Suppose a computer program needs to apply an affine transformation to a complex three-dimensional object made up of 3000 points. The transformation is composed of 8 matrices (call them M¡ through Mg), so for each point (x, y, z) in the object, the following operation is performed. M1 M2 M3 M7 Mg Each multiplication of a matrix times a column vector involves 16 multiplications (of one number by another) and 12 additions, for a total of 28 arithmetic operations. Each multiplication of a matrix times another matrix involves 64 multiplications and 48 additions, for a total of 112 arithmetic operations. (These numbers are not made up or chosen randomly; they are facts about 4x 4 matrix multiplication.)

Suppose a computer program needs to apply an affine transformation to a complex three-dimensional object made up of 3000 points. The transformation is composed of 8 matrices (call them M¡ through Mg), so for each point (x, y, z) in the object, the following operation is performed. M1 M2 M3 M7 Mg Each multiplication of a matrix times a column vector involves 16 multiplications (of one number by another) and 12 additions, for a total of 28 arithmetic operations. Each multiplication of a matrix times another matrix involves 64 multiplications and 48 additions, for a total of 112 arithmetic operations. (These numbers are not made up or chosen randomly; they are facts about 4x 4 matrix multiplication.)

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Concept explainers

Matrices

‘Matrices’ is the plural form of the word matrix, and it is basically a spreadsheet in the form of a box. In mathematics, various functions can be carried out with matrices. Generally, a matrix comes in the shape of a square or rectangle. The elements are distributed horizontally in the form of rows, and vertically in the form of columns.

Question

Suppose a computer program needs to apply an affine transformation to a complex three-dimensional object made up of 3000
points. The transformation is composed of 8 matrices (call them M1 through M3), so for each point (x, y, z) in the object, the
following operation is performed.
y
M1
M2
M3
M7
M8
Each multiplication of a matrix times a column vector involves 16 multiplications (of one number by another) and 12 additions,
for a total of 28 arithmetic operations. Each multiplication of a matrix times another matrix involves 64 multiplications and 48
additions, for a total of 112 arithmetic operations. (These numbers are not made up or chosen randomly; they are facts about
4 x 4 matrix multiplication.)

The most efficient method would be to multiply the 8 matrices together and call that result A (without yet multiplying A by any
column vector). After computing A (just once of course), multiply A by each of the 3000 points, each represented as a column
vector.
How many arithmetic operations would it take to compute A?
How many arithmetic operations would it require to multiply A by one point?
How many would this method require in total?
Using this method would therefore save what percentage of the time of the previous method?
%

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