UTG - Computer Math II

Ch1 Vector
- Ch1 Part I Vector
  - CPE 332Computer Engineering Mathematics II
  - Web Site
  - Today Topics
  - Slide Number 4
  - CPE 332 T1-57 Wk1
  - Definition of Vector
  - Definition of Vector
  - Slide Number 8
  - Slide Number 9
  - Notes
  - Vector Operations
  - Addition and Substraction
  - Slide Number 13
  - Slide Number 14
  - การประยุกต์ใช้ใน Plane Geometry
  - Slide Number 16
  - Slide Number 17
  - Slide Number 18
  - Component Vector
  - Component Vector in Cartesian Coordinate
  - Component Vector in Cartesian Coordinate
  - Component Vector in Cartesian Coordinate
  - Position Vector
  - Position Vector และAddition-Subtraction using Component Vector
  - สรุป
  - Slide Number 26
  - Any vectors in Cartesian Coordinates
  - Any vectors in Cartesian Coordinates
  - Direction Cosine/Ratio
  - Direction Cosine
  - Direction Cosine and Direction Ratio
  - Direction Cosine and Direction Ratio
  - Example
  - Direction Cosine and Direction Ratio
  - Next Week
- CPE 332Computer Engineering Mathematics II
- Web Site
- Today Topics
- Slide Number 4
- CPE 332 T1-57 Wk1
- Definition of Vector
- Definition of Vector
- Slide Number 8
- Slide Number 9
- Notes
- Vector Operations
- Addition and Substraction
- Slide Number 13
- Slide Number 14
- การประยุกต์ใช้ใน Plane Geometry
- Slide Number 16
- Slide Number 17
- Slide Number 18
- Component Vector
- Component Vector in Cartesian Coordinate
- Component Vector in Cartesian Coordinate
- Component Vector in Cartesian Coordinate
- Position Vector
- Position Vector และAddition-Subtraction using Component Vector
- สรุป
- Slide Number 26
- Any vectors in Cartesian Coordinates
- Any vectors in Cartesian Coordinates
- Direction Cosine/Ratio
- Direction Cosine
- Direction Cosine and Direction Ratio
- Direction Cosine and Direction Ratio
- Example
- Direction Cosine and Direction Ratio
- Next Week
- Ch1 Part II Vector Operations
  - CPE 332Computer Engineering Mathematics II
  - Today Topics
  - Component Vector
  - Component Vector in Cartesian Coordinate
  - Component Vector in Cartesian Coordinate
  - Position Vector
  - Position Vector และAddition-Subtraction using Component Vector
  - Any vectors in Cartesian Coordinates
  - Direction Cosine/Ratio
  - Direction Cosine
  - Products of Vectors
  - Scalar Product(DOT)
  - Scalar Product (DOT)
  - Scalar(Dot) Product
  - Scalar(Dot) Product
  - DOT Product
  - Example
  - Vector Product (Cross)
  - Cross Product
  - Slide Number 20
  - 3 Vector Products
  - Examples
  - Plane Equation in 3D
  - Example 1
  - Slide Number 25
  - Example 2
  - Slide Number 27
  - Definition of Matrix
  - Row Matrix, Column Matrix
  - Basic Operations
  - Matrix Multiplication
  - Square Matrix
  - Matrix Transpose
  - Types of Matrix
  - Types of Matrix
  - Types of Matrix
  - Types of Matrix
  - Types of Matrix
  - Matrix Inverse
  - Orthogonal/Unitary Matrix
  - Orthogonal Vector
  - Orthogonal Vector
  - Examples
  - Examples
  - Examples
  - Examples
  - Examples
  - Examples
  - Determinant
  - Determinant of Matrix
  - Sign Matrix
  - Minor aij
  - Cofactor aij
  - Determinant of 2x2 and 3x3
  - Calculation of Determinant using Recursive Expansion
  - Calculation of Determinant using Recursive Expansion
  - ความหมายของ Determinant
  - คุณสมบัติของ Determinant
  - คุณสมบัติของ Determinant
  - Calculation of Determinant (Algorithm)
  - Algorithm การหา Determinant ที่มีประสิทธิภาพ
  - Algorithm การหา Determinant ที่มีประสิทธิภาพ(ต่อ)
  - การหา Determinant
  - Slide Number 64
  - การหา Determinant
  - Inverse of Matrix
  - Inverse of Matrix
  - Inverse of Matrix
  - Inverse of Matrix
  - Inverse of Matrix
  - Matrix Inverse
  - Matrix Norms
  - Norms
  - Homework 1:Vector
- CPE 332Computer Engineering Mathematics II
- Today Topics
- Component Vector
- Component Vector in Cartesian Coordinate
- Component Vector in Cartesian Coordinate
- Position Vector
- Position Vector และAddition-Subtraction using Component Vector
- Any vectors in Cartesian Coordinates
- Direction Cosine/Ratio
- Direction Cosine
- Products of Vectors
- Scalar Product(DOT)
- Scalar Product (DOT)
- Scalar(Dot) Product
- Scalar(Dot) Product
- DOT Product
- Example
- Vector Product (Cross)
- Cross Product
- Slide Number 20
- 3 Vector Products
- Examples
- Plane Equation in 3D
- Example 1
- Slide Number 25
- Example 2
- Slide Number 27
- Definition of Matrix
- Row Matrix, Column Matrix
- Basic Operations
- Matrix Multiplication
- Square Matrix
- Matrix Transpose
- Types of Matrix
- Types of Matrix
- Types of Matrix
- Types of Matrix
- Types of Matrix
- Matrix Inverse
- Orthogonal/Unitary Matrix
- Orthogonal Vector
- Orthogonal Vector
- Examples
- Examples
- Examples
- Examples
- Examples
- Examples
- Determinant
- Determinant of Matrix
- Sign Matrix
- Minor aij
- Cofactor aij
- Determinant of 2x2 and 3x3
- Calculation of Determinant using Recursive Expansion
- Calculation of Determinant using Recursive Expansion
- ความหมายของ Determinant
- คุณสมบัติของ Determinant
- คุณสมบัติของ Determinant
- Calculation of Determinant (Algorithm)
- Algorithm การหา Determinant ที่มีประสิทธิภาพ
- Algorithm การหา Determinant ที่มีประสิทธิภาพ(ต่อ)
- การหา Determinant
- Slide Number 64
- การหา Determinant
- Inverse of Matrix
- Inverse of Matrix
- Inverse of Matrix
- Inverse of Matrix
- Inverse of Matrix
- Matrix Inverse
- Matrix Norms
- Norms
- Homework 1:Vector
Ch1 Part I Vector
- CPE 332Computer Engineering Mathematics II
- Web Site
- Today Topics
- Slide Number 4
- CPE 332 T1-57 Wk1
- Definition of Vector
- Definition of Vector
- Slide Number 8
- Slide Number 9
- Notes
- Vector Operations
- Addition and Substraction
- Slide Number 13
- Slide Number 14
- การประยุกต์ใช้ใน Plane Geometry
- Slide Number 16
- Slide Number 17
- Slide Number 18
- Component Vector
- Component Vector in Cartesian Coordinate
- Component Vector in Cartesian Coordinate
- Component Vector in Cartesian Coordinate
- Position Vector
- Position Vector และAddition-Subtraction using Component Vector
- สรุป
- Slide Number 26
- Any vectors in Cartesian Coordinates
- Any vectors in Cartesian Coordinates
- Direction Cosine/Ratio
- Direction Cosine
- Direction Cosine and Direction Ratio
- Direction Cosine and Direction Ratio
- Example
- Direction Cosine and Direction Ratio
- Next Week
CPE 332Computer Engineering Mathematics II
Web Site
Today Topics
Slide Number 4
CPE 332 T1-57 Wk1
Definition of Vector
Definition of Vector
Slide Number 8
Slide Number 9
Notes
Vector Operations
Addition and Substraction
Slide Number 13
Slide Number 14
การประยุกต์ใช้ใน Plane Geometry
Slide Number 16
Slide Number 17
Slide Number 18
Component Vector
Component Vector in Cartesian Coordinate
Component Vector in Cartesian Coordinate
Component Vector in Cartesian Coordinate
Position Vector
Position Vector และAddition-Subtraction using Component Vector
สรุป
Slide Number 26
Any vectors in Cartesian Coordinates
Any vectors in Cartesian Coordinates
Direction Cosine/Ratio
Direction Cosine
Direction Cosine and Direction Ratio
Direction Cosine and Direction Ratio
Example
Direction Cosine and Direction Ratio
Next Week
Ch1 Part II Vector Operations
- CPE 332Computer Engineering Mathematics II
- Today Topics
- Component Vector
- Component Vector in Cartesian Coordinate
- Component Vector in Cartesian Coordinate
- Position Vector
- Position Vector และAddition-Subtraction using Component Vector
- Any vectors in Cartesian Coordinates
- Direction Cosine/Ratio
- Direction Cosine
- Products of Vectors
- Scalar Product(DOT)
- Scalar Product (DOT)
- Scalar(Dot) Product
- Scalar(Dot) Product
- DOT Product
- Example
- Vector Product (Cross)
- Cross Product
- Slide Number 20
- 3 Vector Products
- Examples
- Plane Equation in 3D
- Example 1
- Slide Number 25
- Example 2
- Slide Number 27
- Definition of Matrix
- Row Matrix, Column Matrix
- Basic Operations
- Matrix Multiplication
- Square Matrix
- Matrix Transpose
- Types of Matrix
- Types of Matrix
- Types of Matrix
- Types of Matrix
- Types of Matrix
- Matrix Inverse
- Orthogonal/Unitary Matrix
- Orthogonal Vector
- Orthogonal Vector
- Examples
- Examples
- Examples
- Examples
- Examples
- Examples
- Determinant
- Determinant of Matrix
- Sign Matrix
- Minor aij
- Cofactor aij
- Determinant of 2x2 and 3x3
- Calculation of Determinant using Recursive Expansion
- Calculation of Determinant using Recursive Expansion
- ความหมายของ Determinant
- คุณสมบัติของ Determinant
- คุณสมบัติของ Determinant
- Calculation of Determinant (Algorithm)
- Algorithm การหา Determinant ที่มีประสิทธิภาพ
- Algorithm การหา Determinant ที่มีประสิทธิภาพ(ต่อ)
- การหา Determinant
- Slide Number 64
- การหา Determinant
- Inverse of Matrix
- Inverse of Matrix
- Inverse of Matrix
- Inverse of Matrix
- Inverse of Matrix
- Matrix Inverse
- Matrix Norms
- Norms
- Homework 1:Vector
CPE 332Computer Engineering Mathematics II
Today Topics
Component Vector
Component Vector in Cartesian Coordinate
Component Vector in Cartesian Coordinate
Position Vector
Position Vector และAddition-Subtraction using Component Vector
Any vectors in Cartesian Coordinates
Direction Cosine/Ratio
Direction Cosine
Products of Vectors
Scalar Product(DOT)
Scalar Product (DOT)
Scalar(Dot) Product
Scalar(Dot) Product
DOT Product
Example
Vector Product (Cross)
Cross Product
Slide Number 20
3 Vector Products
Examples
Plane Equation in 3D
Example 1
Slide Number 25
Example 2
Slide Number 27
Definition of Matrix
Row Matrix, Column Matrix
Basic Operations
Matrix Multiplication
Square Matrix
Matrix Transpose
Types of Matrix
Types of Matrix
Types of Matrix
Types of Matrix
Types of Matrix
Matrix Inverse
Orthogonal/Unitary Matrix
Orthogonal Vector
Orthogonal Vector
Examples
Examples
Examples
Examples
Examples
Examples
Determinant
Determinant of Matrix
Sign Matrix
Minor aij
Cofactor aij
Determinant of 2x2 and 3x3
Calculation of Determinant using Recursive Expansion
Calculation of Determinant using Recursive Expansion
ความหมายของ Determinant
คุณสมบัติของ Determinant
คุณสมบัติของ Determinant
Calculation of Determinant (Algorithm)
Algorithm การหา Determinant ที่มีประสิทธิภาพ
Algorithm การหา Determinant ที่มีประสิทธิภาพ(ต่อ)
การหา Determinant
Slide Number 64
การหา Determinant
Inverse of Matrix
Inverse of Matrix
Inverse of Matrix
Inverse of Matrix
Inverse of Matrix
Matrix Inverse
Matrix Norms
Norms
Homework 1:Vector
Ch2 Matrices and Linear EQ
- CPE 332Computer Engineering Mathematics II
- Today Topics
- Chapter 2: Cont
- Determinant
- Calculation of Determinant
- คุณสมบัติของ Determinant
- คุณสมบัติของ Determinant
- Calculation of Determinant
- Algorithm การหา Determinant ที่มีประสิทธิภาพ
- Algorithm การหา Determinant ที่มีประสิทธิภาพ(ต่อ)
- การหา Determinant
- Slide Number 12
- การหา Determinant
- Inverse of Matrix
- Inverse of Matrix
- Inverse of Matrix
- Inverse of Matrix
- Inverse of Matrix
- สาธิตทดลอง Row Expansion
- Matrix Norms
- Norms
- Break
- System of Linear Equations
- System of Linear Equations
- Reduced Matrix
- Reduced Matrix
- Reduced Matrix
- Solutions of Homogeneous
- Solutions of Homogeneous
- Solutions of Homogeneous
- Non-homogeneous Systems
- Non-homogeneous Systems
- Non-homogeneous Systems
- Non-homogeneous Systems
- Non-homogeneous Systems
- Non-homogeneous Systems
- Slide Number 37
- Slide Number 38
- Chapter 3 Intro
- Slide Number 40
- Slide Number 41
- Slide Number 42
- MATLAB TUTORIAL
- End of Week 3
CPE 332Computer Engineering Mathematics II
Today Topics
Chapter 2: Cont
Determinant
Calculation of Determinant
คุณสมบัติของ Determinant
คุณสมบัติของ Determinant
Calculation of Determinant
Algorithm การหา Determinant ที่มีประสิทธิภาพ
Algorithm การหา Determinant ที่มีประสิทธิภาพ(ต่อ)
การหา Determinant
Slide Number 12
การหา Determinant
Inverse of Matrix
Inverse of Matrix
Inverse of Matrix
Inverse of Matrix
Inverse of Matrix
สาธิตทดลอง Row Expansion
Matrix Norms
Norms
Break
System of Linear Equations
System of Linear Equations
Reduced Matrix
Reduced Matrix
Reduced Matrix
Solutions of Homogeneous
Solutions of Homogeneous
Solutions of Homogeneous
Non-homogeneous Systems
Non-homogeneous Systems
Non-homogeneous Systems
Non-homogeneous Systems
Non-homogeneous Systems
Non-homogeneous Systems
Slide Number 37
Slide Number 38
Chapter 3 Intro
Slide Number 40
Slide Number 41
Slide Number 42
MATLAB TUTORIAL
End of Week 3
Ch3 Eigenvector-Prob
- CPE 332Computer Engineering Mathematics II
- Today Topics
- Determinant
- Slide Number 4
- Chapter 3 Intro
- Slide Number 6
- Slide Number 7
- Slide Number 8
- Slide Number 9
- Slide Number 10
- Slide Number 11
- Slide Number 12
- Slide Number 13
- Slide Number 14
- Diagonalization
- Slide Number 16
- Slide Number 17
- Slide Number 18
- Slide Number 19
- Slide Number 20
- Slide Number 21
- Slide Number 22
- Orthogonal and Symmetric
- Slide Number 24
- Slide Number 25
- Break
- Slide Number 27
- Slide Number 28
- Slide Number 29
- Slide Number 30
- Slide Number 31
- Slide Number 32
- End of Chapter 3
- Chapter 4 Probability
- Definition
- Example: Dice Roll
- Mutually Exclusive
- Mutually Exclusive
- 3 AXIOMS OF Probability
- Conditional Probability
- Bayes Rule
- Properties
- Example 1
- Example 2
- Example3
- Random Variables
- Random Variables
- CDF:Cumulative Distribution Function of RV X
- CDF Properties
- CDF การทอยเหรียญ: Discrete RV
- CDF การทอยลูกเต๋า: Discrete RV
- PDF: Probability Density Function
- Properties of PDF
- Discrete Version
- Slide Number 55
- Chapter 4 Cont. Next week
CPE 332Computer Engineering Mathematics II
Today Topics
Determinant
Slide Number 4
Chapter 3 Intro
Slide Number 6
Slide Number 7
Slide Number 8
Slide Number 9
Slide Number 10
Slide Number 11
Slide Number 12
Slide Number 13
Slide Number 14
Diagonalization
Slide Number 16
Slide Number 17
Slide Number 18
Slide Number 19
Slide Number 20
Slide Number 21
Slide Number 22
Orthogonal and Symmetric
Slide Number 24
Slide Number 25
Break
Slide Number 27
Slide Number 28
Slide Number 29
Slide Number 30
Slide Number 31
Slide Number 32
End of Chapter 3
Chapter 4 Probability
Definition
Example: Dice Roll
Mutually Exclusive
Mutually Exclusive
3 AXIOMS OF Probability
Conditional Probability
Bayes Rule
Properties
Example 1
Example 2
Example3
Random Variables
Random Variables
CDF:Cumulative Distribution Function of RV X
CDF Properties
CDF การทอยเหรียญ: Discrete RV
CDF การทอยลูกเต๋า: Discrete RV
PDF: Probability Density Function
Properties of PDF
Discrete Version
Slide Number 55
Chapter 4 Cont. Next week
Ch4 Probability
- CPE 332Computer Engineering Mathematics II
- Today Topics
- Random Variables
- CDF:Cumulative Distribution Function of RV X
- CDF Properties
- CDF การทอยเหรียญ: Discrete RV
- PDF: Probability Density Function
- Properties of PDF
- Discrete Version
- Slide Number 10
- Statistical Average
- Importance Expectation
- Slide Number 13
- Multivariate: RV มากกว่า 1 ตัว
- Multivariate: RV มากกว่า 1 ตัว
- Multivariate: RV มากกว่า 1 ตัว
- Joint CDF/Marginal Density
- Joint Moment
- Correlation
- Covarience
- Correlation and Covarience
- การประมาณค่า Rxy และ Cxy จาก Samples
- EX.จงหา Rxy และ Cxy จากตัวอย่างในตารางข้างล่าง
- Scatter Diagram
- Calculation Table
- PDF ที่สำคัญ
- Binomial Distribution
- Geometric Distribution
- Uniform Distribution
- Gaussian Distribution
- Gaussian
- Jointly Gaussian: X, Y
- Slide Number 33
- Slide Number 34
- Slide Number 35
- Slide Number 36
- Slide Number 37
- Exponential Distribution
- Exponential Distribution
- Poisson Distribution
- Lambda = 1
- Lambda = 3
- Lambda = 5
- Lambda = 8
- Lambda = 12
- Lambda = 18
- Break
- Random Process
- Random Process (Stochastic Process)
- Statistical Average
- Slide Number 51
- Sample Function: Ensemble Average and Time Average
- Sample Function and Ensemble
- Statistical Average: Correlation/Covariance
- Stationary RP
- Stationary RP
- WIDE SENSE STATIONARY
- Autocorrelation(WSS)
- Ergodic Random Process
- Discrete-Time Random Process
- Discrete-Time Random Process (Limited Samples)
- Ergodic Discrete RP
- Ergodic Discrete RP
- กรณีที่จำกัด Sequence ความยาว N
- จำกัด Sequence ความยาว N
- ความหมายของ x(n)x(n+m)
- ความหมายของ x(n)x(n+m), N Points
- จำกัด Sequence ความยาว N
- Sequence ความยาว N / RAW
- Sequence ทั่วไป
- Example: Rxx
- Example: Rxx
- Example: Rxx
- Example: Rxx
- Example: Rxx
- Example: Rxx
- Example: Rxx
- Example: Rxx
- Rxx(m)
- Example: Rxy
- Example: Rxy
- Example: Rxy
- Example: Rxy
- Example: Rxy
- Example: Rxy
- Example: Rxy
- Rxy(m)
- End of Week 5
CPE 332Computer Engineering Mathematics II
Today Topics
Random Variables
CDF:Cumulative Distribution Function of RV X
CDF Properties
CDF การทอยเหรียญ: Discrete RV
PDF: Probability Density Function
Properties of PDF
Discrete Version
Slide Number 10
Statistical Average
Importance Expectation
Slide Number 13
Multivariate: RV มากกว่า 1 ตัว
Multivariate: RV มากกว่า 1 ตัว
Multivariate: RV มากกว่า 1 ตัว
Joint CDF/Marginal Density
Joint Moment
Correlation
Covarience
Correlation and Covarience
การประมาณค่า Rxy และ Cxy จาก Samples
EX.จงหา Rxy และ Cxy จากตัวอย่างในตารางข้างล่าง
Scatter Diagram
Calculation Table
PDF ที่สำคัญ
Binomial Distribution
Geometric Distribution
Uniform Distribution
Gaussian Distribution
Gaussian
Jointly Gaussian: X, Y
Slide Number 33
Slide Number 34
Slide Number 35
Slide Number 36
Slide Number 37
Exponential Distribution
Exponential Distribution
Poisson Distribution
Lambda = 1
Lambda = 3
Lambda = 5
Lambda = 8
Lambda = 12
Lambda = 18
Break
Random Process
Random Process (Stochastic Process)
Statistical Average
Slide Number 51
Sample Function: Ensemble Average and Time Average
Sample Function and Ensemble
Statistical Average: Correlation/Covariance
Stationary RP
Stationary RP
WIDE SENSE STATIONARY
Autocorrelation(WSS)
Ergodic Random Process
Discrete-Time Random Process
Discrete-Time Random Process (Limited Samples)
Ergodic Discrete RP
Ergodic Discrete RP
กรณีที่จำกัด Sequence ความยาว N
จำกัด Sequence ความยาว N
ความหมายของ x(n)x(n+m)
ความหมายของ x(n)x(n+m), N Points
จำกัด Sequence ความยาว N
Sequence ความยาว N / RAW
Sequence ทั่วไป
Example: Rxx
Example: Rxx
Example: Rxx
Example: Rxx
Example: Rxx
Example: Rxx
Example: Rxx
Example: Rxx
Rxx(m)
Example: Rxy
Example: Rxy
Example: Rxy
Example: Rxy
Example: Rxy
Example: Rxy
Example: Rxy
Rxy(m)
End of Week 5
Ch5 MarKov Process
- CPE 332Computer Engineering Mathematics II
- Today Topics
- Discrete-Time Random Process
- จำกัด Sequence ความยาว N
- Sequence ทั่วไป
- การคำนวณ Correlation
- การคำนวณ Correlation
- Random Process กรณีพิเศษ
- Counting Process
- Poisson Process
- Poisson Process
- Poisson Process
- Birth and Death Process
- State Diagram
- MarKov Process and Markov Chain
- Markov Process and Markov Chain
- MarKov Chain
- Discrete Time Markov Chain
- Discrete Time Markov Chain
- Discrete Time Markov Chain
- Notations: สำหรับ Discrete Time MarKov Chain
- Discrete Time Markov Chain
- Discrete Time Markov Chain
- Discrete Time Markov Chain
- Discrete Time Markov Chain
- Discrete Time Markov Chain
- สรุป MarKov Chain
- Detailed Balance Equation: Simple MarKov Chain
- Markov Chain(Detailed Bal Eq)
- Slide Number 30
- Example
- Example
- Example
- Example
- Example
- Example
- Example
- Example
- Example
- Example
- Example
- Example
- Example
- Example
- Example: Simple MarKov
- Example: Simple MarKov
- Example: Simple MarKov
- Example: Simple MarKov
- Example: Simple MarKov
- End of Chapter 5
- CPE 332Computer Engineering Mathematics II
- Topics
- System
- Queuing System
- Queuing System
- Queuing System
- Queuing SystemSimple MarKov Model
- Queuing SystemSimple MarKov Model
- Queuing SystemSimple MarKov Model
- Queuing SystemSimple MarKov Model
- Queuing System Case 1: Unlimited Server; No Queue
- Queuing System Case 1: Unlimited Server; No Queue
- Queuing System Case 2: Lost SystemLimited Server=N; No Queue
- Queuing System Case 3: Delay SystemLimited Server=N; With Unlimited Queue
- Queuing System Case 3: Delay SystemServer=1; With Unlimited Queue; M/M/1
- Queuing System: Model
- M/M/1: Summary
- Queuing Model(1 Server); M/M/1
- การทำงานของ M/M/1
- เปรียบเทียบ Queuing Model (N Server); M/M/N
- M/M/N
- Network Model using M/M/1
- Network Model (M/M/1)
- Network Model (M/M/1)
- Network Model (M/M/1)
- Kendal Notation
- Kendal Notation
- Kendal Notation
- Next Week
CPE 332Computer Engineering Mathematics II
Today Topics
Discrete-Time Random Process
จำกัด Sequence ความยาว N
Sequence ทั่วไป
การคำนวณ Correlation
การคำนวณ Correlation
Random Process กรณีพิเศษ
Counting Process
Poisson Process
Poisson Process
Poisson Process
Birth and Death Process
State Diagram
MarKov Process and Markov Chain
Markov Process and Markov Chain
MarKov Chain
Discrete Time Markov Chain
Discrete Time Markov Chain
Discrete Time Markov Chain
Notations: สำหรับ Discrete Time MarKov Chain
Discrete Time Markov Chain
Discrete Time Markov Chain
Discrete Time Markov Chain
Discrete Time Markov Chain
Discrete Time Markov Chain
สรุป MarKov Chain
Detailed Balance Equation: Simple MarKov Chain
Markov Chain(Detailed Bal Eq)
Slide Number 30
Example
Example
Example
Example
Example
Example
Example
Example
Example
Example
Example
Example
Example
Example
Example: Simple MarKov
Example: Simple MarKov
Example: Simple MarKov
Example: Simple MarKov
Example: Simple MarKov
End of Chapter 5
CPE 332Computer Engineering Mathematics II
Topics
System
Queuing System
Queuing System
Queuing System
Queuing SystemSimple MarKov Model
Queuing SystemSimple MarKov Model
Queuing SystemSimple MarKov Model
Queuing SystemSimple MarKov Model
Queuing System Case 1: Unlimited Server; No Queue
Queuing System Case 1: Unlimited Server; No Queue
Queuing System Case 2: Lost SystemLimited Server=N; No Queue
Queuing System Case 3: Delay SystemLimited Server=N; With Unlimited Queue
Queuing System Case 3: Delay SystemServer=1; With Unlimited Queue; M/M/1
Queuing System: Model
M/M/1: Summary
Queuing Model(1 Server); M/M/1
การทำงานของ M/M/1
เปรียบเทียบ Queuing Model (N Server); M/M/N
M/M/N
Network Model using M/M/1
Network Model (M/M/1)
Network Model (M/M/1)
Network Model (M/M/1)
Kendal Notation
Kendal Notation
Kendal Notation
Next Week
Ch6 Queuing System
- CPE 332Computer Engineering Mathematics II
- Topics
- Queuing System Case 3: Delay SystemLimited Server=N; With Unlimited Queue
- Queuing System Case 3: Delay SystemServer=1; With Unlimited Queue; M/M/1
- M/M/1: Summary
- Queuing Model(1 Server); M/M/1
- การทำงานของ M/M/1
- Kendal Notation
- Kendal Notation
- Kendal Notation
- Analysis ของ M/M/1
- Queuing in Communication NW and M/M/1
- Example
- Assumption
- Utilization
- Arrival Rate
- Service Time
- Queue Distribution
- ค่าเฉลี่ย Customer ในระบบ, N คือค่าเฉลี่ยของ State, 𝐸[𝑋]
- Queuing Delay
- Queuing Delay
- System Delay
- Little’s Theorem
- สรุป M/M/1
- สรุป M/M/1
- สรุป M/M/1
- M/M/1 Example 1
- M/M/1 Example 1
- M/M/1 Example 1
- M/M/1 Example 1
- M/M/1 Example 1
- M/M/1 Example 2
- M/M/1 Example 2
- 4.ธนาคารต้องจัดที่นั่งกี่ที่(รวมที่นั่งตอนใช้บริการ)เพื่อจะให้แน่ใจว่าอย่างน้อย 80% ของผู้ที่เข้ามาจะได้นั่ง
- 4.ธนาคารต้องจัดที่นั่งกี่ที่(รวมที่นั่งตอนใช้บริการ)เพื่อจะให้แน่ใจว่าอย่างน้อย 80% ของผู้ที่เข้ามาจะได้นั่ง
- M/M/1 Example 3
- M/M/1 Example 3
- M/M/1 Example 3
- End of Chapter 6
- Topics
- Random Number
- การใช้งาน Random Number
- Random Number Generation
- PRNG: Pseudo Random Number Generator: Linear Congruential Generator
- Period of LCG
- Parameters used
- คุณสมบัติ LCG
- Slide Number 48
- PRNG อื่นๆ เช่น
- Randomness Test
- Midterm Preparation
- Slide Number 52
- Review หัวข้อออกสอบ
- Review หัวข้อออกสอบ
- Review หัวข้อออกสอบ
CPE 332Computer Engineering Mathematics II
Topics
Queuing System Case 3: Delay SystemLimited Server=N; With Unlimited Queue
Queuing System Case 3: Delay SystemServer=1; With Unlimited Queue; M/M/1
M/M/1: Summary
Queuing Model(1 Server); M/M/1
การทำงานของ M/M/1
Kendal Notation
Kendal Notation
Kendal Notation
Analysis ของ M/M/1
Queuing in Communication NW and M/M/1
Example
Assumption
Utilization
Arrival Rate
Service Time
Queue Distribution
ค่าเฉลี่ย Customer ในระบบ, N คือค่าเฉลี่ยของ State, 𝐸[𝑋]
Queuing Delay
Queuing Delay
System Delay
Little’s Theorem
สรุป M/M/1
สรุป M/M/1
สรุป M/M/1
M/M/1 Example 1
M/M/1 Example 1
M/M/1 Example 1
M/M/1 Example 1
M/M/1 Example 1
M/M/1 Example 2
M/M/1 Example 2
4.ธนาคารต้องจัดที่นั่งกี่ที่(รวมที่นั่งตอนใช้บริการ)เพื่อจะให้แน่ใจว่าอย่างน้อย 80% ของผู้ที่เข้ามาจะได้นั่ง
4.ธนาคารต้องจัดที่นั่งกี่ที่(รวมที่นั่งตอนใช้บริการ)เพื่อจะให้แน่ใจว่าอย่างน้อย 80% ของผู้ที่เข้ามาจะได้นั่ง
M/M/1 Example 3
M/M/1 Example 3
M/M/1 Example 3
End of Chapter 6
Topics
Random Number
การใช้งาน Random Number
Random Number Generation
PRNG: Pseudo Random Number Generator: Linear Congruential Generator
Period of LCG
Parameters used
คุณสมบัติ LCG
Slide Number 48
PRNG อื่นๆ เช่น
Randomness Test
Midterm Preparation
Slide Number 52
Review หัวข้อออกสอบ
Review หัวข้อออกสอบ
Review หัวข้อออกสอบ
Ch7 Function Approximation and Error
- CPE 332Computer Engineering Mathematics II
- Today Topics
- Numerical Methods
- Numerical Methods
- Y=f(x)=2x3-4x2-7x+5
- Slide Number 6
- Quadratic Equation
- Iterative Technique
- Errors
- Errors
- Errors
- Precision, Accuracy,Significant Digit
- การยิงเป้า
- Significant Digits (Figures)
- Error
- Absolute Error
- Relative Error
- Error Estimation
- Mean Square Error(MSE)
- Functions
- Polynomial Functions
- Slide Number 22
- Transcendental Function
- Transcendental Examples
- การประมาณค่าของ Function
- Series และ Infinite Series
- Convergence of Series
- Power Series
- Taylor Series/Maclaurin Series
- Slide Number 30
- Slide Number 31
- Example 1
- Example 2:
- Taylor (Maclaurin) Series
- MATLAB Program
- END OF WEEK 10
- MATLAB Program
- MATLAB Program
- MATLAB Program
CPE 332Computer Engineering Mathematics II
Today Topics
Numerical Methods
Numerical Methods
Y=f(x)=2x3-4x2-7x+5
Slide Number 6
Quadratic Equation
Iterative Technique
Errors
Errors
Errors
Precision, Accuracy,Significant Digit
การยิงเป้า
Significant Digits (Figures)
Error
Absolute Error
Relative Error
Error Estimation
Mean Square Error(MSE)
Functions
Polynomial Functions
Slide Number 22
Transcendental Function
Transcendental Examples
การประมาณค่าของ Function
Series และ Infinite Series
Convergence of Series
Power Series
Taylor Series/Maclaurin Series
Slide Number 30
Slide Number 31
Example 1
Example 2:
Taylor (Maclaurin) Series
MATLAB Program
END OF WEEK 10
MATLAB Program
MATLAB Program
MATLAB Program
Ch8 Roots of Equations
- CPE 332Computer Engineering Mathematics II
- Today Topics
- Roots of Equations
- Zeroes of Functions
- Zeroes of Functions
- Zeroes of Functions y=x3-2x2+x-5
- Zeroes of Functions y=x3-2x2+x-5
- Roots of Equation
- Methods
- Bracketing Method
- Bracketing Method
- Bracketing Method
- Bracketing Method:Bisection
- Bracketing Method
- Bracketing Method
- Bracketing Method
- Bracketing Method
- Bracketing Method
- Bracketing Method:Bisection
- Bracketing Method:Bisection
- Bracketing Method:Bisection
- Slide Number 22
- Slide Number 23
- Slide Number 24
- Bracketing Method: Bisection
- False-Position Method
- False-Position Method
- False-Position Method
- False-Position Method
- False-Position Method
- False-Position Method
- Open Method: Simple One-Point Iteration
- Open Method: Simple One-Point Iteration
- Open Method: Simple One-Point Iteration
- Open Method: Simple One-Point Iteration
- Slide Number 36
- Open Method: Simple One-Point Iteration
- Open Method: Newton-Ralphson Method
- Open Method: Newton-Ralphson Method
- Open Method: Newton-Ralphson Method
- Open Method: Newton-Ralphson Method
- Open Method: Newton-Ralphson Method
- Open Method: Secant Method
- Open Method: Secant Method
- Open Method: Secant Method
- Slide Number 46
- Multiple Roots
- Multiple Roots
- Multiple Roots
- Multiple Roots
- Multiple Roots
- Multiple Roots
- Multiple Roots
- Comparison
- Comparison
- Comparison
- Homework 8
CPE 332Computer Engineering Mathematics II
Today Topics
Roots of Equations
Zeroes of Functions
Zeroes of Functions
Zeroes of Functions y=x3-2x2+x-5
Zeroes of Functions y=x3-2x2+x-5
Roots of Equation
Methods
Bracketing Method
Bracketing Method
Bracketing Method
Bracketing Method:Bisection
Bracketing Method
Bracketing Method
Bracketing Method
Bracketing Method
Bracketing Method
Bracketing Method:Bisection
Bracketing Method:Bisection
Bracketing Method:Bisection
Slide Number 22
Slide Number 23
Slide Number 24
Bracketing Method: Bisection
False-Position Method
False-Position Method
False-Position Method
False-Position Method
False-Position Method
False-Position Method
Open Method: Simple One-Point Iteration
Open Method: Simple One-Point Iteration
Open Method: Simple One-Point Iteration
Open Method: Simple One-Point Iteration
Slide Number 36
Open Method: Simple One-Point Iteration
Open Method: Newton-Ralphson Method
Open Method: Newton-Ralphson Method
Open Method: Newton-Ralphson Method
Open Method: Newton-Ralphson Method
Open Method: Newton-Ralphson Method
Open Method: Secant Method
Open Method: Secant Method
Open Method: Secant Method
Slide Number 46
Multiple Roots
Multiple Roots
Multiple Roots
Multiple Roots
Multiple Roots
Multiple Roots
Multiple Roots
Comparison
Comparison
Comparison
Homework 8
Ch9 Linear Equations
- CPE 332Computer Engineering Mathematics II
- Today(Week 13) Topics
- MATLAB Programming
- Ex: หารากของ
- เราจะหาคำตอบในช่วง [0, 2]
- MATLAB: Bisection Mtd.
- Bisection Results:>> example91a(0.01)
- Other Results: xt= 0.95774795776341
- MATLAB: False-Position
- FP Results:>> example91b(0.01)
- Other Results: xt= 0.95774795776341
- Newton-Ralphson Method
- MATLAB Program:
- Result: ea=0.01, xo=?
- เราจะหาคำตอบในช่วง [0, 2]
- Ex: หารากของ
- Result: x0=0.5, es = 0.01
- Result: x0=0.5, es = 0.000001
- Compare : xt =0.95774795776341
- Chapter 10: System of Linear Eq.
- System of Linear Equations
- Krammer’s Rule
- Solution ของ AX=C
- Solution by Elimination
- Gauss Elimination
- Gauss Elimination
- Gauss Elimination
- Gauss Elimination
- Gauss Elimination
- Gauss Elimination
- Gauss Elimination
- Back Substitution
- Back Substitution
- Gauss Elimination
- Gauss Elimination
- Gauss Elimination
- Gauss Elimination
- Gauss Elimination
- Gauss Elimination Alg
- Gauss Elimination Alg
- Gauss Elimination Alg
- Example 8.1
- Example 8.1
- Example 8.1
- ปัญหาของ Gauss Elimination
- ปัญหาของ Gauss Elimination
- ปัญหาของ Gauss Elimination
- ปัญหาของ Gauss Elimination
- Gauss-Jordan Method
- Gauss-Jordan Method
- Example 8.2
- Example 8.2
- Example 8.2
- Example 8.2
- Example 8.2
- Example 8.2
- การหา Matrix Inverse ด้วย GJ
- การหา Matrix Inverse ด้วย GJ
- การหา Matrix Inverse ด้วย GJ
- การหา Matrix Inverse ด้วย GJ
- การหา Matrix Inverse ด้วย GJ
- Example 8.3
- Example 8.3
- Example 8.3
- Iterative Method and Gauss-Seidel
- Gauss-Seidel
- Gauss-Seidel
- Gauss-Seidel
- Gauss-Seidel
- Gauss-Seidel
- Gauss-Seidel: Ex 8.4
- Gauss-Seidel: Ex 8.4
- Gauss-Seidel: Ex 8.4
- Gauss-Seidel: Ex 8.4
- Gauss-Seidel: Ex 8.4
- Gauss-Seidel: Ex 8.4
- Gauss-Seidel: Ex 8.4
- Jacobi Method
- Convergence of Iterative Method
- Break
- LU Decomposition
- LU Decomposition
- LU Decomposition
- LU Decomposition
- LU Decomposition
- LU Decomposition
- LU Decomposition
- LU Decomposition
- LU Decomposition
- LU Decomposition
- Crout Decomposition
- Crout Decomposition
- Crout Decomposition
- Crout Decomposition
- Crout Decomposition
- Crout Decomposition
- Crout Decomposition
- Example 8.6
- Crout Decomposition
- Crout Decomposition
- Example 8.7
- Example 8.7
- Slide Number 103
- Summary Chapter 8
- Homework 9, Chapter 10
- End of Chapter 10
CPE 332Computer Engineering Mathematics II
Today(Week 13) Topics
MATLAB Programming
Ex: หารากของ
เราจะหาคำตอบในช่วง [0, 2]
MATLAB: Bisection Mtd.
Bisection Results:>> example91a(0.01)
Other Results: xt= 0.95774795776341
MATLAB: False-Position
FP Results:>> example91b(0.01)
Other Results: xt= 0.95774795776341
Newton-Ralphson Method
MATLAB Program:
Result: ea=0.01, xo=?
เราจะหาคำตอบในช่วง [0, 2]
Ex: หารากของ
Result: x0=0.5, es = 0.01
Result: x0=0.5, es = 0.000001
Compare : xt =0.95774795776341
Chapter 10: System of Linear Eq.
System of Linear Equations
Krammer’s Rule
Solution ของ AX=C
Solution by Elimination
Gauss Elimination
Gauss Elimination
Gauss Elimination
Gauss Elimination
Gauss Elimination
Gauss Elimination
Gauss Elimination
Back Substitution
Back Substitution
Gauss Elimination
Gauss Elimination
Gauss Elimination
Gauss Elimination
Gauss Elimination
Gauss Elimination Alg
Gauss Elimination Alg
Gauss Elimination Alg
Example 8.1
Example 8.1
Example 8.1
ปัญหาของ Gauss Elimination
ปัญหาของ Gauss Elimination
ปัญหาของ Gauss Elimination
ปัญหาของ Gauss Elimination
Gauss-Jordan Method
Gauss-Jordan Method
Example 8.2
Example 8.2
Example 8.2
Example 8.2
Example 8.2
Example 8.2
การหา Matrix Inverse ด้วย GJ
การหา Matrix Inverse ด้วย GJ
การหา Matrix Inverse ด้วย GJ
การหา Matrix Inverse ด้วย GJ
การหา Matrix Inverse ด้วย GJ
Example 8.3
Example 8.3
Example 8.3
Iterative Method and Gauss-Seidel
Gauss-Seidel
Gauss-Seidel
Gauss-Seidel
Gauss-Seidel
Gauss-Seidel
Gauss-Seidel: Ex 8.4
Gauss-Seidel: Ex 8.4
Gauss-Seidel: Ex 8.4
Gauss-Seidel: Ex 8.4
Gauss-Seidel: Ex 8.4
Gauss-Seidel: Ex 8.4
Gauss-Seidel: Ex 8.4
Jacobi Method
Convergence of Iterative Method
Break
LU Decomposition
LU Decomposition
LU Decomposition
LU Decomposition
LU Decomposition
LU Decomposition
LU Decomposition
LU Decomposition
LU Decomposition
LU Decomposition
Crout Decomposition
Crout Decomposition
Crout Decomposition
Crout Decomposition
Crout Decomposition
Crout Decomposition
Crout Decomposition
Example 8.6
Crout Decomposition
Crout Decomposition
Example 8.7
Example 8.7
Slide Number 103
Summary Chapter 8
Homework 9, Chapter 10
End of Chapter 10
Ch10 Numerical Diff & Intg
- CPE 332Computer Engineering Mathematics II
- Today Topics
- Slide Number 3
- Slide Number 4
- Slope of Line
- Definition of Derivative
- Approximation of slope at point xusing secant line
- Derivative(Forward)
- Derivative(Backward)
- Derivative(Central)
- Introduction
- Finite Divided-Difference
- Slide Number 13
- Finite Divided-Difference
- Slide Number 15
- Finite Divided-Difference
- Slide Number 17
- Second Derivative
- Second Derivative
- High Accuracy Finite Divided-Difference
- High Accuracy Finite Divided-Difference
- Summary
- Summary
- Summary
- Numerical Integration
- Slide Number 26
- Slide Number 27
- Integral Approximation
- Integral Approximation
- Zero-Order Approximation
- Zero-Order Approximation
- Zero-Order Approximation
- Zero-Order Approximation
- First-Order Approximation
- First-Order Approximation
- First-Order Approximation
- First-Order Approximation
- Slide Number 38
- Trapezoidal Rule
- Trapezoidal Rule
- Slide Number 41
- Slide Number 42
- Slide Number 43
- Second-Order Approximation
- Second-Order Approximation
- Second-Order Approximation
- Second-Order Approximation
- Second-Order Approximation
- Slide Number 49
- Third-Order Approximation
- Third Degree Approximation
- Simson’s 3/8 Rule
- Slide Number 53
- Romberg Integration
- Romberg Integration
- Romberg Integration
- Romberg Integration
- Romberg Integration
- Romberg Integration
- Romberg Integration
- Romberg Integration
- Romberg Integration
- Romberg Integration
- Romberg Integration
- Romberg Integration
- Summary
- Homework 10: Ch 11
CPE 332Computer Engineering Mathematics II
Today Topics
Slide Number 3
Slide Number 4
Slope of Line
Definition of Derivative
Approximation of slope at point xusing secant line
Derivative(Forward)
Derivative(Backward)
Derivative(Central)
Introduction
Finite Divided-Difference
Slide Number 13
Finite Divided-Difference
Slide Number 15
Finite Divided-Difference
Slide Number 17
Second Derivative
Second Derivative
High Accuracy Finite Divided-Difference
High Accuracy Finite Divided-Difference
Summary
Summary
Summary
Numerical Integration
Slide Number 26
Slide Number 27
Integral Approximation
Integral Approximation
Zero-Order Approximation
Zero-Order Approximation
Zero-Order Approximation
Zero-Order Approximation
First-Order Approximation
First-Order Approximation
First-Order Approximation
First-Order Approximation
Slide Number 38
Trapezoidal Rule
Trapezoidal Rule
Slide Number 41
Slide Number 42
Slide Number 43
Second-Order Approximation
Second-Order Approximation
Second-Order Approximation
Second-Order Approximation
Second-Order Approximation
Slide Number 49
Third-Order Approximation
Third Degree Approximation
Simson’s 3/8 Rule
Slide Number 53
Romberg Integration
Romberg Integration
Romberg Integration
Romberg Integration
Romberg Integration
Romberg Integration
Romberg Integration
Romberg Integration
Romberg Integration
Romberg Integration
Romberg Integration
Romberg Integration
Summary
Homework 10: Ch 11
Ch11 ODE
- CPE 332Computer Engineering Mathematics II
- Today Topics
- Differential Equation
- Example 1
- Example 1
- Example 1
- Differential Equation
- Example 2
- Example 2
- Differential Equation
- Differential Equation
- Chapter 10
- ODE
- ODE
- ODE
- ODE: One Step Method
- ODE: One Step Method
- Euler’s Method
- Euler’s Method
- Euler’s Method
- Euler’s Method
- Euler’s Method
- Euler’s Method
- Heun and Polygon Method
- Heun Method
- Heun Method
- Heun Method
- Heun and Polygon Method
- Heun Method
- Heun and Polygon Method
- Heun Method
- Heun Method
- Improved Polygon Method
- Improved Polygon Method
- Runge-Kutta Method
- Runge-Kutta Method
- Second Order Runge-Kutta Method
- Second Order Runge-Kutta Method
- Second Order Runge-Kutta Method
- Second Order Runge-Kutta Method
- Second Order Runge-Kutta Method
- Second Order Runge-Kutta Method
- Second Order Runge-Kutta Method
- Second Order Runge-Kutta Method
- Second Order Runge-Kutta Method
- Slide Number 46
- Third Order Runge-Kutta Method
- Third Order Runge-Kutta Method
- Forth Order Runge-Kutta Method
- Forth Order Runge-Kutta Method
- Higher Order Runge-Kutta Method
- Comparison
- Chapter 12 Homework (HW 11)
- Final Exam Preparation
- Slide Number 55
- Formulasดูใน MT+ต่อไปนี้
CPE 332Computer Engineering Mathematics II
Today Topics
Differential Equation
Example 1
Example 1
Example 1
Differential Equation
Example 2
Example 2
Differential Equation
Differential Equation
Chapter 10
ODE
ODE
ODE
ODE: One Step Method
ODE: One Step Method
Euler’s Method
Euler’s Method
Euler’s Method
Euler’s Method
Euler’s Method
Euler’s Method
Heun and Polygon Method
Heun Method
Heun Method
Heun Method
Heun and Polygon Method
Heun Method
Heun and Polygon Method
Heun Method
Heun Method
Improved Polygon Method
Improved Polygon Method
Runge-Kutta Method
Runge-Kutta Method
Second Order Runge-Kutta Method
Second Order Runge-Kutta Method
Second Order Runge-Kutta Method
Second Order Runge-Kutta Method
Second Order Runge-Kutta Method
Second Order Runge-Kutta Method
Second Order Runge-Kutta Method
Second Order Runge-Kutta Method
Second Order Runge-Kutta Method
Slide Number 46
Third Order Runge-Kutta Method
Third Order Runge-Kutta Method
Forth Order Runge-Kutta Method
Forth Order Runge-Kutta Method
Higher Order Runge-Kutta Method
Comparison
Chapter 12 Homework (HW 11)
Final Exam Preparation
Slide Number 55
Formulasดูใน MT+ต่อไปนี้
Ch12 Curve Fitting

Computer Math II

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