Method of finding solution of equation and recording medium storing program for implementing the same
View Patent ↗The present invention relates to a solution-finding method, which finds an approximate solution of an equation having difficulty in obtaining an actual solution and a complicated equation in numerical analysis. The method obtains an approximate solution of an equation having a solution in a predetermined interval. Initial values are calculated based on upper and lower limits of the interval. The initial values are applied to a solution-finding equation, including a sign function and the upper and lower limits, and the solution-finding equation is arranged so that a definite integral formula for the sign function is included in the equation. The definite integral formula in the solution-finding equation is calculated using numerical integration, and results of the definite integral formula are applied to the solution-finding equation, thus obtaining an approximate solution. This performance is iterated until the approximate solution satisfies an allowable error.
1. A method of finding a solution of an equation, the method performed by a non-transitory recording medium for obtaining an approximate solution of an equation having a solution in a predetermined interval, the method comprising:
a) calculating, by a processor in the non-transitory recording medium, initial values based on upper and lower limits of the interval;
b) applying the initial values to a solution-finding equation which includes a sign function and the upper and lower limits, and arranging the solution-finding equation so that a definite integral formula for the sign function of the equation is included in the solution-finding equation;
c) calculating the definite integral formula in the arranged solution-finding equation using numerical integration, and inductively applying results of the calculation of the definite integral formula to the arranged solution-finding equation, thus obtaining an approximate solution; and
d) iterating c) until the approximate solution obtained at c) satisfies an allowable error,
wherein
in a), when the initial values are p (0) =(a+b)/2 and δ (0) =(b −a)/2 when the upper and lower limits of the interval are a and b, respectively.
2. The method of claim 1 , wherein in b), the solution-finding equation including the sign function and the upper and lower limits is given by the following equation:
p
*
=
p
(
0
)
+
sgn
f
(
a
)
2
J
δ
(
0
)
where
J
δ
(
0
)
=
∫
p
(
0
)
-
σ
(
0
)
p
(
0
)
+
δ
(
0
)
sgn
f
(
x
)
ⅆ
x
.
3. The method of claim 1 , wherein in c), the numerical integration is numerical integration using a trapezoidal rule.
4. A non-transitory recording medium storing a program for implementing the method of claim 1 .
5. A non-transitory recording medium storing a program for implementing the method of claim 2 .
6. A non-transitory recording medium storing a program for implementing the method of claim 3 .
7. A method of finding a solution of an equation, the method performed by a non-transitory recording medium for obtaining an approximate solution of an equation having a solution in a predetermined interval, the method comprising:
a) calculating, by a processor in the non-transitory recording medium, initial values based on upper and lower limits of the interval;
b) applying the initial values to a solution-finding equation which includes a sign function and the upper and lower limits, and arranging the solution-finding equation so that a definite integral formula for the sign function of the equation is included in the solution-finding equation;
c) calculating the definite integral formula in the arranged solution-finding equation using numerical integration, and inductively applying results of the calculation of the definite integral formula to the arranged solution-finding equation, thus obtaining an approximate solution; and
d) obtaining a resulting approximate solution based on a secant method which uses the approximate solution, calculated by iterating c) a preset number of times, as an initially estimated value,
wherein, in a), when the initial values are p (0) =(a+b)/2 and δ (0) =(b −a)/2 when the upper and lower limits of the interval are a and b, respectively.
8. The method of claim 7 , wherein in b), the solution-finding equation including the sign function and the upper and lower limits is given by the following equation:
p
*
=
p
(
0
)
+
sgn
f
(
a
)
2
J
δ
(
0
)
where
J
δ
(
0
)
=
∫
p
(
0
)
-
δ
(
0
)
p
(
0
)
+
δ
(
0
)
sgn
f
(
x
)
ⅆ
x
.
9. The method of claim 7 , wherein in c), the numerical integration is numerical integration using a trapezoidal rule.
10. A non-transitory recording medium storing a program for implementing the method of claim 7 .
11. A method of finding a solution of an equation, the method being performed by a non-transitory recording medium for obtaining an approximate solution of an equation having a solution in a predetermined interval, the method comprising:
a) calculating, by a processor in the non-transitory recording medium, initial values based on upper and lower limits of the interval;
b) applying the initial values to a solution-finding equation which includes a sign function and the upper and lower limits, and arranging the solution-finding equation so that a definite integral formula for the sign function of the equation is included in the solution-finding equation;
c) calculating the definite integral formula in the arranged solution-finding equation using numerical integration, and inductively applying results of the calculation of the definite integral formula to the arranged solution-finding equation, thus obtaining an approximate solution; and
d) obtaining a resulting approximate solution based on Newton's method which uses the approximate solution, calculated by iterating c) a preset number of times, as an initially estimated value,
wherein in a), when the initial values are p (0) =(a+b)/2 and δ (0) =(b −a)/2 when the upper and lower limits of the interval are a and b, respectively.
12. The method of claim 11 , wherein in b), the solution-finding equation including the sign function and the upper and lower limits is given by the following equation:
p
*
=
p
(
0
)
+
sgn
f
(
a
)
2
J
δ
(
0
)
where
J
δ
(
0
)
=
∫
p
(
0
)
-
δ
(
0
)
p
(
0
)
+
δ
(
0
)
sgn
f
(
x
)
ⅆ
x
.
13. The method of claim 11 , wherein in c), the numerical integration is numerical integration using a trapezoidal rule.
14. A non-transitory recording medium storing a program for implementing the method of claim 11 .