Study Guide
Chapter 10 Matrix Multiplication is Distribution
-
University:
Kentucky State University -
Course:
COS 232| Programming I Academic year:
2021
-
Views:
422
Pages:
12
Author:
Kundulu8mbj
10.14
Option Explicit
Sub
Dim
Dim
Dim
LUDTest()
n As Integer, er As Integer, i As Integer, j As Integer
a(3, 3) As Single, b(3) As Single, x(3) As Single
tol As Single
n = 3
a(1, 1) = 3: a(1, 2) = -0.1: a(1, 3) = -0.2
a(2, 1) = 0.1: a(2, 2) = 7: a(2, 3) = -0.3
a(3, 1) = 0.3: a(3, 2) = -0.2: a(3, 3) = 10
b(1) = 7.85: b(2) = -19.3: b(3) = 71.4
tol = 0.000001
Call LUD(a(), b(), n, x(), tol, er)
'output results to worksheet
Sheets("Sheet1").Select
Range("a3").Select
For i = 1 To n
ActiveCell.Value = x(i)
ActiveCell.Offset(1, 0).Select Next i
Range("a3").Select
End Sub
Sub LUD(a, b, n, x, tol, er)
Dim i As Integer, j As Integer
Dim o(3) As Single, s(3) As Single
Call Decompose(a, n, tol, o(), s(), er)
If er = 0 Then
Call Substitute(a, o(), n, b, x)
Else
MsgBox "ill-conditioned system"
End
End If
End Sub
Sub Decompose(a, n, tol, o, s, er)
Dim i As Integer, j As Integer, k As Integer
Dim factor As Single
For i = 1 To n
o(i) = i
s(i) = Abs(a(i, 1))
For j = 2 To n
If Abs(a(i, j)) > s(i) Then s(i) = Abs(a(i, j))
Next j
Next i
For k = 1 To n - 1
Call Pivot(a, o, s, n, k)
If Abs(a(o(k), k) / s(o(k))) < tol Then
er = -1
Exit For
End If
For i = k + 1 To n
factor = a(o(i), k) / a(o(k), k)
a(o(i), k) = factor
For j = k + 1 To n
a(o(i), j) = a(o(i), j) - factor * a(o(k), j)
Next j
Next i
Next k
If (Abs(a(o(k), k) / s(o(k))) < tol) Then er = -1
End Sub
Sub Pivot(a, o, s, n, k)
Dim ii As Integer, p As Integer
Dim big As Single, dummy As Single
p = k
big = Abs(a(o(k), k) / s(o(k)))
For ii = k + 1 To n
dummy = Abs(a(o(ii), k) / s(o(ii)))
If dummy > big Then
big = dummy
p = ii
End If
Next ii
dummy = o(p)
o(p) = o(k)
o(k) = dummy
End Sub
Sub Substitute(a, o, n, b, x)
Dim k As Integer, i As Integer, j As Integer
Dim sum As Single, factor As Single
For k = 1 To n - 1
For i = k + 1 To n
factor = a(o(i), k) b(o(i)) = b(o(i)) - factor * b(o(k))
Next i
Next k
x(n) = b(o(n)) / a(o(n), n)
For i = n - 1 To 1 Step -1
sum = 0
For j = i + 1 To n
sum = sum + a(o(i), j) * x(j)
Next j
x(i) = (b(o(i)) - sum) / a(o(i), i)
Next i
End Sub
10.15
Option Explicit
Sub LUGaussTest()
Dim n As Integer, er As Integer, i As Integer, j As Integer
Dim a(3, 3) As Single, b(3) As Single, x(3) As Single
Dim tol As Single, ai(3, 3) As Single
n = 3
a(1, 1) = 3: a(1, 2) = -0.1: a(1, 3) = -0.2
a(2, 1) = 0.1: a(2, 2) = 7: a(2, 3) = -0.3
a(3, 1) = 0.3: a(3, 2) = -0.2: a(3, 3) = 10
tol = 0.000001
Call LUDminv(a(), b(), n, x(), tol, er, ai())
If er = 0 Then
Range("a1").Select
For i = 1 To n
For j = 1 To n
ActiveCell.Value = ai(i, j)
ActiveCell.Offset(0, 1).Select
Next j
ActiveCell.Offset(1, -n).Select
Next i
Range("a1").Select
Else
MsgBox "ill-conditioned system"
End If
End Sub
Sub LUDminv(a, b, n, x, tol, er, ai)
Dim i As Integer, j As Integer
Dim o(3) As Single, s(3) As Single
Call Decompose(a, n, tol, o(), s(), er)
If er = 0 Then
For i = 1 To n
For j = 1 To n
If i = j Then
b(j) = 1
Else
b(j) = 0
End If
Next j
Call Substitute(a, o, n, b, x)
For j = 1 To n
ai(j, i) = x(j)
Next j
Next i
End If
End Sub
Sub Decompose(a, n, tol, o, s, er)
Dim i As Integer, j As Integer, k As Integer Dim factor As Single
For i = 1 To n
o(i) = i
s(i) = Abs(a(i, 1))
For j = 2 To n
If Abs(a(i, j)) > s(i) Then s(i) = Abs(a(i, j))
Next j
Next i
For k = 1 To n - 1
Call Pivot(a, o, s, n, k)
If Abs(a(o(k), k) / s(o(k))) < tol Then
er = -1
Exit For
End If
For i = k + 1 To n
factor = a(o(i), k) / a(o(k), k)
a(o(i), k) = factor
For j = k + 1 To n
a(o(i), j) = a(o(i), j) - factor * a(o(k), j)
Next j
Next i
Next k
If (Abs(a(o(k), k) / s(o(k))) < tol) Then er = -1
End Sub
Sub Pivot(a, o, s, n, k)
Dim ii As Integer, p As Integer
Dim big As Single, dummy As Single
p = k
big = Abs(a(o(k), k) / s(o(k)))
For ii = k + 1 To n
dummy = Abs(a(o(ii), k) / s(o(ii)))
If dummy > big Then
big = dummy
p = ii
End If
Next ii
dummy = o(p)
o(p) = o(k)
o(k) = dummy
End Sub
Sub Substitute(a, o, n, b, x)
Dim k As Integer, i As Integer, j As Integer
Dim sum As Single, factor As Single
For k = 1 To n - 1
For i = k + 1 To n
factor = a(o(i), k)
b(o(i)) = b(o(i)) - factor * b(o(k))
Next i
Next k
x(n) = b(o(n)) / a(o(n), n)
For i = n - 1 To 1 Step -1
sum = 0
For j = i + 1 To n
sum = sum + a(o(i), j) * x(j)
Next j
x(i) = (b(o(i)) - sum) / a(o(i), i)
Next i
End Sub 10.17
A ⋅ B = 0 ⇒ − 4a + 2b = 3 (1)
A ⋅ C = 0 ⇒ 2a − 3c = −6 (2)
B ⋅ C = 2 ⇒ 3b + c = 10
( 3)
Solve the three equations using Matlab:
>> A=[-4 2 0; 2 0 –3; 0 3 1]
b=[3; -6; 10]
x=inv(A)*b
x = 0.525
2.550
2.350
Therefore, a = 0.525, b = 2.550, and c = 2.350.
10.18
i
( A × B) = a
−2
j k
b c = ( −4b − c)i − ( −4 a + 2c ) j + (a + 2b)k
1 −4
i
( A × C) = a
1
j
b
3
k
c = (2b − 3c )i − (2a − c ) j + (3a + b) k
2
( A × B ) + ( A × C ) = (−2b − 4c)i − ( −2a + c ) j + ( 4a + b)k Therefore,
(−2b − 4c)i + (−2a − c) j + (4a + b)r = (5a + 6)i + (3b − 2) j + (−4c + 1) k
We get the following set of equations ⇒
− 2b − 4c = 5a + 6
2a − c = 3b − 2
4a + b = −4c + 1
⇒ − 5a − 2b − 4c = 6
⇒ 2a − 3b − c = −2
⇒ 4 a + b − 4c = 1
(1)
(2)
(3)
In Matlab:
A=[-5 -5 -4 ; 2 -3 -1 ; 4 1 -4]
B=[ 6 ; -2 ; 1] ; x = inv (A) * b
Answer ⇒ x
= -3.6522
-3.3478
4.7391
a = -3.6522, b = -3.3478, c = 4.7391
10.19
(I)
f ( 0) = 1 ⇒ a ( 0) + b = 1 ⇒ b = 1
f ( 2) = 1 ⇒ c ( 2) + d = 1 ⇒ 2c + d = 1
(II) If f is continuous, then at x = 1
ax + b = cx + d ⇒ a(1) + b = c (1) + d ⇒ a + b − c − d = 0
(III)
a+b = 4
0 1 0 0
0 0 2 2
1 1 − 1 − 1
1 1 0 0
Solve using Matlab ⇒
a
1
b
= 1
c
0
d
4
a=3
b=1
c = -3
d=7
10.20 MATLAB provides a handy way to solve this problem. (a)
>> a=hilb(3)
a =
1.0000
0.5000
0.3333
0.5000
0.3333
0.2500
0.3333
0.2500
0.2000
>> b=[1 1 1]'
b =
1
1
1
>> c=a*b
c =
1.8333
1.0833
0.7833
>> format long e
>> x=a\b
>> x =
9.999999999999991e-001
1.000000000000007e+000
9.999999999999926e-001
(b)
>> a=hilb(7);
>> b=[1 1 1 1 1 1 1]';
>> c=a*b;
>> x=a\b
x =
9.999999999914417e-001
1.000000000344746e+000
9.999999966568566e-001
1.000000013060454e+000
9.999999759661609e-001
1.000000020830062e+000
9.999999931438059e-001
(c)
>>
>>
>>
>>
x =
a=hilb(10);
b=[1 1 1 1 1 1 1 1 1 1]';
c=a*b;
x=a\b
9.999999987546324e-001
1.000000107466305e+000
9.999977129981819e-001
1.000020777695979e+000
9.999009454847158e-001
1.000272183037448e+000
9.995535966572223e-001
1.000431255894815e+000
9.997736605804316e-001
1.000049762292970e+000
Chapter 10 Matrix Multiplication is Distribution
of 12
Report
Tell us what’s wrong with it:
Thanks, got it!
We will moderate it soon!
Free up your schedule!
Our EduBirdie Experts Are Here for You 24/7! Just fill out a form and let us know how we can assist you.
Take 5 seconds to unlock
Enter your email below and get instant access to your document