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Left and right methods make the approximation using the right and left endpoints of each subinterval, respectively. Upper and lower methods make the approximation using the largest and smallest endpoint values of each subinterval, respectively. The values of the sums converge as the subintervals halve from top-left to bottom-right.
If the tangent line at the right end point is considered (which can be estimated using Euler's Method), it has the opposite problem. [3] The points along the tangent line of the left end point have vertical coordinates which all underestimate those that lie on the solution curve, including the right end point of the interval under consideration.
This exact rule is known as the Gauss–Legendre quadrature rule. The quadrature rule will only be an accurate approximation to the integral above if f (x) is well-approximated by a polynomial of degree 2n − 1 or less on [−1, 1]. The Gauss–Legendre quadrature rule is not typically used for integrable functions with endpoint singularities ...
The trapezoidal rule may be viewed as the result obtained by averaging the left and right Riemann sums, and is sometimes defined this way. The integral can be even better approximated by partitioning the integration interval, applying the trapezoidal rule to each subinterval, and summing the results. In practice, this "chained" (or "composite ...
It is said left-open or right-open depending on whether the excluded endpoint is on the left or on the right. These intervals are denoted by mixing notations for open and closed intervals. [ 3 ] For example, (0, 1] means greater than 0 and less than or equal to 1 , while [0, 1) means greater than or equal to 0 and less than 1 .
In summary, the Tanh-Sinh quadrature scheme is designed so that it gives the most accurate result for the minimum number of function evaluations. In practice, the Tanh-Sinh quadrature rule is almost invariably the best rule and is often the only effective rule when extended precision results are sought [citation needed].
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This can be seen in the following tables, the left of which shows Newton's method applied to the above f(x) = x + x 4/3 and the right of which shows Newton's method applied to f(x) = x + x 2. The quadratic convergence in iteration shown on the right is illustrated by the orders of magnitude in the distance from the iterate to the true root (0,1 ...