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  2. Euler method - Wikipedia

    en.wikipedia.org/wiki/Euler_method

    For this reason, the Euler method is said to be a first-order method, while the midpoint method is second order. We can extrapolate from the above table that the step size needed to get an answer that is correct to three decimal places is approximately 0.00001, meaning that we need 400,000 steps.

  3. List of Runge–Kutta methods - Wikipedia

    en.wikipedia.org/wiki/List_of_Runge–Kutta_methods

    The Runge–Kutta–Fehlberg method has two methods of orders 5 and 4; it is sometimes dubbed RKF45 . Its extended Butcher Tableau is: / / / / / / / / / / / / / / / / / / / / / / / / / / The first row of b coefficients gives the fifth-order accurate solution, and the second row has order four.

  4. Formulas for generating Pythagorean triples - Wikipedia

    en.wikipedia.org/wiki/Formulas_for_generating...

    Leonardo of Pisa (c. 1170 – c. 1250) described this method [1] [2] for generating primitive triples using the sequence of consecutive odd integers ,,,,, … and the fact that the sum of the first n terms of this sequence is .

  5. Trigonometric tables - Wikipedia

    en.wikipedia.org/wiki/Trigonometric_tables

    A simple recurrence formula to generate trigonometric tables is based on Euler's formula and the relation: (+) = This leads to the following recurrence to compute trigonometric values s n and c n as above: c 0 = 1 s 0 = 0 c n+1 = w r c n − w i s n s n+1 = w i c n + w r s n

  6. Numerical methods for ordinary differential equations - Wikipedia

    en.wikipedia.org/wiki/Numerical_methods_for...

    This is the Euler method (or forward Euler method, in contrast with the backward Euler method, to be described below). The method is named after Leonhard Euler who described it in 1768. The Euler method is an example of an explicit method. This means that the new value y n+1 is defined in terms of things that are already known, like y n.

  7. Adaptive step size - Wikipedia

    en.wikipedia.org/wiki/Adaptive_step_size

    Let us now apply Euler's method again with a different step size to generate a second approximation to y(t n+1). We get a second solution, which we label with a (). Take the new step size to be one half of the original step size, and apply two steps of Euler's method. This second solution is presumably more accurate.

  8. Semi-implicit Euler method - Wikipedia

    en.wikipedia.org/wiki/Semi-implicit_Euler_method

    However, the semi-implicit Euler method is a symplectic integrator, unlike the standard method. As a consequence, the semi-implicit Euler method almost conserves the energy (when the Hamiltonian is time-independent). Often, the energy increases steadily when the standard Euler method is applied, making it far less accurate.

  9. Primitive root modulo n - Wikipedia

    en.wikipedia.org/wiki/Primitive_root_modulo_n

    And then, Euler's theorem says that a φ (n) ≡ 1 (mod n) for every a coprime to n; the lowest power of a that is congruent to 1 modulo n is called the multiplicative order of a modulo n. In particular, for a to be a primitive root modulo n , a φ ( n ) has to be the smallest power of a that is congruent to 1 modulo n .