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- Toshio Fukushima
- J. Computational Applied Mathematics
- 2012

- T. FUKUSHIMA
- 2003

K e y w o r d s O b r e c h k o f f methods, Super-implicit method, Initial value problems. 1: I N T R O D U C T I O N In th is paper , we discuss t h e numer ica l so lu t ion of f i rs t -order ini t ia l va lue p rob l ems ( IVPs) y'(x) = f ( x , y(~)) , y(0) = y0, (1) and a specia l class (for which yl is miss ing) of second-order I V P s y ' ( x ) = f… (More)

We present a time ephemeris of the Earth, TE405, which approximates a relativistic time-dilation integral from 1600 to 2200 using numerical quadrature of quantities supplied by the recent JPL ephemeris, DE405. The integral is required to transform between terrestrial time, TT, and the (solar-system) barycentric time scales Teph or TCB. Teph is a linear… (More)

- Toshio Fukushima
- J. Computational Applied Mathematics
- 2011

- Toshio Fukushima
- 2012

As a preparation step to compute Jacobian elliptic functions efficiently, we created a fast method to calculate the complete elliptic integral of the first and second kinds, K(m) and E(m), for the standard domain of the elliptic parameter, 0 < m < 1. For the case 0 < m < 0.9, the method utilizes 10 pairs of approximate polynomials of the order of 9 to 19… (More)

- Toshio Fukushima
- Numerische Mathematik
- 2010

- Toshio Fukushima
- J. Computational Applied Mathematics
- 2013

We present a numerical method to invert a general incomplete elliptic integral with respect to its argument and/or amplitude. The method obtains a solution by bisection accelerated by the half argument formulas and the addition theorems to evaluate the incomplete elliptic integrals and Jacobian elliptic functions required in the course. If a faster… (More)

- Toshio Fukushima
- Numerische Mathematik
- 2013

We developed a new method to compute the cosine amplitude function, c ≡ cn(u|m), by using its double argument formula. The accumulation of roundoff errors is effectively suppressed by the introduction of a complementary variable, b ≡ 1− c, and a conditional switch between the duplication of b and c. The sine and delta amplitude functions, s ≡ sn(u|m) and d… (More)

- Toshio Fukushima
- J. Computational Applied Mathematics
- 2013

- Toshio Fukushima
- J. Computational Applied Mathematics
- 2013

We developed a novel method to calculate an associate complete elliptic integral of the third kind, J(n|m) ≡ [Π (n|m) − K(m)]/n. The key idea is the double argument formula of J(n|m) with respect to n. We derived it from the not-so-popular addition theorem of Jacobi’s complete elliptic integral of the third kind, Π1(a|m), with respect to a, which is a real… (More)