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A trigonometric equation is an equation g = 0 where g is a trigonometric polynomial. Such an equation may be converted into a polynomial system by expanding the sines and cosines in it (using sum and difference formulas), replacing sin(x) and cos(x) by two new variables s and c and adding the new equation s 2 + c 2 – 1 = 0.
A concrete example in the case of a one-step method of order can illustrate the general procedure of extrapolation. With such a method, the calculated approximation for small step sizes ℎ can be easily described by a polynomial of the form
Polynomials of degree one, two or three are respectively linear polynomials, quadratic polynomials and cubic polynomials. [8] For higher degrees, the specific names are not commonly used, although quartic polynomial (for degree four) and quintic polynomial (for degree five) are sometimes used.
In mathematics, an algebraic equation or polynomial equation is an equation of the form =, where P is a polynomial with coefficients in some field, often the field of the rational numbers. For example, x 5 − 3 x + 1 = 0 {\displaystyle x^{5}-3x+1=0} is an algebraic equation with integer coefficients and
Polynomial equations of degree up to four can be solved exactly using algebraic methods, of which the quadratic formula is the simplest example. Polynomial equations with a degree of five or higher require in general numerical methods (see below) or special functions such as Bring radicals, although some specific cases may be solvable ...
Polynomial curves fitting points generated with a sine function. The black dotted line is the "true" data, the red line is a first degree polynomial, the green line is second degree, the orange line is third degree and the blue line is fourth degree. The first degree polynomial equation = + is a line with slope a. A line will connect any two ...
Combining two consecutive steps of these methods into a single test, one gets a rate of convergence of 9, at the cost of 6 polynomial evaluations (with Horner's rule). On the other hand, combining three steps of Newtons method gives a rate of convergence of 8 at the cost of the same number of polynomial evaluation.
The next step is to multiply the above value by the step size , which we take equal to one here: h ⋅ f ( y 0 ) = 1 ⋅ 1 = 1. {\displaystyle h\cdot f(y_{0})=1\cdot 1=1.} Since the step size is the change in t {\displaystyle t} , when we multiply the step size and the slope of the tangent, we get a change in y {\displaystyle y} value.
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