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Transcript of [email protected] ENGR-25_Linear_Equations-1.ppt 1 Bruce Mayer, PE Engineering/Math/Physics...
[email protected] • ENGR-25_Linear_Equations-1.ppt1
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Engr/Math/Physics 25
Chp8 LinearAlgebraic
Eqns-2
[email protected] • ENGR-25_Linear_Equations-1.ppt2
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Learning Goals
Define Linear Algebraic Equations Solve Systems of Linear Equation by
Hand using• Gaussian Elimination• Cramer’s Method
Distinguish between Equation System Conditions: Exactly Determined, Overdetermined, Underdetermined
Use MATLAB to Solve Systems of Eqns
[email protected] • ENGR-25_Linear_Equations-1.ppt3
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Cramer’s Method for Eqn Sys
Solves equations using determinants. Gives insight into the
• existence and uniqueness of solutions– Identifies SINGULAR (a.k.a. Divide by Zero)
Systems
• effects of numerical inaccuracy– Identifies ILL-CONDITIONED
(a.k.a. Stiff) Systems
[email protected] • ENGR-25_Linear_Equations-1.ppt4
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Cramer’s Method – Illustrated-1
Cramer’s Method can Solve “Square” Systems; • i.e., [No. Eqns] =
[No. Unknowns]
Consider Sq Sys
502248
3263
3312921
zyx
zyx
zyx
Calc Cramer’s Determinant, Dc
• Also Called the “Characteristic” or “Denominator” Determinant
Dc Determinant of the Coefficients
2248
263
12921
cD
[email protected] • ENGR-25_Linear_Equations-1.ppt5
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Cramer’s Method – Illustrated-2
Now, to Find The Individual Solns, Sub The Constraint Vector for the Variable Coefficients and Compute the Determinant for Each unknown, Dk
In this Example Find Dx, Dy, Dz as
22450
263
12933
xD
502248
3263
3312921
zyx
zyx
zyx
22508
233
123321
yD
5048
363
33921
zD
[email protected] • ENGR-25_Linear_Equations-1.ppt6
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Cramer’s Method – Illustrated-3
Once We’ve Calculated all these Determinants, The Rest is Easy
These Eqns Ilustrate the most UseFul Feature of Cramer’s Method
c
x
D
Dx
c
y
D
Dy
c
z
D
Dz
Dc appears in all THREE
Denominators
[email protected] • ENGR-25_Linear_Equations-1.ppt7
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Cramer’s Method – Illustrated-4
Since – However, We can ANTICIPATE Problems if |Dc| << than the SMALLEST Coefficient
Completing the Example
c
k
D
Dk
Can ID “Condition” by Calculating Dc
• SINGULAR Systems → Dc = 0
• ILL-CONDITIONED Systems → Dc = “Small”– Small is technically
relative to the Dk
502248
3263
3312921
zyx
zyx
zyx
[email protected] • ENGR-25_Linear_Equations-1.ppt8
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Cramer’s Method – Illustrated-6 Calc the
Determinants First Recall
The SIGN pattern for Determinants
48
63112
228
2319
224
26121
cD
Find Dc
1146
7207382604
c
c
D
D
6*84*312
2*822*39
4*222*621
cD
Dc is LARGE → WELL Conditioned System
[email protected] • ENGR-25_Linear_Equations-1.ppt9
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Cramer’s Method – Illustrated-7 Solve using MATLAB’s det Function
>> EqnSys = [21 -9 -12 -33; -3 6 -2 3; -8 -4 22 50];
>> Dc = det(EqnSys(:,1:3))Dc = 1146
>> Dx = det([EqnSys(:,4),EqnSys(:,2:3)])
Dx = 1146
>> Dy = det([EqnSys(:,1),EqnSys(:,4), EqnSys(:,3)])Dy = 2292
All Row Elements of Cols 1-3
All Row Elements of Cols 4, 2-3
All Row Elements of Cols 1, 4, 3
[email protected] • ENGR-25_Linear_Equations-1.ppt10
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Cramer’s Method – Illustrated-8 Solve using MATLAB’s det Function
>> Dz = det([EqnSys(:,1:2),EqnSys(:,4)])Dz = 3438
>> x = Dx/Dcx = 1
>> y = Dy/Dcy = 2
>> z = Dz/Dcz = 3
All Row Elements of Cols 1-2, 4
[email protected] • ENGR-25_Linear_Equations-1.ppt11
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Cramer vs Homogenous: 0bAx
In general, for a set of HOMOGENEOUS linear algebraic equations that contains the same number of equations as unknowns• a nonzero solution exists only if the
set is singular; that is, if Cramer’s determinant is zero
• furthermore, the solution is not unique.• If Cramer’s determinant is not zero, the
homogeneous set has a zero solution; that is, all the unknowns are zero
[email protected] • ENGR-25_Linear_Equations-1.ppt12
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Cramer’s Rule Summary
Cramer’s determinant gives some insight into ill-conditioned problems, which are close to being singular.
A Cramer’s determinant close to zero indicates an ill-conditioned problem.
[email protected] • ENGR-25_Linear_Equations-1.ppt13
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
UnderDetermined Systems An UNDERdetermined system does not
contain enough information to solve for ALL of the unknown variables • Usually because it has fewer equations
than unknowns. In this case an INFINITE number of
solutions can exist, with one or more of the unknowns dependent on the remaining unknowns.• For such systems the Matrix-Inverse
and Cramer’s methods will NOT work
[email protected] • ENGR-25_Linear_Equations-1.ppt14
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
UnderDetermined Example-1
A simple UnderDetermined system is the equation 63 yx
All we can do is solve for one of the unknowns in terms of the other; for example, x = 6 – 3y OR y = −x/3 + 2 • An INFINITE number of (x,y) solutions
satisfy this equation
[email protected] • ENGR-25_Linear_Equations-1.ppt15
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
UnderDetermined Example-2
When there are more Unknowns than Equations, the LEFT-DIVISION method will give a solution with some of the unknowns set equal to ZERO• For Example
which corresponds to • x = 0 • y = 2
>>A = [1, 3]; b = 6;>>solution = A\bsolution =
02
63163
y
xyx
[email protected] • ENGR-25_Linear_Equations-1.ppt16
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
More UnderDetermined Systems
An infinite number of solutions might exist EVEN when the number of equations EQUALS the number of knowns• Predict by Cramer as: 0det cDA
For such systems the Matrix Inverse method and Cramer’s method will also NOT work• MATLAB’s left-division method generates
an error message warning us that the matrix A is singular
[email protected] • ENGR-25_Linear_Equations-1.ppt17
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Minimum Norm Solution
When det(A) = 0, We can use the PSEUDOINVERSE method to find ONE Solution, x, such that the Euclidean (or Pythagorean) Length of x is MINIMIZED
In MATLAB:
223
22
21min nxxxx x
MATLAB will return the MINIMUM NORM SOLUTION →
x = pinv(A)*b
[email protected] • ENGR-25_Linear_Equations-1.ppt18
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Existence and Uniqueness
The set Ax = b with m equations and n unknowns has solutions if and only if
1rankrank AbA Rank[A] is the maximum number of
LINEARLY INDEPENDENT rows of A• Linear Independence → No Row of A
is a SCALAR multiple of ANY OTHER Combinations of Rows
[email protected] • ENGR-25_Linear_Equations-1.ppt19
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
An INconsistent Example
Consider Ax = b Since The Ranks are Unequal → this system of equations is NOT solvable
Graphically
5
4
42
21
2
1
x
x
ERO: Multiply the 1st row by −2 and add to the 2nd row
00
21Rank[A]=1
Rank[Ab]=2
34
0
2
0
1
[email protected] • ENGR-25_Linear_Equations-1.ppt20
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Existence and Uniqueness
Recall Rank for m-Eqns & n-Unknwns
1rankrank AbA Now Let r = rank[A]
• If condition (1) is satisfied and if r = n, then the solution is unique
• If condition (1) is satisfied but r < n, an infinite number of solutions exists and – r unknown variables can be expressed as linear
combinations of the other n−r unknown variables, whose values are ARBITRARY
[email protected] • ENGR-25_Linear_Equations-1.ppt21
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Homogenous Case The homogeneous set Ax = 0 is a
special case in which b = 0 For this case rank[A] = rank[Ab] always,
and thus the system in all cases has the trivial solution x = 0
A nonzero solution, in which at least one unknown is nonzero, exists if and only if rank[A] < n (n No. Unknowns)
If m < n, the homogeneous set always has a nonzero solution
[email protected] • ENGR-25_Linear_Equations-1.ppt22
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
ILLconditioned Systems An ill-conditioned set of equations is a
set that is CLOSE to being SINGULAR The ill-conditioned status depends on
the ACCURACY with which the solution calculations are made• When the internal numerical accuracy used
by MATLAB is INsufficient to obtain a solution, then MATLAB prints a message to warn you that the matrix is close to singular and that the results might be INaccurate
[email protected] • ENGR-25_Linear_Equations-1.ppt23
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
OVERdetermined Systems - 1 By Contrast an OVERdetermined
system is a set of equations that has MORE independent equations than unknowns
For such a system the Matrix Inverse method and Cramer’s method will NOT work because the A matrix is not square
However, SOME overdetermined systems have exact solutions, and they can be obtained with the left division method x = A\b
[email protected] • ENGR-25_Linear_Equations-1.ppt24
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
OVERdetermined Systems - 2 For OTHER OverDetermined systems,
NO exact solution exists.• In some of these cases, the left-division
method does not yield an answer, while in other cases the left-division method gives an answer that satisfies the equation set only in a “LEAST SQUARES” sense– When MATLAB gives an answer to an
overdetermined set, it does NOT tell us whether the answer is Exact or Least-Squares in NatureWe need to check the ranks of A and [Ab] to
know whether the answer is the exact solution.
[email protected] • ENGR-25_Linear_Equations-1.ppt25
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Example Under/Over Determined% Bruce Mayer, PE * 27Feb08% ENGR25 * Under/OverDetermined Linear Systems% file = Over_Under_Lin_Sys_Example_0904.m% Solve Under & Over Determined Systems for [x1; x2; x3; x4]%% An UNDERdetermined case => 3-Eqns in 4-UnknwnsAu = [2 -3 7 4; -9 2 -2 -7; 7 6 -5 -4]bu = [5; -9; 7]disp('Showing UNDERdetermined solution & check - Hit Any Key to continue')xu = Au\bubu_chk = Au*xupause%% An OVERdetermined case => 5-Eqns in 4-UnknwnsAo = [2 -3 7 4; -9 2 -2 -7; 7 6 -5 -4; 1 9 0 -6; 2 -5 -8 1]bo = [5; -9; 7; 2; -6]disp('Showing OVERdetermined solution & check')xo = Ao\bobo_chk= Ao*xo
[email protected] • ENGR-25_Linear_Equations-1.ppt26
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Uniqueness of solutions
A system has a unique solution If, and Only If (IFF): Rank[A]=Rank[Ab] = n• Where n is the ORDER of the system;
i.e., the Number of UNKNOWNS
Such systems are called full-rank systems
[email protected] • ENGR-25_Linear_Equations-1.ppt27
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
OverDetermined Rank
To interpret MATLAB answers correctly for an OVERdetermined system, first check the ranks of [A] and [Ab] to see whether an exact solution exists r[Ab] = r[A] = n (the no. of Unknowns)
If r[Ab] = r[A] > n then a unique soln does not exist, then you know that the left-division answer is a least squares solution• Can easily Chk: Axsoln = b (exactly)?
– This is at least as easy as the rank calc
[email protected] • ENGR-25_Linear_Equations-1.ppt28
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Solving Linear Eqns Summary - 1
If the number of equations (m) in the set equals the number of unknown variables (n), the matrix A is square and MATLAB provides two ways of solving the equation set Ax = b:
1. The matrix inverse method; solve for x by typing x = inv(A)*b.
2. The matrix left-division method; solve for x by typing x = A\b.
[email protected] • ENGR-25_Linear_Equations-1.ppt29
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Solving Linear Eqns Summary - 2
If A is square and if MATLAB does not generate an error message when you use one of these methods, then the set has a unique solution, which is given by the left-division method.
You can always CHECK the solution for x by typing A*x to see if the result is the same as b.
• Use this to avoid the rank calcs
[email protected] • ENGR-25_Linear_Equations-1.ppt30
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Solving Linear Eqns Summary - 3
If when you type A*x you receive an error message, the set is UNDERdetermined, and either it does not have a solution or it has more than one solution
In such a case, if you need more information, you must use the following procedures.
[email protected] • ENGR-25_Linear_Equations-1.ppt31
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Solving Linear Eqns Summary - 4 For UNDERdetermined and
OVERdetermined sets, MATLAB provides three ways of dealing with the equation set Ax = b. (Note that the matrix inverse method will NEVER work with such sets.)
1. The matrix left-division method; solve for x by typing x = A\b
2. The pseudoinverse method; solve for x by typing x = pinv(A)*b
3. the Reduced Row Echelon Form (RREF) method. This method uses the MATLAB function rref to obtain a solution.
[email protected] • ENGR-25_Linear_Equations-1.ppt32
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Solving Linear Eqns Summary - 5
UNDERdetermined Systems• In an underdetermined system NOT
enough information is given to determine the values of all the unknown variables.
– An INFINITE number of solutions might exist in which one or more of the unknowns are dependent on the remaining unknowns
– For such systems Cramer’s method and the matrix inverse method will not work because either A is not square or because |A| = 0.
[email protected] • ENGR-25_Linear_Equations-1.ppt33
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Solving Linear Eqns Summary - 6• Underdetermined Systems cont
– The left-division method will give a solution with some of the unknowns arbitrarily set equal to zero, but this solution is NOT the general solution; it’s just ONE solution (of Trillions)
– An infinite number of solutions might exist even when the number of equations equals the number of unknowns. The left-division method fails to give a solution in such cases
– In cases that have an infinite number of solutions, some of the unknowns can be expressed in terms of the remaining unknowns, whose values are arbitrary. The rref function can be used to find these relations.
[email protected] • ENGR-25_Linear_Equations-1.ppt34
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Solving Linear Eqns Summary - 7
OVERdetermined Systems• An overdetermined system is a set of
equations that has more independent equations than unknowns
– For such a system Cramer’s method and the matrix inverse method will not work because the A matrix is NOT SQUARE
– Some overdetermined systems have exact solutions, which can be obtained with the left-division method A\b.
[email protected] • ENGR-25_Linear_Equations-1.ppt35
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Solving Linear Eqns Summary - 8• OVERdetermined systems cont.
– For overdetermined systems that have no exact solution, the estimated-answer given by the left-division method satisfies the equation set only in a LEAST SQUARES sense
– When we use MATLAB to solve an overdetermined set, the program does not tell us whether the solution is exact. We must determine this information ourselves. The first step is to check the ranks of A and [A b] to see whether a solution exists; if no solution exists, then we know that the left-division solution is a least squares answer.
[email protected] • ENGR-25_Linear_Equations-1.ppt36
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
FlowChart for LinSys Solution
Can Construct Least-Squares Approximation
[email protected] • ENGR-25_Linear_Equations-1.ppt37
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Descison Tree for LinSys Solns
if Rank(A)==Rank(Ab) & Rank(A)==NoUnknowns disp('EXACT Solution')
elseif Rank(A)==Rank(Ab) & Rank(A)<NoUnknowns disp('INFINITE Solutions')
elseif Rank(Ab) > NoUnknowns disp('LEAST SQS Approximation')
else disp('NO solution; can least-sqs est.')
end
[email protected] • ENGR-25_Linear_Equations-1.ppt38
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
All Done for Today
GabrielCramer
Born: 31 July 1704 in Geneva, SwitzerlandDied: 4 Jan 1752 in Bagnols-sur-Cèze, France
[email protected] • ENGR-25_Linear_Equations-1.ppt39
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Engr/Math/Physics 25
Appendix 6972 23 xxxxf
[email protected] • ENGR-25_Linear_Equations-1.ppt40
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
MATLAB LinSys Solver Code - 1 % Script file lineq.m% Solves the set Ax = b, given A and b.% Check the ranks of A and [A b].if rank(A) == rank([A b]) % Then The ranks are equal. Size_A = size(A); % Does the rank of A equal the number of
unknowns? if rank(A) == size_A(2) % Then the Rank of A equals the number of unknowns. disp(’There is a unique solution,
which is:’) x = A\b % Solve using left division.
[email protected] • ENGR-25_Linear_Equations-1.ppt41
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
MATLAB LinSys Solver Code - 2 else % Rank of A does not equal the number
of unknowns. disp(’There is an infinite number of
solutions.’) disp(’The augmented matrix of the
reduced system is:’) rref([A b]) % Compute the augmented
matrix. endelse % The ranks of A and [A b] are not equal. disp(’There are no solutions.’)end
[email protected] • ENGR-25_Linear_Equations-1.ppt42
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
MATLAB rank
rank • Rank of matrix
Syntax• k = rank(A)
Description• The rank function provides an estimate of
the number of linearly independent rows or columns of a full matrix.