Open Library - открытая библиотека учебной информации

Открытая библиотека для школьников и студентов. Лекции, конспекты и учебные материалы по всем научным направлениям.


Образование Solution
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Example 1

The upward velocity of a rocket is given at three different times in the following table

Time, t Velocity, v
s m/s

The velocity data is approximated by a polynomial as

The coefficients for the above expression were found in Chapter 5 to be given by

Find the values of a1, a2, a3 using Naïve Guass Elimination. Find the velocity at seconds.

Forward Elimination of Unknowns: Since there are three equations, there will be two steps of forward elimination of unknowns.

First step: Divide Row 1 by 25 and then multiply it by 64

[Row 1] 2.56 gives Row 1 as

Subtract the result from Row 2

Divide Row 1 by 25 and then multiply it by 144

[Row 1] 5.76 gives Row 1 as

Subtract the result from Row 3

Second step: We now divide Row 2 by –4.8 and then multiply by –16.8

[Row 2] 3.5 gives Row 2 as

Subtract the result from Row 3

Back substitution: From the third equation

Substituting the value of a3 in the second equation,

Substituting the value of in the first equation,

Hence the solution vectoris

The polynomial that passes through the three data points is then

Since we want to find the velocity at seconds, we could simply substitute each value of in and find the corresponding velocity. For example, at

However we could also find all the needed values of velocity at t = 6, 7.5, 9 and 11 seconds using matrixmultiplication.

So if we want to find it is given by

Example 2

Use Naïve GaussEliminationto solve

Use six significant digits with chopping in your calculations.

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