Finite

The Runge-Kutta method achieves the accuracy of the Taylor series method without the need to calculate higher-order derivatives. Although various variations exist, it can be generally expressed in the following forms.  Here, $\phi ({x_i},{y_i},h)$ is referred to as the increment function, and it can typically be written as follows.  The second-order version of the above formula can be expressed ..

To improve the calculation of simulaneous linear equations, iterative refinement methods have been used. Error correction techniques are used to enhance numerical integration results based on the integral estimates themselves. Commonly known as Richardson's extrapolation, this method calculates a more accurate third estimate using two initial integral estimates. The estimates and errors associat..

As explained in the previous posting, Simpson's $1/3$ rule can only be used when the intervals are equidistant and there are an even number of segments with an odd number of points. To address scenarios that don't meet these requirement, this posting will introduce Simpson's $3/8$ rule, which can be used alongside the $1/3$ rule to calculate cases with odd segments and an even number of points. ..

Similarly to the trapezoidal rule, Simpson's rule can also imrpove results by dividing the integration interval into multiple segements of equal width.  Therefore,  To implement this method, an even number of segments must be used, as the odd points represent the middle terms in each application and thus carry a weight of $4$. Even points are used commonly between adjacent segments and are thus ..

Continuing from the previous posting, another method to estimate a more accurate integral value is by connecting points using higher-degree polynomials. The formula for integrating under such polynomials is known as Simpson's rule. Simpson's $1/3$ rule can be expressed using a second-degree interpolating polynomial as follows.  If $a$ and $b$ are designated as ${x_0}$ and ${x_2}$, respectively, ..

One method to enhance the accuracy of the trapezoidal rule, as explained the previous posting, involves dividing the interval into multiple segments and applying the method to each segment. Subsequently, by summing the area of each segment, the integral for the entire interval can be obtained. There are $n$ eqaully spaced reference points, thus it is assumed that there are $n$ segments of the sa..

The trapezoidal rule is the first formula in the Newton-Cotes integration formulas, and it corresponds to the case where the polynomial in the equation below is of the first degree.  Here, the straight line can be expressed as follows.  The area under this straight line is an estimate of the integral between the interval $a$ and $b$.  This is called the trapezoidal rule. The derivation process c..

The above equation can be considered as a linear least squares model.  Since all other $z$ values are zero, the goal is to determine the coefficient values that minimize the following.  To minimize this, the normal equations can be expressed in matrix form.  These equations can be used to determine the unknown coefficients. Additionally, in the special case where $N$ observations form equally sp..

In this post, I'll solve the same example as in the previous post using the cubic spline process derived in the last posting. The first step is to generate a set of simultaneous equations to determine the second derivatives at the nodes. For example, the following data is used for the first internal node.  When these values are substituted into the equation, it results in the following.    Due t..

Based on the theoretical part previously explained, let's solve an example of estimating the value at $x=5$ using quadratic splines from the data provided below.  In the current problem, there are four data points and $n=3$ intervals, so we need to determine 9 unknowns. The following conditions can be applied.  By substituting the initial and final values into the first and last functions, two m..