Numerical Differentiation Chapter 5
Nizar Salim 1 lecture 1
5.1 INTRODUCTION
Figure 5.1 presents a set of tabular data in the form of a set of [x,f(x)] pairs. The function
f(x) is known only at discrete values of x. Interpolation within a set of discrete data is
discussed in Chapter 3. Differentiation within a set of discrete data is presented in this
chapter. The discrete data presented in Figure 5.1 are values of the function f(x) = l/x,
which are used as the example problem in this chapter.
The evaluation of a derivative is required in many problems in engineering and
science:
where the alternate notations f (x) and fx(x) are used for the derivatives. The function f(x),
which is to be differentiated, may be a know function or a set of discrete data. In general,
known functions can be differentiated exactly. Differentiation of discrete data, however,
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requires an approximate numerical procedure. The evaluation of derivatives by
approximate numerical procedures is the subject of this chapter.
Numerical differentiation formulas can be developed by fitting approximating
functions (e.g., polynomials) to a set of discrete data and differentiating the
approximating function. Thus,
This process is illustrated in Figure 5.2. As illustrated in Figure 5.2, even though the
approximating polynomial Pn(x) passes through the discrete data points exactly, the
derivative of the polynomial P n(x) may not be a very accurate approximation of the
derivative of the exact function f(x) even at the known data points themselves. In general,
numerical differentiation is an inherently inaccurate process.
To perform numerical differentiation, an approximating polynomial is fit to the
discrete data, or a subset of the discrete data, and the approximating polynomial is
differentiated. The polynomial may be fit exactly to a set of discrete data by the methods
presented in Sections 4.3 to 4.9, or approximately by a least squares fit as described in
Section 4.10. In both cases, the degree of the approximating polynomial chosen to
represent the discrete data is the only parameter under our control.
Several numerical differentiation procedures are presented in this chapter.
Differentiation of direct fit polynomials, Lagrange polynomials, and divided difference
polynomials can be applied to both unequally spaced data and equally spaced data.
Differentiation formulas based on both Newton forward-difference polynomials and
Newton backward-difference polynomials can be applied to equally spaced data.
Numerical differentiation formulas can also be developed using Taylor series. This
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approach is quite useful for developing difference formulas for approximating exact
derivatives in the numerical solution of differential equations.
The simple function
5.2 UNEQUALLY SPACED DATA
Three straightforward numerical differentiation procedures that can be used for both
unequally spaced data and equally spaced data are presented in this section:
1. Direct fit polynomials
2. Lagrange polynomials
3. Divided difference polynomials
5.2.1 Direct Fit Polynomials
A direct fit polynomial procedure is based on fitting the data directly by a polynomial and
differentiating the polynomial. Recall the direct fit polynomial, Eq. (4.34):
where Pn(x) is determined by one of the following methods:
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1. Given N = n + I points, [xi, f(xi)], determine the exact nth-degree polynomial that
passes through the data points, as discussed in Section 4.3.
2. Given N > n + 1 points, [xi, f(xi)], determine the least squares nth-degree polynomial
that best fits the data points, as discussed in Section 4.10.3.
After the approximating polynomial has been fit, the derivatives are determined by
differentiating the approximating polynomial. Thus,
Equations (5.7a) and (5.7b) are illustrated in Example 5.1.
5.2.2. Lagrange Polynomials
The second procedure that can be used for both unequally spaced data and equally spaced
data is based on differentiating a Lagrange polynomial. For example, consider the
second- degree Lagrange polynomial, Eq. (4.45):
5.2.3. Divided Difference Polynomials
The third procedure that can be used for both unequally spaced data and equally spaced
data is based on differentiating a divided difference polynomial, Eq. (4.65):
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Example 5.1. Direct fit, Lagrange, and divided difference polynomials.
Let s solve the example problem presented in Section 5.1 by the three procedures
presented above. Consider the following three data points:
Solving for ao,a1 and a2 by Gauss elimination gives a0 = 0.858314, a1 = -0.245500, and
a2 =0.023400.
Substituting these values into Eqs. (5.7a) and (5.7b) and evaluating at x = 3.5 yields the
solution for the direct fit polynomial:
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Substituting the tabular values into Eqs. (5.9a) and (5.9b) and evaluating at x = 3.5
yields the solution for the Lagrange polynomial:
A divided difference table must be constructed for the tabular data to use the divided
difference polynomial. Thus,
Substituting these values into Eqs. (5.11a) and (5.11b) yields the solution for the divided
difference polynomial:
The results obtained by the three procedures are identical since the same three
points are used in all three procedures.
The error in f (3.5) is Error =f (3.5) P 2(3.5) = -0.081700 -(-0.081633) = -0.000067,
and the error in f "(3.5) is Error =f "(3.5)- P 2 (3.5) = 0.046800 (.046647) = 0.000153.
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5.3 EQUALLY SPACED DATA
When the tabular data to be differentiated are known at equally spaced points, the
Newton forward-difference and backward-difference polynomials, presented in Section
4.6, can be fit to the discrete data with much less effort than a direct fit polynomial, a
Lagrange polynomial, or a divided difference polynomial. This can significantly decrease
the amount of effort required to evaluate derivatives. Thus,
where n( ) is either the Newton forward-difference or backward-difference polynomial.
5.3.1. Newton Forward-Difference Polynomial
Recall the Newton forward-difference polynomial, Eq. (4.88):
Equation (5.15) requires that the approximating polynomial be an explicit function of x,
whereas Eq. (5.16) is implicit in x. Either Eq. (5.16) must be made explicit in x by
introducing Eq. (5.18) into Eq. (5.16), or the differentiation operations in Eq. (5.15) must
be transformed into explicit operations in terms of s, so that Eq. (5.16) can be used
directly.
The first approach leads to a complicated procedure, so the second approach is taken.
From Eq. (5.18), x = x(s). Thus,
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Higher-order derivatives can be obtained in a similar manner. Recall that nf becomes
less and less accurate as n increases. Consequently, higher-order derivatives become
increasingly less accurate.
At x = x0, s = 0.0, and Eqs. (5.23) and (5.25) becomes
Numerical Differentiation Chapter 5
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Example 5.2. Newton forward-difference polynomial, one-sided.
Let s solve the example problem presented in Section 5.1 using a Newton forwarddifference
polynomial with base point x0 = 3.5, so x0 = 3.5 in Eqs. (5.26) and (5.27).
Selecting data points from Figure 5.1 and constructing the difference table gives
The order of the approximation of P n(x) is the same as the order of the highest-order
difference included in the evaluation. The first term in Eq. (5.32) is fo, so evaluating that
term gives an 0(h) result. The second term in Eq. (5.32) is 2 fo , so evaluating that term
yields an 0(h2) result, etc. Evaluating Eq. (4.32) term by term yields
P n(3.5) = -0.07936_ first order Error = 0.00227_
= -0.08150. second order = 0.00013_
= -0.08161_ third order = 0.00002_
The first-order result is quite inaccurate. The second- and third-order results are quite
good. In all cases, only five significant digits after the decimal place are obtained.
Equation (5.27) gives
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The order of the approximation of P (x) is one less than the order of the highest-order
difference included in the evaluation. The first term in Eq. (5.33) is 2 fo, so evaluating
that term gives an 0(h) result. The second term in Eq. (5.33) is 3 fo, so evaluating that
term yields an 0(h2) result, etc. Evaluating Eq. (5.33) term by term yields
P (3.5) = 0.0428__ first order Error = -0.0038__
= 0.0460__ second order = -0.0006__
The first-order result is very poor. The second-order result, although much more accurate,
has only four significant digits after the decimal place.
5.3.2. Newton Backward-Difference Polynomial
Recall the Newton backward-difference polynomial, Eq. (4.101):
Higher-order derivatives can be obtained in a similar manner. Recall that V"f becomes
less and less accurate as n increases. Consequently higher-order derivatives become
increasingly less accurate.
At x = xo, s = 0.0, and Eqs. (5.40) and (5.41) become
Numerical Differentiation Chapter 5
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