Linear Program Polynomial Interpolation Matlab
In everyday life, sometimes we may require finding some unknown value with the given set of observations. For example, the data available for the premium, payable for a policy of Rs.1000 at age x, is for every fifth year. Suppose, the data given is for the ages 30, 35, 40, 45, 50 and we are required to find the value of the premium at the age of 42 years, which is not directly given in the table. Here we use the method of estimating an unknown value within the range with the help of given set of observation which is known as interpolation.
Definition of Interpolation Given the set of tabular values (x0, y0), (x1, y1),,(xn, yn) satisfying the relation y=f(x) where the explicit nature of f(x) is not known, it is required to find a simpler function say?(x), such that f(x) and?(x) agree at the set of tabulated points. Such a process is called as interpolation.
The function polyval is used to evaluate polynomials in the Matlab. Looks like the polynomial interpolation problem reduces to a linear equation problem. As an application, here is a script that displays cubic interpolants of sin(x) on [0, 2π]. Piecewise Linear Interpolation Exercise 7 Exercise 8 Approximating the derivative (Extra) Exercise 9 Exercise 10 Exercise 11 Exercise 12 Exercise 13 Exercise 14 Extra Credit 1 Introduction We saw in the last lab that the interpolating polynomial could get worse (in the sense that values at interme-diate points are far from the function) as its degree increased.
If we know ‘n’ values of a function, we can get a polynomial of degree (n-1) whose graph passes through the corresponding points. Such a polynomial is used to estimate the values of the function at the values of x.
We will study two different interpolation formula based on finite differences when the values of x are equally spaced. The first formula is: Newton’s forward difference interpolation formula: The formula is stated as: Where ‘a+ph’ is the value for which the value of the function f(x) is to be estimated. Here ‘a’ is the initial value of x and ‘h’ is the interval of differencing. Question The table gives the distance between nautical miles of the visible horizon for the given height in feet above the earth surface. Find the value of y when x= 218 feet.
• • • • • Interpolation Engineering problems often require the analysis of data pairs. For example, the paired data might represent a cause and effect, or input-output relationship, such as the current produced in a resistor as a result of an applied voltage, or a time history, such as the temperature of an object as a function of time. Another type of paired data represents a profile, such as a road profile (which shows the height of the road along its length).
In some applications we want to estimate a variable’s value between the data points. This process is called interpolation. In other cases we might need to estimate the variable’s value outside of the given data range. This process is extrapolation.
Interpolation and extrapolation are greatly aided by plotting the data. Such plots, some perhaps using logarithmic axes, often help to discover a functional description of the data. Suppose that x represents the independent variable in the data (such as the applied voltage in the preceding example), and y represents the dependent variable (such as the resistor current). In some applications the data set will contain only. One value of y for each value of x. In other cases there will be several measured values of y for a particular value of x.
This condition is called data scatter. For example, suppose we apply 10 V to a resistor, and measure 3.1 mA of current. Then, repeating the experiment, suppose We measure 3.3 mA the second time. Motivational videos for student success. If we average the two results, the resulting data point will be x = 10V,y = 3.2 mA, which is an example of aggregating the data, In this section we assume that the. data have been aggregated if necessary, so only one value of y corresponds to a specific value of x. You can use the methods of Sections 7.1 and 7.2 to aggregate the data by computing its mean. The data’s standard deviation indicates how much the data is spread around the aggregated point.