Finite

Cheney and Kincaid(Cheney, Ward, and David Kincaid. Numerical mathematics and computing. International Thomson Publishing, 1998.) based the derivation of cubic splines on the concept that each pair of intervals is connected by a cubic polynomial, which means the second derivative within each interval is linear. The goal of cubic splines is to derive a cubic polynomial for each interval between t..

The Lagrange interpolation polynomial reformulates the Newton polynomial while avoiding the calculation of divided differences, and it can be succinctly expressed as follows.  The linear and secon-order versions of the above formula are as follows.  The succinctly expressed equation can be directly derived from Newton's polynomial. However, in the Lagrange form each term of ${L_i}(x)$ becomes $1..

As explained in the previous post, an $n$th degree polynomial can be expressed as follows.  As with linear and quadratic interpolation, data points can be used to calculate the coefficients ${b_0},{b_1}, \cdots ,{b_n}$, and for an $n$th degree polynomial, $n+1$ data points are required. These data points can be used to calculated the coefficients through the equation below.  Here, the function i..

In the previous post, errors occurred because curves were approximated as straight lines. One way to improve the estimates is to introduce some curvature to the line connecting the points. When three data points. When three data points are available, this can be accomplished using a second-degree polynomial.  The equation can be summarized as follows.  Therefore, the above equation represents th..

In engineering problems, there are instances when it's necessary to estimate values between data points. The most commonly used method for this purpose is polynomial interpolation. Typically, an $n$-th degree polynomial is expressed as follows.  For $n+1$ data points, there exists only one $n$-th degree polynomial that passes through all these points, just as there is only one straight line that..

In engineering problems, nonlinear models oftern fit data better than linear models. These models can be defined as having nonlinear dependencies on their parameters.  For instance, such an equation typically cannot be transformed into the following form.  Like linear least squares, nonlinear regression also determines parameter values that minimize the sum of the squares of the residuals. Howev..

MathJax = { tex: {inlineMath: [['$', '$'], ['\\(', '\\)']]} }; There exists a procedure for deriving the equation of a line using the least squares criterion. However, some engineering data, while showing clear patterns, often cannot be accurately represented by a straight line. In such cases, using a curve rather than a line is more appropriate to represent the data. Therefore, transformatio..

The Newton-Raphson method is a technique for finding the roots of $x$ where $f(x)=0$. This method can be summarized as follows. A similar approach can be used to find the minimum or maximum values of $f(x)$. To do this, a new function is defined. This method, similar to the Newton-Raphson approach, is an open method, so it does not requre initial estimates that bracket the optimum. Additionally,..

Parabolic interpolation leverages the existence of a single quadratic polynomial or parabola that connects three points, similar to how two points define a line. Therefore, if there are three points that collectively include an extremum, a parabola can be fitted to these points. Then, by differentiating this parabola and setting the derivative to zero, the optimal $x$ estimate can be found. Here..

In this post, apply the golden-section search method to an example, as discussed in the previous post(142_Golden-Section Search). Since the method is straightforward, today's post will be brief. Use the golden-section search method to find the maximum value of a function, setting ${x_l}=0$ and ${x_u}=4$. First, create two internal points using the golden ratio. Calculate the function values at t..