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Simultaneous Linear Equations. A farmer had 196 cows. When he rounded them up, he had 200 cows. ( Reader’s Digest). The name of the person in the picture is. A$AP Rocky Kid Cudi MC Hammer T.I. Vanilla Ice. The size of matrix. is. 10. The c 32 entity of the matrix. 2 3 6.3
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Simultaneous Linear Equations http://nm.mathforcollege.com
A farmer had 196 cows. When he rounded them up, he had 200 cows. (Reader’s Digest) http://nm.mathforcollege.com
The name of the person in the picture is • A$AP Rocky • Kid Cudi • MC Hammer • T.I. • Vanilla Ice
The size of matrix is 10 http://nm.mathforcollege.com
The c32 entity of the matrix • 2 • 3 • 6.3 • does not exist 10 http://nm.mathforcollege.com
Given • -3 • 3 • 9 then if [C]=[A]-[B], c23= http://nm.mathforcollege.com
Given • -57 • -45 • 57 • does not exist then if [C]=[A][B], then c31=. 10 http://nm.mathforcollege.com
A square matrix [A] is upper triangular if http://nm.mathforcollege.com
An identity matrix [I] needs to satisfy the following matrix is square all of the above 10 http://nm.mathforcollege.com
The following system of equations x + y=26x + 6y=12has solution(s). • no • one • more than one but a finite number of • infinite 10 http://nm.mathforcollege.com
PHYSICAL PROBLEMS http://nm.mathforcollege.com
Truss Problem http://nm.mathforcollege.com
a a c b b Pressure vessel problem http://nm.mathforcollege.com
Polynomial Regression We are to fit the data to the polynomial regression model http://nm.mathforcollege.com
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Simultaneous Linear EquationsGaussian Elimination(Naïve and the Not That So Innocent Also) http://nm.mathforcollege.com
You know Lady Gaga; Who is Shady Gaga • Lady Gaga’s sister! • A person who looks bad with their sunglasses on. • A person who looks good with sunglasses on but bad once he/she takes the sunglasses off • That is what Alejandro calls Lady Gaga • Slim Shady’s sister! 10 http://nm.mathforcollege.com
The goal of forward elimination steps in Naïve Gauss elimination method is to reduce the coefficient matrix to a (an) _________ matrix. • diagonal • identity • lower triangular • upper triangular http://nm.mathforcollege.com
One of the pitfalls of Naïve Gauss Elimination method is • large truncation error • large round-off error • not able to solve equations with a noninvertible coefficient matrix http://nm.mathforcollege.com
Increasing the precision of numbers from single to double in the Naïve Gaussian elimination method • avoids division by zero • decreases round-off error • allows equations with a noninvertible coefficient matrix to be solved http://nm.mathforcollege.com
Division by zero during forward elimination steps in Naïve Gaussian elimination for [A][X]=[C] implies the coefficient matrix [A] • is invertible • is not invertible • cannot be determined to be invertible or not http://nm.mathforcollege.com
Division by zero during forward elimination steps in Gaussian elimination with partial pivoting of the set of equations [A][X]=[C] implies the coefficient matrix [A] • is invertible • is not invertible • cannot be determined to be invertible or not http://nm.mathforcollege.com
Using 3 significant digit with chopping at all stages, the result for the following calculation is • -0.0988 • -0.0978 • -0.0969 • -0.0962 http://nm.mathforcollege.com
Using 3 significant digits with rounding-off at all stages, the result for the following calculation is • -0.0988 • -0.0978 • -0.0969 • -0.0962 http://nm.mathforcollege.com
Determinants If a multiple of one row of [A]nxn is added or subtracted to another row of [A]nxn to result in [B]nxn then det(A)=det(B) • The determinant of an upper triangular matrix [A]nxnis given by Using forward elimination to transform [A]nxnto an upper triangular matrix, [U]nxn. http://nm.mathforcollege.com
Simultaneous Linear EquationsLU Decomposition http://nm.mathforcollege.com
You thought you have parking problems. Frank Ocean is scared to park when __________ is around. • A$AP Rocky • Adele • Chris Brown • Hillary Clinton http://nm.mathforcollege.com
Truss Problem http://nm.mathforcollege.com
If you have n equations and n unknowns, the computation time for forward substitution is approximately proportional to • 4n • 4n2 • 4n3 http://nm.mathforcollege.com
If you have a nxn matrix, the computation time for decomposing the matrix to LU is approximately proportional to • 8n/3 • 8n2/3 • 8n3/3 http://nm.mathforcollege.com
LU decomposition method is computationally more efficient than Naïve Gauss elimination for solving • a single set of simultaneous linear equations • multiple sets of simultaneous linear equations with different coefficient matrices and same right hand side vectors. • multiple sets of simultaneous linear equations with same coefficient matrix and different right hand side vectors http://nm.mathforcollege.com
For a given 1700 x 1700 matrix [A], assume that it takes about 16 seconds to find the inverse of [A] by the use of the [L][U] decomposition method. The approximate time in seconds that all the forward substitutions take out of the 16 seconds is • 4 • 6 • 8 • 12 http://nm.mathforcollege.com
The following data is given for the velocity of the rocket as a function of time. To find the velocity at t=21s, you are asked to use a quadratic polynomial v(t)=at2+bt+c to approximate the velocity profile. http://nm.mathforcollege.com
THE END http://nm.mathforcollege.com
4.09Adequacy of Solutions http://nm.mathforcollege.com
The well or ill conditioning of a system of equations [A][X]=[C] depends on the • coefficient matrix only • right hand side vector only • number of unknowns • coefficient matrix and the right hand side vector http://nm.mathforcollege.com
The condition number of a n×n diagonal matrix [A] is http://nm.mathforcollege.com
The adequacy of a simultaneous linear system of equations [A][X]=[C] depends on (choose the most appropriate answer) • condition number of the coefficient matrix • machine epsilon • product of the condition number of coefficient matrix and machine epsilon • norm of the coefficient matrix http://nm.mathforcollege.com
If , then in [A][X]=[C], at least these many significant digits are correct in your solution, • 0 • 1 • 2 • 3 http://nm.mathforcollege.com
THE END http://nm.mathforcollege.com
