Unit 5: Interpolation, Differentiation and Integration - Subjective Questions

Interpolation is the process of estimating the value of a function at an intermediate point using a set of known data points.

If the known data are , interpolation constructs a function such that

Purpose:

• It estimates unknown values between measured or tabulated values.
• It replaces a complicated function with a simpler polynomial.
• It is useful when the analytical form of a function is unavailable.

Applications in robotics:

• Estimating robot joint positions between sampled time instants.
• Generating smooth trajectories for the position, velocity, and orientation of a robot manipulator.
• Approximating sensor data and calibration curves.

For three data points, assume a polynomial of the form

Using gives

Using gives

Using gives

Therefore, Newton's divided-difference polynomial is

The divided differences are defined by

and

A divided-difference table is constructed in successive columns.

For data , the first divided differences are

Higher-order divided differences are calculated using

The table has the following general structure:

• First column: values.
• Second column: function values .
• Third column: first divided differences.
• Fourth column: second divided differences.
• Subsequent columns: higher-order divided differences.

The first entry of each divided-difference column becomes a coefficient in Newton's polynomial:

This form is computationally convenient because a new data point can be added without completely recalculating the existing polynomial.

The data points are , , and .

The first divided differences are

The second divided difference is

Newton's polynomial is therefore

On simplifying,

Thus, the interpolating polynomial is

Verification:

• .
• .
• .

For distinct data points , the Lagrange interpolating polynomial is

where the Lagrange basis polynomial is

Each basis polynomial has the property

Therefore, at ,

Advantages:

• It gives a direct formula for interpolation.
• It does not require a divided-difference table.
• It is suitable for a small number of data points.

Limitation: If a new data point is added, the entire expression generally has to be reconstructed.

For three points, construct three basis polynomials:

The quadratic interpolating polynomial is

Hence,

At each data point, one basis polynomial becomes and the others become , so the polynomial passes through all three points.

Both methods produce the unique polynomial of degree at most passing through distinct data points, but their computational forms differ.

Newton's form is

Lagrange's form is

Conclusion: Lagrange interpolation is convenient for a fixed small data set, whereas Newton interpolation is preferable when data points may be added successively.

Let the data points be equally spaced with spacing , and define

For equally spaced data, the divided differences can be expressed in terms of forward differences. The Newton forward interpolation formula is

In general,

Here:

• is the first forward difference.
• is the second forward difference.
• is the uniform interval between successive values of .
• The formula is most accurate when is near the beginning of the table.

Newton's backward formula is used for equally spaced data when the required value of lies near the end of the tabulated interval. Define

where is the last tabulated value and is the equal spacing.

The formula is

In general,

Here, the backward difference operator is defined by

It is preferred when:

• The data points are equally spaced.
• The interpolation point is close to the last entry in the table.
• Backward differences near the end of the table are easier to calculate than forward differences near the beginning.

The main differences are as follows:

The forward formula is

whereas the backward formula is

Both require equally spaced values of and represent the same interpolating polynomial when all terms are retained.

The spacing is , and choose . Thus,

The forward-difference table is:

• , , .
• .
• .
• .

Using the quadratic forward formula,

we obtain

Therefore,

Hence, the estimated value is

Numerical integration is the process of approximating a definite integral using a finite number of function values.

For a function , the integral

may be difficult or impossible to evaluate analytically. Numerical integration replaces the function by an approximating polynomial or geometric shape and integrates that approximation.

Importance in robotics:

• Computing displacement from velocity data:

• Computing robot joint position from measured angular velocity.
• Estimating energy or work from force and displacement data.
• Integrating equations of motion in robot simulation.
• Processing sampled sensor data when only discrete measurements are available.

Common methods include the trapezoidal rule and Simpson's rules.

Newton-Cotes formulae approximate a definite integral by replacing the function with an interpolation polynomial constructed at equally spaced points.

If the interval is divided into equal subintervals, the integral is approximated by

where is an interpolation polynomial.

Closed Newton-Cotes formulae:

• Include both endpoints and .
• Use points such as and .
• The trapezoidal rule and Simpson's rules are closed formulas.

Open Newton-Cotes formulae:

• Do not include the endpoints.
• Use interior points only.
• They can be useful when the function is undefined or difficult to evaluate at an endpoint.

For equally spaced points, the general approximation has the form

where is the spacing and are Newton-Cotes weights.

Consider two adjacent points and , where

The linear interpolating polynomial between the points is

Integrating from to gives

For equal subintervals, summing all individual trapezoids gives

where

The composite trapezoidal rule gives greater accuracy than a single trapezoid because it uses several smaller intervals.

For one interval , the trapezoidal rule is

For equal subintervals, with , the composite rule is

If has a continuous second derivative, the composite truncation error is

for some .

Therefore, the error is of order . If the step size is reduced by half, the approximate error is reduced by a factor of four, provided the function is sufficiently smooth.

The rule is exact for all polynomials of degree one or less.

Given

we have

The function values are

Using the composite trapezoidal rule,

Substitution gives

Thus,

Therefore,

The exact value is , so the trapezoidal rule overestimates this convex function.

Consider three equally spaced points , , and . A quadratic polynomial is fitted through the function values , , and .

Using a local coordinate centered at , let . The three points correspond to . The quadratic interpolation formula is

Integrating from to gives

After integration,

This is Simpson's rule. It is exact for polynomials of degree three or less, even though it is derived from a quadratic interpolating polynomial.

For equal subintervals, where is even, define

The composite Simpson's rule is

The weights follow the pattern

Condition: The number of subintervals must be even because Simpson's rule applies a quadratic approximation over pairs of subintervals.

For two subintervals, the formula reduces to

Here,

The step size is

The function values are

Using the composite Simpson's rule,

Therefore,

Thus,

Hence,

Since Simpson's rule is exact for cubic polynomials, the exact integral is also .

For one application of Simpson's rule over two subintervals of width ,

If has a continuous fourth derivative, the truncation error is

for some .

For the composite Simpson's rule over with even and , the error is

Thus, Simpson's rule has order , whereas the composite trapezoidal rule has order .

The rule is exact for polynomials of degree three or less, because their fourth derivative is zero.