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Movie Title Year Distributor Notes Rev Formats Awakening of Salley 1984 VCR LezOnly DO Personal Touch 2 1983 Arrow Productions MastOnly O Personal Touch 2 (new) 1993 Arrow Productions MastOnly O Sex Star 1983 Caballero Home Video LezOnly DRO Women in Love 1985 4-Play Video LezOnly DRO X-Rated Bloopers 1984 Arrow Productions NonSex In numerical analysis, Ridders' method is a root-finding algorithm based on the false position method and the use of an exponential function to successively approximate a root of a continuous function{\displaystyle f(x)}f(x) . The method is due to C. Ridders.
Similar concepts are used for discretization methods. The solution of the discretized problem converges to the solution of the continuous problem as the grid size goes to zero, and the speed of convergence is one of the factors of the efficiency of the method. However, the terminology in this case is different from the terminology for iterative methods. Series acceleration is a collection of techniques for improving the rate of convergence of a series discretization. Such acceleration is commonly accomplished with sequence transformations. Contents 1 Convergence speed for iterative methods 1.1 Basic definition 1.2 Extended definition 1.3 Examples 2 Convergence speed for discretization methods 2.1 Examples (continued) 3 Acceleration of convergence 4 References 5 Literature Convergence speed for iterative methods Basic definition

Ridders' method is simpler than Muller's method or Brent's method but with similar performance.[3] The formula below converges quadratically when the function is well-behaved, which implies that the number of additional significant digits found at each step approximately doubles; but the function has to be evaluated twice for each step, so the overall order of convergence of the method is {\displaystyle {\sqrt {2}}}{\sqrt {2}} . If the function is not well-behaved, the root remains bracketed and the length of the bracketing interval at least halves on each iteration, so convergence is guaranteed. Method Given two values of the independent variable, {\displaystyle x_{0}}x_{0} and {\displaystyle x_{2}}x_{2}, which are on two different sides of the root being sought, i.e.,{\displaystyle f(x_{0})f(x_{2})<0}{\displaystyle f(x_{0})f(x_{2})<0}, the method begins by evaluating the function at the midpoint {\displaystyle x_{1}=(x_{0}+x_{2})/2}{\displaystyle x_{1}=(x_{0}+x_{2})/2}. One then finds the unique exponential function {\displaystyle e^{ax}}e^{{ax}} such that function {\displaystyle h(x)=f(x)e^{ax}}{\displaystyle h(x)=f(x)e^{ax}} satisfies {\displaystyle h(x_{1})=(h(x_{0})+h(x_{2}))/2}{\displaystyle h(x_{1})=(h(x_{0})+h(x_{2}))/2}. Specifically, parameter {\displaystyle a}a is determined by {\displaystyle e^{a(x_{1}-x_{0})}={\frac {f(x_{1})-\operatorname {sign} [f(x_{0})]{\sqrt {f(x_{1})^{2}-f(x_{0})f(x_{2})}}}{f(x_{2})}}.}{\displaystyle e^{a(x_{1}-x_{0})}={\frac {f(x_{1})-\operatorname {sign} [f(x_{0})]{\sqrt {f(x_{1})^{2}-f(x_{0})f(x_{2})}}}{f(x_{2})}}.} The false position method is then applied to the points {\displaystyle (x_{0},h(x_{0}))}{\displaystyle (x_{0},h(x_{0}))} and {\displaystyle (x_{2},h(x_{2}))}{\displaystyle (x_{2},h(x_{2}))}, leading to a new value {\displaystyle x_{3}}{\displaystyle x_{3}} between {\displaystyle x_{0}}x_0 and {\displaystyle x_{2}}{\displaystyle x_{2}}, {\displaystyle x_{3}=x_{1}+(x_{1}-x_{0}){\frac {\operatorname {sign} [f(x_{0})]f(x_{1})}{\sqrt {f(x_{1})^{2}-f(x_{0})f(x_{2})}}},}{\displaystyle x_{3}=x_{1}+(x_{1}-x_{0}){\frac {\operatorname {sign} [f(x_{0})]f(x_{1})}{\sqrt {f(x_{1})^{2}-f(x_{0})f(x_{2})}}},} which will be used as one of the two bracketing values in the next step of the iteration. The other bracketing value is taken to be {\displaystyle x_{1}}x_{1} if {\displaystyle f(x_{1})f(x_{3})<0}{\displaystyle f(x_{1})f(x_{3})<0} (well-behaved case), or otherwise whichever of {\displaystyle x_{0}}x_0 and {\displaystyle x_{2}}{\displaystyle x_{2}} has function value of opposite sign to {\displaystyle f(x_{3})}{\displaystyle f(x_{3})}. The procedure can be terminated when a given accuracy is obtained In numerical analysis, the speed at which a convergent sequence approaches its limit is called the rate of convergence. Although strictly speaking, a limit does not give information about any finite first part of the sequence, the concept of rate of convergence is of practical importance when working with a sequence of successive approximations for an iterative method, as then typically fewer iterations are needed to yield a useful approximation if the rate of convergence is higher. This may even make the difference between needing ten or a million iterations. Suppose that the sequence {\displaystyle (x_{k})}(x_k) converges to the number {\displaystyle L}L. The sequence is said to converge linearly to {\displaystyle L}L, if there exists a number {\displaystyle \mu \in (0,1)}{\displaystyle \mu \in (0,1)} such that {\displaystyle \lim _{k\to \infty }{\frac {|x_{k+1}-L|}{|x_{k}-L|}}=\mu }{\displaystyle \lim _{k\to \infty }{\frac {|x_{k+1}-L|}{|x_{k}-L|}}=\mu } where the number {\displaystyle \mu }\mu is called the rate of convergence. The sequence is said to converge superlinearly (i.e. faster than linearly) to {\displaystyle L}L, if {\displaystyle \lim _{k\to \infty }{\frac {|x_{k+1}-L|}{|x_{k}-L|}}=0.}{\displaystyle \lim _{k\to \infty }{\frac {|x_{k+1}-L|}{|x_{k}-L|}}=0.} The sequence is said to converge sublinearly (i.e. slower than linearly) to {\displaystyle L}L, if {\displaystyle \lim _{k\to \infty }{\frac {|x_{k+1}-L|}{|x_{k}-L|}}=1.}{\displaystyle \lim _{k\to \infty }{\frac {|x_{k+1}-L|}{|x_{k}-L|}}=1.} If the sequence converges sublinearly and additionally {\displaystyle \lim _{k\to \infty }{\frac {|x_{k+2}-x_{k+1}|}{|x_{k+1}-x_{k}|}}=1,}{\displaystyle \lim _{k\to \infty }{\frac {|x_{k+2}-x_{k+1}|}{|x_{k+1}-x_{k}|}}=1,} then it is said that the sequence {\displaystyle (x_{k})}(x_k) converges logarithmically to {\displaystyle L}L. The next definition is used to distinguish superlinear rates of convergence. The sequence converges with order {\displaystyle q}q to {\displaystyle L}L for {\displaystyle q>1}{\displaystyle q>1}[1] if {\displaystyle \lim _{k\to \infty }{\frac {|x_{k+1}-L|}{|x_{k}-L|^{q}}}

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