Multiple quadrature with central differences on one line
Authors: Salzer Herbert E. 
Published: June 2010

The coefficients

A

2
m

n

A_{2m}^n

in the n-fold quadrature formulas for the stepwise integration of (1)

y

(
n
)

=
f
(
x
,
y
,

y
′

,
⋯
,

y

(
n
−
1
)

)

{y^{(n)}} = f(x,y,y’, \cdots , {y^{(n - 1)}})

, at intervals of h, namely, for n even, (2)
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="delta Superscript n Baseline y 0 equals h Superscript n Baseline sigma-summation Underscript m equals 1 Overscript 10 Endscripts left-parenthesis 1 plus upper A Subscript 2 m Superscript n Baseline delta Superscript 2 m Baseline right-parenthesis f 0 plus midline-horizontal-ellipsis">

δ
n

y
0

=

h
n

∑

m
=
1

10

(
1
+

A

2
m

n

δ

2
m

)

f
0

+
⋯

{\delta ^n}{y_0} = {h^n}\sum \nolimits _{m = 1}^{10} {(1 + A_{2m}^n{\delta ^{2m}}){f_0} + \cdots }

, for n odd, (3)
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu delta Superscript n Baseline y 0 equals h Superscript n Baseline sigma-summation Underscript m equals 1 Overscript 10 Endscripts left-parenthesis 1 plus upper A Subscript 2 m Superscript n Baseline delta Superscript 2 m Baseline right-parenthesis f 0 plus midline-horizontal-ellipsis">

μ

δ
n

y
0

=

h
n

∑

m
=
1

10

(
1
+

A

2
m

n

δ

2
m

)

f
0

+
⋯

\mu {\delta ^n}{y_0} = {h^n}\sum \nolimits _{m = 1}^{10} {(1 + A_{2m}^n{\delta ^{2m}}){f_0} + \cdots }

, are tabulated exactly for n = 1(1)6, m = 1(1)10. They were calculated from the well-known symbolic formulas (4)

δ

n

y
=

(

δ

/

D

)

n

f

{{\delta }^{n}}y = {{({\delta }/{D})}^{n}}f

, (5)
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis delta slash upper D right-parenthesis Superscript n Baseline equals left-parenthesis delta h slash 2 hyperbolic sine Superscript negative 1 Baseline left-parenthesis delta slash 2 right-parenthesis right-parenthesis Superscript n">

(

δ

/

D

)

n

=

(

δ
h

/

2

sinh

−
1

(

δ

/

2

)
)

n

{{({\delta }/{D})}^{n}} = {{({{{\delta h}/{2\,\sinh }}^{-1}}({\delta }/{2}))}^{n}}

and (6)
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu equals left-parenthesis 1 plus delta squared slash 4 right-parenthesis Superscript 1 slash 2 Baseline equals 1 plus StartFraction delta squared Over 8 EndFraction minus StartFraction delta Superscript 4 Baseline Over 128 EndFraction plus StartFraction delta Superscript 6 Baseline Over 1024 EndFraction minus StartFraction 5 delta Superscript 8 Baseline Over 32768 EndFraction plus midline-horizontal-ellipsis">

μ
=

(
1
+

δ

2

/

4

)

1

/

2

=
1
+

δ

2

8

−

δ

4

128

+

δ

6

1024

−

5

δ

8

32768

+
⋯

\mu = {{(1 + {{{\delta }^{2}}}/{4})}^{{1}/{2}}} = 1 + \frac {{{\delta }^{2}}}{8}\,-\,\frac {{{\delta }^{4}}}{128} + \frac {{{\delta }^{6}}}{1024}\,-\,\frac {5{{\delta }^{8}}}{32768} + \cdots

. For calculating

y

(
r
)

{y^{(r)}}

, replace n by n - r in (2) and (3). Use of (2) and (3) avoids the solution of (1) by simultaneous lower-order systems for

n
>
1

n > 1

, as well as mid-interval tabular arguments, requires only even-order differences, on a single line, and provides great accuracy due to rapid decrease of

A

2
m

n

A_{2m}^n

as m increases. However, the integration may be slowed down by the need to estimate and refine iteratively the later values of

y
,

y
′

,
⋯
,

y

(
n
−
1
)

y,y’, \cdots , {y^{(n - 1)}}

required in

δ

2
m

f
0

{\delta ^{2m}}{f_0}

. Reference to earlier collected formulas of Legendre, Oppolzer, Thiele, Lindow, Salzer, Milne and Buckingham, reveals that Thiele and Buckingham come closest to (2), (3), as their works contain schemes that involve just tabular arguments throughout. For n odd, they give formulas that are based upon the series in

δ

2
m

{\delta ^{2m}}

for

(

1

/

μ

)

(

δ

/

D

)

n

({1}/{\mu }){{({\delta }/{D})}^{n}}

instead of

μ

(

δ

/

D

)

n

\mu {{({\delta }/{D})}^{n}}

as in the present arrangement.