Interpolation is a technique for discovering new values for any function by using a collection of values. We may use this formula to find the unknown value of a point. 3 x + 2 y = 7 are the two lines of expression for a distribution (x,y). 3x+2y equals 7 3x+2y = 7 and x + 4 y = 9 x+4y = 9 x+4y = 9 x+4y = 9 x+4y = 9 x+4y = 9 x+4y = 9 x+4y = 9 x+4y = 9 x+4y = 9 x+4y = 9 x+4y = 9 x+4y = 9 x+4y = 9.. This is called a grid. You can see that there are nine points on each line, so we can divide through by 9 to get the x-value of the point.

In mathematics, interpolation is the process of finding a value for **a variable quantity** that lies between the values known or assumed from facts or circumstances. In statistics, interpolation is the process of estimating a value for a variable when actual observations are not available. Interpolation is also used in **numerical differentiation** and integration techniques where approximations are made to obtain **updated values** at points between those at which original data is recorded.

The three main types of interpolation are linear interpolation, quadratic interpolation, and cubic interpolation. These methods are used to estimate the value of a variable when actual measurements are not possible. For example, if we have observed that the temperature is 70 degrees Fahrenheit at 8:00 AM and 75 degrees at 12:00 PM, it can be inferred that the temperature was 70 degrees at 11:00 AM. Because no measurement was taken at this time, we need an approximation.

Interpolation is a statistical strategy for estimating an unknown price or possible yield of an asset using related known variables. Interpolation is accomplished by employing other proven values that are positioned in the same sequence as the uncertain value. The interpolated result is then used in place of the original unknown value.

There are several methods available for performing interpolation. Two common methods are linear interpolation and non-linear interpolation. Linear interpolation assumes that the position being interpolated has a value that lies on a line segment joining **two known points**. Thus, the estimated value will be located directly between the known points. Non-linear interpolation uses more than one point to estimate the value of **the unknown point**. These methods include **spline interpolation** and regression analysis.

Interpolation can also refer to the process of estimating the value of an asset that cannot be observed directly. For example, if there was a market price for assets with the same characteristics as the asset being valued, then the value of the unobservable asset could be estimated by looking up its price in a table or graph. Such tables are commonly called "valuation tables" or "asset valuation models." The actual process by which this is done varies depending on the type of asset being valued.

As previously stated, interpolation is the process of approximating a given function, the values of which are known at tabular points, by a suitable polynomial of the degree that it takes the values at the tabular points. It should be noted that any inaccuracies in the input data will be reflected in the resulting polynomial. For example, if the original function were to be slightly undershot at one point, there would be a corresponding overshoot on the reconstructed curve.

The Newton interpolation method uses the concept of derivatives to determine **the best possible polynomial approximation** to a given function at a set of points. Starting with an initial approximation, the method uses this approximation and its first derivative at the points to compute a second-derivative estimate. This process is then repeated until the desired accuracy is achieved.

Newton's method requires only two inputs: a starting value and a stopping criterion. It calculates an approximate value for the function between **each pair** of iterations, and stops when the relative change in this value falls below **some small threshold**. The method is very efficient at finding a good approximation to a function, but it can get stuck in a loop if the starting value is inappropriate. For example, if the starting value is too high or too low, it will not reach an accurate conclusion about the function's behavior.

Newton's method was invented by **Isaac Newton** around 1665.

Interpolation is a method of fitting the data points to reflect the value of a function that is defined as utilizing the data to forecast the data inside the dataset. Extrapolation is the process of using a data set to produce an estimate that is larger than the data set. For example, if you have observed that all of your students who applied for financial aid last year were awarded some form of assistance, you could use this information to estimate what percentage of applicants would receive funding. This represents one use of interpolation and extrapolation in forecasting.

Forecasting is used by businesses and government agencies to plan activities and make decisions. For example, economists use forecasts when determining how changes in economic conditions will affect future events. Forecasters also help companies make plans by identifying trends that will influence market behavior.

The two most common types of forecasts are subjective and objective. Subjective forecasts rely on **expert judgment** to determine what factors will be important in the future and how those factors will impact **the predicted outcome**. Objective forecasts use data to project what might happen in the future based on what has happened in the past. For example, an objective forecast would calculate the percentage of students who received financial aid by looking at the data from **the previous year**. Subjective predictions may differ between experts. Objective predictions will usually be consistent across different people because they are based on facts and figures relating to the past.