12-02-2022, 09:33 AM
Numerical Integration of the Reciprocal Function
We use several numerical methods to calculate the integral in this formula for the logarithm:
\(
\begin{aligned}
\log(1+x) = \int_{1}^{1+x}\frac{1}{t}\,dt
\end{aligned}
\)
These approximations are then compared to its Taylor series:
\(
\begin{aligned}
\log(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \frac{x^5}{5} - \frac{x^6}{6} + \frac{x^7}{7} + \mathcal{O}(n^8) \\
\end{aligned}
\)
Anyone familiar with the HP-41C/HP-42S will recognise the LN1+X function.
The difference in the coefficients of the first deviating term is calculated and can be used to compare the methods.
Also for the example \( x = 0.2 \) the approximation is compared with the true value:
0.18232155679395462
Furthermore, for each approximation, a program for the HP-42S is provided with the result of the calculation for the same value.
Conclusions
Adding intervals in the case of the trapezoidal or midpoint rule brings us closer to the true coefficients of the third-order term.
However, Simpson's 1/3 rule reduces the error to a 5th order term, while its complexity is similar to the others.
We can improve this a bit with Simpson's 3/8 rule, but we can't change the order of magnitude.
Using Gaussian quadrature, the constants are not integers anymore, but for 2 points the formula is similar.
Adding one more point reduces the error to a 7th order term.
When integrating a function numerically, choose your sampling points carefully.
References
Trapezoidal Rule
\(
\begin{aligned}
\int_{a}^{b}f(x)\,dx
&\approx \sum_{k=1}^{N}\frac{f(x_{k-1})+f(x_{k})}{2}\Delta x_{k} \\
&= \frac{\Delta x}{2}\left(f(x_{0})+2f(x_{1})+2f(x_{2})+2f(x_{3})+2f(x_{4})+\cdots +2f(x_{N-1})+f(x_{N})\right) \\
\end{aligned}
\)
1 Interval
\(
\int_{a}^{b}f(x)\,dx \approx (b-a)\cdot {\tfrac {1}{2}}(f(a)+f(b))
\)
\(
\begin{aligned}
\int_{1}^{1+x}\frac{1}{t}\,dt
&\approx \frac{x}{2}\,\left[1 + \frac{1}{1+x}\right] \\
&= x - \frac{x^2}{2} + \frac{x^3}{2} + \mathcal{O}(x^4) \\
\end{aligned}
\)
Difference
\(
\begin{aligned}
\frac{1}{2} - \frac{1}{3} = \frac{1}{6} \approx 0.166667
\end{aligned}
\)
Example
(0.18333333333333335, 0.18232155679395462, 0.005549407086964257)
Program
0.183333333333
0.182321556794
2 Intervals
\(
\begin{aligned}
\int_{a}^{b}f(x)\,dx \approx \frac{1}{2}\frac{b-a}{2} \left[f(a) + 2f\left(\frac{a+b}{2}\right) + f(b)\right]
\end{aligned}
\)
\(
\begin{aligned}
\int_{1}^{1+x}\frac{1}{t}\,dt
&\approx \frac{x}{2 \cdot 2}\,\left[1 + 2 \cdot \frac{2}{1+1+x} + \frac{1}{1+x}\right] \\
&= \frac{x}{4}\,\left[1 + \frac{4}{2+x} + \frac{1}{1+x}\right] \\
&= x - \frac{x^2}{2} + \frac{3 x^3}{8} + \mathcal{O}(x^4) \\
\end{aligned}
\)
Difference
\(
\frac{3}{8} - \frac{1}{3} = \frac{1}{24} \approx 0.0416667
\)
Example
(0.1825757575757576, 0.18232155679395462, 0.0013942442477620518)
Program
0.182575757576
0.182321556794
Midpoint Rule
\(
\begin{aligned}
\int_{a}^{b}f(x)\,dx \approx \Delta x\left[f\left(a+{\tfrac {\Delta x}{2}}\right)+f\left(a+{\tfrac {3\Delta x}{2}}\right)+\ldots +f\left(b-{\tfrac {\Delta x}{2}}\right)\right]
\end{aligned}
\)
1 Interval
\(
\begin{aligned}
\int_{a}^{b}f(x)\,dx\approx (b-a)\cdot f\left(\frac{a+b}{2}\right)
\end{aligned}
\)
\(
\begin{aligned}
\int_{1}^{1+x}\frac{1}{t}\,dt
&\approx x \, \frac{2}{2+x} \\
&= \frac{2x}{2+x} \\
&= x - \frac{x^2}{2} + \frac{x^3}{4} + \mathcal{O}(x^4) \\
\end{aligned}
\)
Difference
\(
\frac{1}{4} - \frac{1}{3} = - \frac{1}{12} \approx -0.0833333
\)
Example
(0.18181818181818182, 0.18232155679395462, -0.0027609185914404585)
Program
0.181818181818
0.182321556794
2 Intervals
\(
\begin{aligned}
\int_{a}^{b}f(x)\,dx\approx \frac{b-a}{2}\,\left[f\left(\frac{3a+b}{4}\right) + f\left(\frac{a+3b}{4}\right)\right]
\end{aligned}
\)
\(
\begin{aligned}
\int_{1}^{1+x}\frac{1}{t}\,dt
&\approx \frac{x}{2}\,\left[\frac{4}{3+1+x} + \frac{4}{1+3+3x}\right] \\
&= 2x\,\left[\frac{1}{4+x} + \frac{1}{4+3x}\right] \\
&= x - \frac{x^2}{2} + \frac{5 x^3}{16} + \mathcal{O}(x^4) \\
\end{aligned}
\)
Difference
\(
\begin{aligned}
\frac{5}{16} - \frac{1}{3} = - \frac{1}{48} \approx -0.0208333
\end{aligned}
\)
Example
(0.1821946169772257, 0.18232155679395462, -0.0006962414042590937)
Program
0.182194616977
0.182321556794
Simpson's Rule
1/3 rule
\(
\begin{aligned}
\int_{a}^{b}f(x)\,dx
&\approx \frac{1}{3} h \,\left[f(a)+4f\left(\frac {a+b}{2}\right)+f(b)\right]\\
&=\frac{b-a}{6}\,\left[f(a)+4f\left(\frac{a+b}{2}\right)+f(b)\right]
\end{aligned}
\)
\(
\begin{aligned}
\int_{1}^{1+x}\frac{1}{t}\,dt
&\approx \frac{x}{6}\,\left[1 + 4 \frac{2}{1+1+x} + \frac{1}{1+x}\right] \\
&= \frac{x}{6}\,\left[1 + \frac{8}{2+x} + \frac{1}{1+x}\right] \\
&= x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \frac{5 x^5}{24} + \mathcal{O}(x^6) \\
\end{aligned}
\)
Difference
\(
\begin{aligned}
\frac{5}{24} - \frac{1}{5} = \frac{1}{120} \approx 0.00833333
\end{aligned}
\)
Example
(0.18232323232323233, 0.18232155679395462, 9.189968027780061e-06)
Program
0.182323232323
0.182321556794
3/8 rule
\(
\begin{aligned}
\int _{a}^{b}f(x)\,dx
&\approx \frac{3}{8}h\,\left[f(a)+3f\left(\frac{2a+b}{3}\right)+3f\left(\frac{a+2b}{3}\right)+f(b)\right]\\
&= \frac{b-a}{8}\,\left[f(a)+3f\left(\frac{2a+b}{3}\right)+3f\left(\frac{a+2b}{3}\right)+f(b)\right] \\
\end{aligned}
\)
\(
\begin{aligned}
\int_{1}^{1+x}\frac{1}{t}\,dt
&\approx \frac{x}{8}\,\left[1 + 3 \frac{3}{2+1+x} + 3 \frac{3}{1+2+2x} + \frac{1}{1+x}\right] \\
&= \frac{x}{8}\,\left[1 + \frac{9}{3+x} + \frac{9}{3+2x} + \frac{1}{1+x}\right] \\
&= x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \frac{11 x^5}{54} + \mathcal{O}(x^6) \\
\end{aligned}
\)
Difference
\(
\frac{11}{54} - \frac{1}{5} = \frac{1}{270} \approx 0.00370370
\)
Example
(0.18232230392156862, 0.18232155679395462, 4.097856705131325e-06)
Program
0.182322303922
0.182321556794
Gauss–Legendre Quadrature
\(
\begin{aligned}
\int_{-1}^{1}f(x)\,dx\approx \sum _{i=1}^{n}w_{i}f(x_{i})
\end{aligned}
\)
2 Point
\(
\begin{array}{|c|c|}
\hline
\text{Point} & \text{Weight} \\
\hline
-\frac{1}{\sqrt{3}} & 1 \\
\hline
+\frac{1}{\sqrt{3}} & 1 \\
\hline
\end{array}
\)
\(
\begin{aligned}
\int_{1}^{1+x}\frac{1}{t}\,dt
&\approx \frac{x}{2}\,\left[\frac{1}{1 + \left(\frac{1}{2} - \frac{1}{\sqrt{12}}\right)x} + \frac{1}{1 + \left(\frac{1}{2} + \frac{1}{\sqrt{12}}\right)x}\right] \\
&= x\,\left[\frac{1}{2 + \left(1 - \frac{1}{\sqrt{3}}\right)x} + \frac{1}{2 + \left(1 + \frac{1}{\sqrt{3}}\right)x}\right] \\
&= x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \frac{7 x^5}{36} + \mathcal{O}(x^6) \\
\end{aligned}
\)
Difference
\(
\begin{aligned}
\frac{7}{36} - \frac{1}{5} = - \frac{1}{180} \approx -0.00555556
\end{aligned}
\)
Example
(0.18232044198895028, 0.18232155679395462, -6.114499151613408e-06)
Program
0.182320441989
0.182321556794
This variant uses registers:
1 ENTER 3 SQRT 1/X - STO 00
1 LASTX + STO 01
0.182320441989
0.182321556794
3 Point
\(
\begin{array}{|c|c|}
\hline
\text{Point} & \text{Weight} \\
\hline
-\sqrt{\frac{3}{5}} & \frac{5}{9} \\
\hline
0 & \frac{8}{9} \\
\hline
\sqrt{\frac{3}{5}} & \frac{5}{9} \\
\hline
\end{array}
\)
\(
\begin{aligned}
\int_{1}^{1+x}\frac{1}{t}\,dt
&\approx \frac{x}{2 \cdot 9}\,\left[\frac{5}{1 + \left(\frac{1}{2} - \sqrt{\frac{3}{20}}\right)x} + \frac{8 \cdot 2}{1+1+x} + \frac{5}{1 + \left(\frac{1}{2} + \sqrt{\frac{3}{20}}\right)x}\right] \\
&= \frac{x}{9}\,\left[\frac{5}{2 + \left(1 - \sqrt{\frac{3}{5}}\right)x} + \frac{8}{2+x} + \frac{5}{2 + \left(1 + \sqrt{\frac{3}{5}}\right)x}\right] \\
&= x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \frac{x^5}{5} - \frac{x^6}{6} + \frac{57 x^7}{400} + \mathcal{O}(x^8) \\
\end{aligned}
\)
Difference
\(
\begin{aligned}
\frac{57}{400} - \frac{1}{7} = - \frac{1}{2800} \approx -0.000357143
\end{aligned}
\)
Example
(0.18232155441457765, 0.18232155679395462, -1.3050442389081796e-08)
Program
0.182321554415
0.182321556794
This variant uses registers:
1 ENTER 0.6 SQRT - STO 00
1 LASTX + STO 01
0.182321554415
0.182321556794
We use several numerical methods to calculate the integral in this formula for the logarithm:
\(
\begin{aligned}
\log(1+x) = \int_{1}^{1+x}\frac{1}{t}\,dt
\end{aligned}
\)
These approximations are then compared to its Taylor series:
\(
\begin{aligned}
\log(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \frac{x^5}{5} - \frac{x^6}{6} + \frac{x^7}{7} + \mathcal{O}(n^8) \\
\end{aligned}
\)
Anyone familiar with the HP-41C/HP-42S will recognise the LN1+X function.
The difference in the coefficients of the first deviating term is calculated and can be used to compare the methods.
Also for the example \( x = 0.2 \) the approximation is compared with the true value:
Code:
from math import log1p
x = 0.2
L = log1p(x)
L
0.18232155679395462
Furthermore, for each approximation, a program for the HP-42S is provided with the result of the calculation for the same value.
Conclusions
Adding intervals in the case of the trapezoidal or midpoint rule brings us closer to the true coefficients of the third-order term.
However, Simpson's 1/3 rule reduces the error to a 5th order term, while its complexity is similar to the others.
We can improve this a bit with Simpson's 3/8 rule, but we can't change the order of magnitude.
Using Gaussian quadrature, the constants are not integers anymore, but for 2 points the formula is similar.
Adding one more point reduces the error to a 7th order term.
When integrating a function numerically, choose your sampling points carefully.
References
Trapezoidal Rule
\(
\begin{aligned}
\int_{a}^{b}f(x)\,dx
&\approx \sum_{k=1}^{N}\frac{f(x_{k-1})+f(x_{k})}{2}\Delta x_{k} \\
&= \frac{\Delta x}{2}\left(f(x_{0})+2f(x_{1})+2f(x_{2})+2f(x_{3})+2f(x_{4})+\cdots +2f(x_{N-1})+f(x_{N})\right) \\
\end{aligned}
\)
1 Interval
\(
\int_{a}^{b}f(x)\,dx \approx (b-a)\cdot {\tfrac {1}{2}}(f(a)+f(b))
\)
\(
\begin{aligned}
\int_{1}^{1+x}\frac{1}{t}\,dt
&\approx \frac{x}{2}\,\left[1 + \frac{1}{1+x}\right] \\
&= x - \frac{x^2}{2} + \frac{x^3}{2} + \mathcal{O}(x^4) \\
\end{aligned}
\)
Difference
\(
\begin{aligned}
\frac{1}{2} - \frac{1}{3} = \frac{1}{6} \approx 0.166667
\end{aligned}
\)
Example
Code:
I = x/2*(1+1/(1+x))
I, L, (I-L)/L
(0.18333333333333335, 0.18232155679395462, 0.005549407086964257)
Program
Code:
00 { 13-Byte Prgm }
01 1
02 RCL+ ST Y
03 1/X
04 1
05 +
06 ×
07 2
08 ÷
09 END
0.183333333333
0.182321556794
2 Intervals
\(
\begin{aligned}
\int_{a}^{b}f(x)\,dx \approx \frac{1}{2}\frac{b-a}{2} \left[f(a) + 2f\left(\frac{a+b}{2}\right) + f(b)\right]
\end{aligned}
\)
\(
\begin{aligned}
\int_{1}^{1+x}\frac{1}{t}\,dt
&\approx \frac{x}{2 \cdot 2}\,\left[1 + 2 \cdot \frac{2}{1+1+x} + \frac{1}{1+x}\right] \\
&= \frac{x}{4}\,\left[1 + \frac{4}{2+x} + \frac{1}{1+x}\right] \\
&= x - \frac{x^2}{2} + \frac{3 x^3}{8} + \mathcal{O}(x^4) \\
\end{aligned}
\)
Difference
\(
\frac{3}{8} - \frac{1}{3} = \frac{1}{24} \approx 0.0416667
\)
Example
Code:
I = x/4*(1+4/(2+x)+1/(1+x))
I, L, (I-L)/L
(0.1825757575757576, 0.18232155679395462, 0.0013942442477620518)
Program
Code:
00 { 22-Byte Prgm }
01 4
02 2
03 RCL+ ST Z
04 ÷
05 1
06 RCL+ ST Z
07 1/X
08 +
09 1
10 +
11 ×
12 4
13 ÷
14 END
0.182575757576
0.182321556794
Midpoint Rule
\(
\begin{aligned}
\int_{a}^{b}f(x)\,dx \approx \Delta x\left[f\left(a+{\tfrac {\Delta x}{2}}\right)+f\left(a+{\tfrac {3\Delta x}{2}}\right)+\ldots +f\left(b-{\tfrac {\Delta x}{2}}\right)\right]
\end{aligned}
\)
1 Interval
\(
\begin{aligned}
\int_{a}^{b}f(x)\,dx\approx (b-a)\cdot f\left(\frac{a+b}{2}\right)
\end{aligned}
\)
\(
\begin{aligned}
\int_{1}^{1+x}\frac{1}{t}\,dt
&\approx x \, \frac{2}{2+x} \\
&= \frac{2x}{2+x} \\
&= x - \frac{x^2}{2} + \frac{x^3}{4} + \mathcal{O}(x^4) \\
\end{aligned}
\)
Difference
\(
\frac{1}{4} - \frac{1}{3} = - \frac{1}{12} \approx -0.0833333
\)
Example
Code:
I = 2*x/(2+x)
I, L, (I-L)/L
(0.18181818181818182, 0.18232155679395462, -0.0027609185914404585)
Program
Code:
00 { 9-Byte Prgm }
01 2
02 RCL+ ST Y
03 ÷
04 2
05 ×
06 END
0.181818181818
0.182321556794
2 Intervals
\(
\begin{aligned}
\int_{a}^{b}f(x)\,dx\approx \frac{b-a}{2}\,\left[f\left(\frac{3a+b}{4}\right) + f\left(\frac{a+3b}{4}\right)\right]
\end{aligned}
\)
\(
\begin{aligned}
\int_{1}^{1+x}\frac{1}{t}\,dt
&\approx \frac{x}{2}\,\left[\frac{4}{3+1+x} + \frac{4}{1+3+3x}\right] \\
&= 2x\,\left[\frac{1}{4+x} + \frac{1}{4+3x}\right] \\
&= x - \frac{x^2}{2} + \frac{5 x^3}{16} + \mathcal{O}(x^4) \\
\end{aligned}
\)
Difference
\(
\begin{aligned}
\frac{5}{16} - \frac{1}{3} = - \frac{1}{48} \approx -0.0208333
\end{aligned}
\)
Example
Code:
I = 2*x*(1/(4+x)+1/(4+3*x))
I, L, (I-L)/L
(0.1821946169772257, 0.18232155679395462, -0.0006962414042590937)
Program
Code:
00 { 20-Byte Prgm }
01 4
02 RCL+ ST Y
03 1/X
04 3
05 RCL× ST Z
06 4
07 +
08 1/X
09 +
10 ×
11 2
12 ×
13 END
0.182194616977
0.182321556794
Simpson's Rule
1/3 rule
\(
\begin{aligned}
\int_{a}^{b}f(x)\,dx
&\approx \frac{1}{3} h \,\left[f(a)+4f\left(\frac {a+b}{2}\right)+f(b)\right]\\
&=\frac{b-a}{6}\,\left[f(a)+4f\left(\frac{a+b}{2}\right)+f(b)\right]
\end{aligned}
\)
\(
\begin{aligned}
\int_{1}^{1+x}\frac{1}{t}\,dt
&\approx \frac{x}{6}\,\left[1 + 4 \frac{2}{1+1+x} + \frac{1}{1+x}\right] \\
&= \frac{x}{6}\,\left[1 + \frac{8}{2+x} + \frac{1}{1+x}\right] \\
&= x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \frac{5 x^5}{24} + \mathcal{O}(x^6) \\
\end{aligned}
\)
Difference
\(
\begin{aligned}
\frac{5}{24} - \frac{1}{5} = \frac{1}{120} \approx 0.00833333
\end{aligned}
\)
Example
Code:
I = x/6*(1+8/(2+x)+1/(1+x))
I, L, (I-L)/L
(0.18232323232323233, 0.18232155679395462, 9.189968027780061e-06)
Program
Code:
00 { 22-Byte Prgm }
01 8
02 2
03 RCL+ ST Z
04 ÷
05 1
06 RCL+ ST Z
07 1/X
08 +
09 1
10 +
11 ×
12 6
13 ÷
14 END
0.182323232323
0.182321556794
3/8 rule
\(
\begin{aligned}
\int _{a}^{b}f(x)\,dx
&\approx \frac{3}{8}h\,\left[f(a)+3f\left(\frac{2a+b}{3}\right)+3f\left(\frac{a+2b}{3}\right)+f(b)\right]\\
&= \frac{b-a}{8}\,\left[f(a)+3f\left(\frac{2a+b}{3}\right)+3f\left(\frac{a+2b}{3}\right)+f(b)\right] \\
\end{aligned}
\)
\(
\begin{aligned}
\int_{1}^{1+x}\frac{1}{t}\,dt
&\approx \frac{x}{8}\,\left[1 + 3 \frac{3}{2+1+x} + 3 \frac{3}{1+2+2x} + \frac{1}{1+x}\right] \\
&= \frac{x}{8}\,\left[1 + \frac{9}{3+x} + \frac{9}{3+2x} + \frac{1}{1+x}\right] \\
&= x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \frac{11 x^5}{54} + \mathcal{O}(x^6) \\
\end{aligned}
\)
Difference
\(
\frac{11}{54} - \frac{1}{5} = \frac{1}{270} \approx 0.00370370
\)
Example
Code:
I = x/8*(1+9/(3+x)+9/(3+2*x)+1/(1+x))
I, L, (I-L)/L
(0.18232230392156862, 0.18232155679395462, 4.097856705131325e-06)
Program
Code:
00 { 33-Byte Prgm }
01 2
02 RCL× ST Y
03 3
04 +
05 1/X
06 3
07 RCL+ ST Z
08 1/X
09 +
10 9
11 ×
12 1
13 RCL+ ST Z
14 1/X
15 +
16 1
17 +
18 ×
19 8
20 ÷
21 END
0.182322303922
0.182321556794
Gauss–Legendre Quadrature
\(
\begin{aligned}
\int_{-1}^{1}f(x)\,dx\approx \sum _{i=1}^{n}w_{i}f(x_{i})
\end{aligned}
\)
2 Point
\(
\begin{array}{|c|c|}
\hline
\text{Point} & \text{Weight} \\
\hline
-\frac{1}{\sqrt{3}} & 1 \\
\hline
+\frac{1}{\sqrt{3}} & 1 \\
\hline
\end{array}
\)
\(
\begin{aligned}
\int_{1}^{1+x}\frac{1}{t}\,dt
&\approx \frac{x}{2}\,\left[\frac{1}{1 + \left(\frac{1}{2} - \frac{1}{\sqrt{12}}\right)x} + \frac{1}{1 + \left(\frac{1}{2} + \frac{1}{\sqrt{12}}\right)x}\right] \\
&= x\,\left[\frac{1}{2 + \left(1 - \frac{1}{\sqrt{3}}\right)x} + \frac{1}{2 + \left(1 + \frac{1}{\sqrt{3}}\right)x}\right] \\
&= x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \frac{7 x^5}{36} + \mathcal{O}(x^6) \\
\end{aligned}
\)
Difference
\(
\begin{aligned}
\frac{7}{36} - \frac{1}{5} = - \frac{1}{180} \approx -0.00555556
\end{aligned}
\)
Example
Code:
from math import sqrt
k_1 = 1 - 1/sqrt(3)
k_2 = 1 + 1/sqrt(3)
I = x*(1/(2+k_1*x)+1/(2+k_2*x))
I, L, (I-L)/L
(0.18232044198895028, 0.18232155679395462, -6.114499151613408e-06)
Program
Code:
00 { 29-Byte Prgm }
01 3
02 SQRT
03 1/X
04 1
05 RCL- ST Y
06 RCL× ST Z
07 2
08 +
09 1/X
10 X<>Y
11 1
12 +
13 RCL× ST Z
14 2
15 +
16 1/X
17 +
18 ×
19 END
0.182320441989
0.182321556794
This variant uses registers:
1 ENTER 3 SQRT 1/X - STO 00
1 LASTX + STO 01
Code:
00 { 18-Byte Prgm }
01 RCL 00
02 RCL× ST Y
03 2
04 +
05 1/X
06 RCL 01
07 RCL× ST Z
08 2
09 +
10 1/X
11 +
12 ×
13 END
0.182320441989
0.182321556794
3 Point
\(
\begin{array}{|c|c|}
\hline
\text{Point} & \text{Weight} \\
\hline
-\sqrt{\frac{3}{5}} & \frac{5}{9} \\
\hline
0 & \frac{8}{9} \\
\hline
\sqrt{\frac{3}{5}} & \frac{5}{9} \\
\hline
\end{array}
\)
\(
\begin{aligned}
\int_{1}^{1+x}\frac{1}{t}\,dt
&\approx \frac{x}{2 \cdot 9}\,\left[\frac{5}{1 + \left(\frac{1}{2} - \sqrt{\frac{3}{20}}\right)x} + \frac{8 \cdot 2}{1+1+x} + \frac{5}{1 + \left(\frac{1}{2} + \sqrt{\frac{3}{20}}\right)x}\right] \\
&= \frac{x}{9}\,\left[\frac{5}{2 + \left(1 - \sqrt{\frac{3}{5}}\right)x} + \frac{8}{2+x} + \frac{5}{2 + \left(1 + \sqrt{\frac{3}{5}}\right)x}\right] \\
&= x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \frac{x^5}{5} - \frac{x^6}{6} + \frac{57 x^7}{400} + \mathcal{O}(x^8) \\
\end{aligned}
\)
Difference
\(
\begin{aligned}
\frac{57}{400} - \frac{1}{7} = - \frac{1}{2800} \approx -0.000357143
\end{aligned}
\)
Example
Code:
from math import sqrt
k_1 = 1 - sqrt(3/5)
k_2 = 1 + sqrt(3/5)
I = x/9*(5/(2+k_1*x)+8/(2+x)+5/(2+k_2*x))
I, L, (I-L)/L
(0.18232155441457765, 0.18232155679395462, -1.3050442389081796e-08)
Program
Code:
00 { 46-Byte Prgm }
01 0.6
02 SQRT
03 1
04 RCL- ST Y
05 RCL× ST Z
06 2
07 +
08 1/X
09 X<>Y
10 1
11 +
12 RCL× ST Z
13 2
14 +
15 1/X
16 +
17 5
18 ×
19 2
20 RCL+ ST Z
21 8
22 X<>Y
23 ÷
24 +
25 ×
26 9
27 ÷
28 END
0.182321554415
0.182321556794
This variant uses registers:
1 ENTER 0.6 SQRT - STO 00
1 LASTX + STO 01
Code:
00 { 34-Byte Prgm }
01 RCL 00
02 RCL× ST Y
03 2
04 +
05 1/X
06 RCL 01
07 RCL× ST Z
08 2
09 +
10 1/X
11 +
12 5
13 ×
14 2
15 RCL+ ST Z
16 8
17 X<>Y
18 ÷
19 +
20 ×
21 9
22 ÷
23 END
0.182321554415
0.182321556794