracies as intersections of curves. Accurate values of degenerate roots with the corresponding values of

α
\alpha

are exhibited in tables. Roots of the third equation for

l
=
1
(
1
)
15

l = 1(1)15

,

n
=
1
(
1
)
30

n = 1(1)30

,

α
=
0.1
(
0.1
)
0.7

\alpha = 0.1(0.1)0.7

, together with their minima and associated values of the parameter

α
\alpha

, are given in the microfiche supplement accompanying this issue. Roots of the first and the second equation and minima of roots of the second equation are published in v. 31 and v. 32 of this journal. The roots of the first two equations determine the eigenfrequencies of the transverse electric and the transverse magnetic normal modes of an ideal cavity resonator bounded by two concentric spheres (

r
=
α
R

r = \alpha R

and

r
=
R

r = R

). The roots of the third equation determine the frequencies of the irrotational magnetic eigenfields.
On some orthogonal polynomial integrals
Authors: Gatteschi Luigi 
Published: June 2010

The modified moments of the weight functions

w
(
x
)
=

x
ρ

(
1
−
x

)
a

ln
⁡
(
1

/

x
)

w(x) = {x^\rho }{(1 - x)^a}\ln (1/x)

, on [0, 1], with respect to the shifted Jacobi polynomials

P
n

∗
(
α
,
β
)

(
x
)
=

P
n

(
α
,
β
)

(
2
x
−
1
)

P_n^{\ast (\alpha ,\beta )}(x) = P_n^{(\alpha ,\beta )}(2x - 1)

, and

w
p

(
x
)
=

x
ρ

e

−
x

(
ln
⁡
x

)
p

{w_p}(x) = {x^\rho }{e^{ - x}}{(\ln x)^p}

,

p
=
1
,
2

p = 1,2

, on

[
0
,
∞
)

[0,\infty )

, with respect to the generalized La guerre polynomials

L
n

(
α
)

(
x
)

L_n^{(\alpha )}(x)

, are explicitly evaluated.
The finite element method with nonuniform mesh sizes for unbounded domains
Authors: Goldstein C. I. 
Published: June 2010

The finite element method with nonuniform mesh sizes is employed to approximately solve elliptic boundary value problems in unbounded domains. Consider the following model problem:
\[

−
Δ
u
=
f

in

Ω
C

,

u
=
g

on

∂
Ω
,

∂
u

∂
r

+

1
r

u
=
o

(

1
r

)

as

r
=

|

x

|

→
∞
,

- \Delta u = f\quad {\text {in}}\;{\Omega ^C},\quad u = g\quad {\text {on}}\;\partial \Omega ,\quad \frac {{\partial u}}{{\partial r}} + \frac {1}{r}u = o\left ( {\frac {1}{r}} \right )\quad {\text {as}}\;r = |x| \to \infty ,

\]
where

Ω
C

{\Omega ^C}

is the complement in

R
3

{R^3}

(three-dimensional Euclidean space) of a bounded set

Ω
\Omega

with smooth boundary

∂
Ω

\partial \Omega

, f and g are smooth functions, and f has bounded support. This problem is approximately solved by introducing an artificial boundary

Γ
R

{\Gamma _R}

near infinity, e.g. a sphere of sufficiently large radius R. The intersection of this sphere with

Ω
C

{\Omega ^C}

is denoted by

Ω
R
C

\Omega _R^C

and the given problem is replaced by
\[

−
Δ

u
R

=
f

in

Ω
R
C

,

u
R

=
g

on

∂
Ω
,

∂

u
R

∂
r

+

1
r

u
R

=
0

on

Γ
R

.

- \Delta {u_R} = f\quad {\text {in}}\;\Omega _R^C,\quad {u_R} = g\quad {\text {on}}\;\partial \Omega ,\quad \frac {{\partial {u_R}}}{{\partial r}} + \frac {1}{r}{u_R} = 0\quad {\text {on}}\;{\Gamma _R}.

\]
This problem is then solved approximately by the finite element method, resulting in an approximate solution

u
R
h

u_R^h

for each

h
>
0

h > 0

. In order to obtain a reasonably small error for

u
−

u
R
h

=
(
u
−

u
R

)
+
(

u
R

−

u
R
h

)

u - u_R^h = (u - {u_R}) + ({u_R} - u_R^h)

, it is necessary to make R large. This necessitates the solution of a large number of linear equations, so that this method is often not very good when a uniform mesh size h is employed. It is shown that a nonuniform mesh may be introduced in such a way that optimal error estimates hold and the number of equations is bounded by

C

h

−
3

C{h^{ - 3}}

with C independent of h and R.
Quadrature methods for integral equations of the second kind over infinite intervals
Authors: Sloan Ian H. 
Published: June 2010

Convergence results are proved for a class of quadrature methods for integral equations of the form

y
(
t
)
=
f
(
t
)
+

∫
0
∞

k
(
t
,
s
)
y
(
s
)

d
s

y(t) = f(t) + \smallint _0^\infty \;k(t,s)y(s)\,ds

. An important special case is the Nyström method, in which the integral term is approximated by an ordinary quadrature rule. For all of the methods considered here, the rate of convergence is the same, apart from a constant factor, as that of the quadrature approximation to the integral term.
Relative distance—an error measure in round-off error analysis
Authors: Ziv Abraham 
Published: June 2010

Olver (SIAM J. Numer. Anal., v. 15, 1978, pp. 368-393) suggested relative precision as an attractive substitute for relative error in round-off error analysis. He remarked that in certain respects the error measure

d
(

x
¯

,
x
)
=
min
{
α

|

1
−
α
⩽
x

/

x
¯

⩽
1

/

(
1
−
α
)
}

d(\bar x,x) = \min \{ \alpha |1 - \alpha \leqslant x/\bar x \leqslant 1/(1 - \alpha )\}

,

x
¯

≠
0

\bar x \ne 0

,

x

/

x
¯

>
0

x/\bar x > 0

is even more favorable, through it seems to be inferior because of two drawbacks which are not shared by relative precision: (i) the inequality

d
(

x
¯

k

,

x
k

)
⩽

|

k

|

d
(

x
¯

,
x
)

d({\bar x^k},{x^k}) \leqslant |k|d(\bar x,x)

is not true for

0
>

|

k

|

>
1

0 > |k| > 1

. (ii)

d
(

x
¯

,
x
)

d(\bar x,x)

is not defined for complex

x
¯

,
x

\bar x,x

. In this paper the definition of

d
(
⋅
,
⋅
)

d( \cdot , \cdot )

is replaced by

d
(

x
¯

,
x
)
=

|

x
¯

−
x

|

/

max
{

|

x
¯

|

,

|

x

|

}

d(\bar x,x) = |\bar x - x|/\max \{ |\bar x|,|x|\}

. This definition is equivalent to the first in case

x
¯

≠
0

\bar x \ne 0

,

x

/

x
¯

>
0

x/\bar x > 0

, and is free of (ii). The inequality

d
(

x
¯

k

,

x
k

)
⩽

|

k

|

d
(

x
¯

,
x
)

d({\bar x^k},{x^k}) \leqslant |k|d(\bar x,x)

is replaced by the more universally valid inequality

d
(

x
¯

k

,

x
k

)
⩽

|

k

|

d
(

x
¯

,
x
)

/

(
1
−
δ
)
,
δ
=
max
{
d
(

x
¯

,
x
)
,

|

k

|

d
(

x
¯

,
x
)
}

d({\bar x^k},{x^k}) \leqslant |k|d(\bar x,x)/(1 - \delta ),\delta = \max \{ d(\bar x,x),|k|d(\bar x,x)\}

. The favorable properties of

d
(
⋅
,
⋅
)

d( \cdot , \cdot )

are preserved in the complex case. Moreover, its definition may be generalized to linear normed spaces by

d
(

x
¯

,
x
)
=

‖

x
¯

−
x

‖

/

max
{

‖

x
¯

‖

,

‖
x
‖

}

d(\bar x,x) = \left \| {\bar x - x} \right \|/\max \{ \left \| {\bar x} \right \|,\left \| x \right \|\}

. Its properties in such spaces raise the possibility that with further investigation it might become the basis for error analysis in some vector, matrix, and function spaces.
On the existence and computation of 𝐿𝑈-factorizations with small pivots
Authors: Chan Tony F. 
Published: June 2010

Let A be an n by n matrix which may be singular with a one-dimensional null space, and consider the LU-factorization of A. When A is exactly singular, we show conditions under which a pivoting strategy will produce a zero nth pivot. When A is not singular, we show conditions under which a pivoting strategy will produce an nth pivot that is

O
(

σ
n

)

O({\sigma _n})

or

O
(

κ

−
1

(
A
)
)

O({\kappa ^{ - 1}}(A))

, where

σ
n

{\sigma _n}

is the smallest singular value of A and

κ
(
A
)

\kappa (A)

is the condition number of A. These conditions are expressed in terms of the elements of

A

−
1

{A^{ - 1}}

in general but reduce to conditions on the elements of the singular vectors corresponding to

σ
n

{\sigma _n}

when A is nearly or exactly singular. They can be used to build a 2-pass factorization algorithm which is guaranteed to produce a small nth pivot for nearly singular matrices. As an example, we exhibit an LU-factorization of the n by n upper triangular matrix
\[

T
=

[

1

a
m
p
;

−
1

a
m
p
;

a
m
p
;

−
1

a
m
p
;

a
m
p
;
⋅

a
m
p
;
⋅

a
m
p
;

−
1

a
m
p
;
1

a
m
p
;

a
m
p
;
⋅

a
m
p
;

a
m
p
;

a
m
p
;

a
m
p
;
⋅

a
m
p
;

a
m
p
;

a
m
p
;
1

a
m
p
;
⋅

a
m
p
;

a
m
p
;

a
m
p
;

a
m
p
;

a
m
p
;

a
m
p
;

a
m
p
;
⋅

a
m
p
;
⋅

a
m
p
;

a
m
p
;
⋅

a
m
p
;

a
m
p
;
0

a
m
p
;

a
m
p
;

a
m
p
;
⋅

a
m
p
;
⋅

a
m
p
;
⋅

a
m
p
;

a
m
p
;

a
m
p
;

a
m
p
;

a
m
p
;

a
m
p
;
⋅

a
m
p
;

−
1

a
m
p
;

a
m
p
;

a
m
p
;

a
m
p
;

a
m
p
;

a
m
p
;

a
m
p
;
1

]

T = \left [ {\begin {array}{*{20}{c}} 1 & { - 1} & {} & { - 1} & {} & \cdot & \cdot & { - 1} \\ {} & 1 & {} & \cdot & {} & {} & {} & \cdot \\ {} & {} & {} & 1 & \cdot & {} & {} & {} \\ {} & {} & {} & {} & \cdot & \cdot & {} & \cdot \\ {} & {} & 0 & {} & {} & \cdot & \cdot & \cdot \\ {} & {} & {} & {} & {} & {} & \cdot & { - 1} \\ {} & {} & {} & {} & {} & {} & {} & 1 \\ \end {array} } \right ]

\]
that has an nth pivot equal to

2

−
(
n
−
2
)

2^{-(n-2)}

.