Mathematics
Linear Interpolation
Linear interpolation is a method of estimating a value between two known values by drawing a straight line between them. It is commonly used in mathematics and other fields to estimate values that fall within a given range. The method is based on the assumption that the relationship between the two known values is linear.
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5 Key excerpts on "Linear Interpolation"
- K. Akbar Ansari Ph.D., P.E., Bonni Dichone Ph.D.(Authors)
- 2019(Publication Date)
- SDC Publications(Publisher)
Chapter 5 Numerical Interpolation In any field where measured or statistical data is involved, a need to interpolate between given data points is bound to exist. Because measured or available data is typically not provided in the form of an analytically determined function, the process known as interpolation must be resorted to in order to obtain function values at points other than the given data points. This process involves the generation of a curve that must pass through the given data points and its use in determining the function value at any intermediate point on this curve. As will be seen in the sections following, a polynomial fitting n data points will be of order n - 1, that is one less than the number of data points given. Thus, four data points will generate a cubic while three data points will give a quadratic and so on. 5.1 Linear Interpolation Given two data points ( x k , y k ) and ( x k +1 , y k +1 ) , as shown in the figure below, the connection between the coordinates of a point ( x, y ) lying between the data points can be expressed by the linear relationship y = y k ( x k - x k +1 ) - y k +1 ( x - x k ) x k - x k +1 . (5.1) ( ) f x ( ) , x y ( ) , k k x y ( ) 1 1 , k k x y + + 121 122 5 Numerical Interpolation 5.2 The Method of Undetermined Coefficients Given the data points ( x k , y k ) , k = 0 ,..., n, it is required to find an interpolating polynomial P n ( x ) such that the following constraint equations are satisfied P n ( x 0 ) = y 0 , P n ( x 1 ) = y 1 , . . . P n ( x n ) = y n . (5.2) Assuming the interpolating polynomial P n ( x ) is of the form P n ( x ) = a 0 + a 1 x + a 2 x 2 + ··· + a n x n , (5.3) Equations (5.2) yield the following matrix equation * . . . . . , 1 x 0 x 2 0 ··· x n 0 1 x 1 x 2 1 ··· x n 1 . . . . . . . . . . . . . . . 1 x n x 2 n ··· x n n + / / / / / - * . . . . . , a 0 a 1 . . . a n + / / / / / - = * . . . .
eBook - PDFNumerical Methods for Engineers and Scientists
An Introduction with Applications Using MATLAB
- Amos Gilat, Vish Subramaniam(Authors)
- 2013(Publication Date)
- Wiley(Publisher)
6.6.1 Linear Splines With linear splines, interpolation is carried out by using a first-order polynomial (linear function) between the points (the points are con- nected with straight lines), as shown in Fig. 6-18. Using the Lagrange form, the equation of the straight line that connects the first two points is given by: (6.64) For n given points, there are intervals. The interpolation in inter- val i, which is between points and ( ), is done by using the equation of the straight line that connects point with point : for (6.65) 0 5 10 15 0 5 10 15 20 25 30 35 x y Figure 6-17: Fitting 16 data points with a 15th order polynomial. f 1 x x x 2 – x 1 x 2 – -------------------- y 1 x x 1 – x 2 x 1 – -------------------- y 2 + = n 1 – x i x i 1 + x i x x i 1 + x i y i x i 1 + y i 1 + f i x x x i 1 + – x i x i 1 + – ------------------------- y i x x i – x i 1 + x i – ------------------------- y i 1 + + = i 1 2 n 1 – = 224 Chapter 6 Curve Fitting and Interpolation It is obvious that linear splines give continuous interpolation since the two adjacent polynomials have the same value at a common knot. There is, however, a discontinuity in the slope of the linear splines at the knots. Interpolation with linear splines is easy to calculate and program, and gives good results when the data points are closely spaced. Exam- ple 6-6 shows a numerical application of linear splines by hand and by using a user-defined MATLB function. Example 6-6: Linear splines. The set of the following four data points is given: x 8 11 15 18 y 5 9 10 8 (a) Determine the linear splines that fit the data. (b) Determine the interpolated value for . (c) Write a MATLAB user-defined function for interpolation with linear splines. The inputs to the function are the coordinates of the given data points and the x coordinate of the point at which y is to be interpolated. The output from the function is the interpolated y value at the given point.
eBook - PDF- M. Atteia(Author)
- 2014(Publication Date)
- North Holland(Publisher)
95 Chapter II INTERPOLATION. APPROXIMATION OF LINEAR FUNCTIONALS In this chapter we have collected, some fundamental results of Linear Interpolation theory which will be useful in the following* Firstly, we present general properties of linear interpo-lation and we formulate the dual problem of Linear Interpolation. Next, to evaluate the interpolation error, we establish a theorem due to Sard. We emphasize two fundamental techniques, those of Lagrange and Newton. In the case of finite number of data, the existence of an interpolatory function can be reduced to a classical algebraical problem. In the case of infinite number of data, some non-trivial considerations from analysis are needed. So, it's often a difficult problem to study the behavior of interpolation error when the number of data increases to infinity. The chapter entitled internal and external convergence of spline functions is devoted to that question. In the last paragraph of this chapter we introduce the theory of frames in a Hilbert space. 96 Chapter II A frame is a denumerable family of elements in a Hilbert space not necessarily free but which possesses similar properties to those of orthonormal bases. In a lot of applied Mathematics problems, it is necessary to know how to calculate an approximate value of a real-valued function f at a point χ of its domain of definition Ω, from the values of f at (n+1) points x 0 ,...,x of Ω· For that, f is replaced by a real-valued function g which have the two following properties: (i) g is a linear combination of (n+1) appropriate real-valued functions g Q , g
eBook - PDFNumerical Algorithms
Methods for Computer Vision, Machine Learning, and Graphics
- Justin Solomon(Author)
- 2015(Publication Date)
- A K Peters/CRC Press(Publisher)
For example, bicubic interpolation requires values from more grid points than just the four closest to ~x needed for biLinear Interpolation. This additional expense can slow down image processing tools for which every lookup in memory incurs significant computation time. 13.3 THEORY OF INTERPOLATION Our treatment of interpolation has been fairly heuristic. While relying on our intuition for what a “reasonable” interpolation for a set of function values for the most part is acceptable, subtle issues can arise with different interpolation methods that should be acknowledged. 13.3.1 Linear Algebra of Functions We began our discussion by posing interpolation strategies using different bases for the set of functions f : R → R . This analogy to vector spaces extends to a complete linear-algebraic theory of functions, and in many ways the field of functional analysis essentially extends the geometry of R n to sets of functions. Here, we will discuss functions of one variable, although many aspects of the extension to more general functions are easy to carry out. Just as we can define notions of span and linear combination for functions, for fixed a, b ∈ R we can define an inner product of functions f ( x ) and g ( x ) as follows: h f, g i ≡ Z b a f ( x ) g ( x ) dx. We then can define the norm of a function f ( x ) to be k f k 2 ≡ p h f, f i . These constructions parallel the corresponding constructions on R n ; both the dot product ~x · ~ y and the inner product h f, g i are obtained by multiplying the “elements” of the two multiplicands and summing—or integrating. Example 13.8 (Functional inner product) . Take p n ( x ) = x n to be the n -th monomial. Then, for a = 0 and b = 1, h p n , p m i = Z 1 0 x n · x m dx = Z 1 0 x n + m dx = 1 n + m + 1 . This shows: p n k p n k , p m k p m k = h p n , p m i k p n kk p m k = p (2 n + 1)(2 m + 1) n + m + 1 .
- Yogesh Jaluria(Author)
- 2011(Publication Date)
- CRC Press(Publisher)
One will expect this value to be the best Numerical Curve Fitting and Interpolation 281 representation of the data if the measurements are all taken with equal care and are thus of comparable accuracy. This example provides a physical basis for the method of least squares and may easily be extended to a function f ( x ), using the consideration of a single unknown variable L given above. 7.6.2 L INEAR R EGRESSION The procedure of obtaining a best fit to a given data set is often known as regression. Let us first consider fitting a straight line to a set of data points denoted by ( x 1 , y 1 ), ( x 2 , y 2 ), . . ., ( x n , y n ), where x is the independent variable and y the dependent variable. Although engineering applications usually lead to nonlinear functions, there are sev-eral circumstances where a linear variation closely approximates the measurements. Moreover, exponential and power-law forms, which are very frequently encountered in practical problems, can often be reduced to linear variations, as illustrated later in this section. Consequently, linear regression is very important in a wide variety of engineering applications, particularly in the derivation of correlating equations from experimental data. The equation of the straight line for curve fitting may be taken as f ( x ) = a + bx (7.46) where a and b are the coefficients that must be determined from the given set of n data points. Thus, a and b are to be chosen such that the sum S of the squares of the deviations of the data points from the values obtained from the equation of the straight line, Equation 7.46, is a minimum. This implies that S y a bx i i i n = --→ = ∑ ( ) 2 1 minimum (7.47) The minimum occurs when the partial derivatives of S with respect to a and b are both zero.
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