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(Fall EE103 2008-09) 3. Linear equations linear equations example: polynomial interpolation applications geometrical interpretation left and right inverse range and nullspace 31 Linear equations m equations in n variables x1, x2, . . . , xn: a11x1 + a12x2 + + a1nxn = b1 a21x1 + a22x2 + + a2nxn = b2 . . am1x1 + am2x2 + + amnxn = bm in matrix form: Ax = b, where a11 a12 a1n a21 a22 a2n A=...
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(Fall EE103 2008-09)
3. Linear equations
linear equations example: polynomial interpolation applications geometrical interpretation left and right inverse range and nullspace
31
Linear equations
m equations in n variables x1, x2, . . . , xn: a11x1 + a12x2 + + a1nxn = b1 a21x1 + a22x2 + + a2nxn = b2 . . am1x1 + am2x2 + + amnxn = bm
in matrix form: Ax = b, where a11 a12 a1n a21 a22 a2n A= . . , . ... . . . am1 am2 amn x1 x2 x = . , . xn b1 b2 b= . . bm
Linear equations
32
Example: polynomial interpolation
t a polynomial p(t) = x1 + x2t + x3t2 + + xntn1 through n points (t1, y1), . . . , (tn, yn)
t1
t2
t3
t4
t5
problem data: t1, . . . , tn, y1, . . . , yn; problem variables: x1, . . . , xn
Linear equations 33
...
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UCLA - EE - 103
EE103 (Fall 2008-09)4. Triangular matrices terminology forward and backward substitution inverse41Terminologya square matrix A is lower triangular if aij = 0 for j > i a11 a21 . . 0 a22 . . 0 0 0 0 . 0 0 an1,n1 0 an,n1 ann
UCLA - EE - 103
EE103 (Fall 2008-09)Terminologya square matrix A is lower triangular if aij = 0 for j > i a11 a21 . . 0 a22 . . 0 0 0 0 . 0 0 an1,n1 0 an,n1 ann 4. Triangular matrices terminology forward and backward substitution inverse A=
UCLA - EE - 103
EE103 (Fall 2008-09)5. The Cholesky factorization positive (semi-)denite matrices examples the Cholesky factorization solving Ax = b with A positive denite inverse of a positive denite matrix permutation matrices sparse Cholesky factorizatio
UCLA - EE - 103
EE103 (Fall 2008-09)Positive (semi-)denite matrices A is positive denite if A is symmetric and xT Ax > 0 for all x = 0 A is positive semidenite if A is symmetric and xT Ax 0 for all x Note: if A is symmetric of order n, thenn n n5. The Choles
UCLA - EE - 103
EE103 (Fall 2008-09)6. The LU factorization factor-solve method nonsingular matrices LU factorization solving Ax = b with A nonsingular the inverse of a nonsingular matrix LU factorization algorithm eect of rounding error sparse LU factoriz
UCLA - EE - 103
EE103 (Fall 2008-09)Factor-solve approachto solve Ax = b, rst write A as a product of simple matrices A = A1 A2 Ak6. The LU factorization factor-solve method nonsingular matrices LU factorization solving Ax = b with A nonsingular the inv
UCLA - EE - 103
EE103 (Fall 2008-09)7. The condition number Ax = b when A is singular condition of a set of linear equations matrix norm condition number71Linear equations with singular coecient matrixif A is nonsingular, then Ax = b has a unique solutio
UCLA - EE - 103
EE103 (Fall 2008-09)Linear equations with singular coecient matrixif A is nonsingular, then Ax = b has a unique solution for every b if A is singular, then Ax = b has zero or innitely many solutions:7. The condition number Ax = b when A is sin
UCLA - EE - 103
EE103 (Fall 2008-09)8. Linear least-squares denition examples and applications81DenitionOverdetermined linear equations Ax = b for most b, cannot solve for x Least-squares formulation m n(A is m n with m > n)minimizeAx b = (i=1
UCLA - EE - 103
EE103 (Fall 2008-09)DenitionOverdetermined linear equations Ax = b for most b, cannot solve for x (A is m n with m > n)8. Linear least-squares denition examples and applicationsLeast-squares formulation m nminimizeAx b = (i=1 j=1
UCLA - EE - 103
EE103 (Fall 2008-09)9. The solution of a least-squares problem geometric interpretation left inverse of a zero nullspace matrix the solution of a least-squares problem the normal equations91Geometric interpretation of a LS problemminimiz
UCLA - EE - 103
EE103 (Fall 2008-09)Geometric interpretation of a LS problem9. The solution of a least-squares problemminimize A is m n with columns a1, . . . , an geometric interpretation left inverse of a zero nullspace matrix the solution of a least-squa
UCLA - EE - 103
EE103 (Fall 2008-09)10. The QR factorization solving the normal equations the QR factorization orthogonal matrices modied Gram-Schmidt algorithm Cholesky factorization versus QR factorization101Least-squares methodsminimize (A is m n w
UCLA - EE - 103
EE103 (Fall 2008-09)Least-squares methods10. The QR factorizationminimize solving the normal equations the QR factorization orthogonal matrices modied Gram-Schmidt algorithm Cholesky factorization versus QR factorization (A is m n with a z
UCLA - EE - 103
EE103 (Fall 2008-09)11. Least-norm problems denition and examples right inverse of a full range matrix the least-norm solution computing the least-norm solution111DenitionUnderdetermined linear equations Ax = b (A is m n with m < n) x
UCLA - EE - 103
EE103 (Fall 2008-09)12. Nonlinear equations with one variable denition and examples bisection method Newtons method secant method121Denition and examplesx is a zero (or root) of a function f if f (x) = 0 Examples f (x) = ex has no zeros
UCLA - EE - 103
EE103 (Fall 2008-09)Denition and examplesx is a zero (or root) of a function f if f (x) = 0 Examples12. Nonlinear equations with one variable denition and examples bisection method Newtons method secant method f (x) = ex has no zeros f (
UCLA - EE - 103
EE103 (Fall 2008-09)13. Newtons method for sets of nonlinear equations sets of nonlinear equations the derivative matrix and linearization Newtons method examples131Sets of nonlinear equationsn nonlinear equations in n variables f1(x1, .
UCLA - EE - 103
EE103 (Fall 2008-09)Sets of nonlinear equationsn nonlinear equations in n variables f1(x1, . . . , xn) = 0 f2(x1, . . . , xn) = 0 . . fn(x1, . . . , xn) = 0 in vector notation: f (x) = 0 where x Rn and f : Rn Rn are dened as f1(x1, . . . , xn)
UCLA - EE - 103
EE103 (Fall 2008-09)14. Unconstrained minimization terminology gradient and Hessian Newtons method141Unconstrained minimization problemminimize g(x1, x2, . . . , xn) g : Rn R (a function that maps n-vectors to scalars) x = (x1, x2, . .
UCLA - EE - 103
EE103 (Fall 2008-09)Unconstrained minimization problem14. Unconstrained minimizationminimize g(x1, x2, . . . , xn) g : Rn R (a function that maps n-vectors to scalars) terminology gradient and Hessian x = (x1, x2, . . . , xn) are the optimiz
UCLA - EE - 103
EE103 (Fall 2008-09)15. Nonlinear least-squares denition Newtons method Gauss-Newton method151Nonlinear least-squaresmminimizei=1ri(x)2 = r(x)2 ri is a nonlinear function of the n-vector of variables x r(x) = (r1(x), r2(x), . .
UCLA - EE - 103
EE103 (Fall 2008-09)15. Nonlinear least-squares denition Newtons method Gauss-Newton method151Nonlinear least-squaresmminimizei=1ri(x)2 = r(x)2 ri is a nonlinear function of the n-vector of variables x r(x) = (r1(x), r2(x), . .
UCLA - EE - 103
EE103 (Fall 2008-09)16. IEEE oating point numbers oating point numbers with base 10 oating point numbers with base 2 IEEE oating point standard machine precision rounding error161Floating point numbers with base 10x = (.d1d2 . . . dn)1
UCLA - EE - 103
EE103 (Fall 2008-09)Floating point numbers with base 1016. IEEE oating point numbersx = (.d1d2 . . . dn)10 10e oating point numbers with base 10 oating point numbers with base 2 IEEE oating point standard Interpretation: x = (d1101 + d2102
UCLA - EE - 103
EE103 (Fall 2008-09)17. Problem conditioning and stability of algorithms the conditioning of a problem the numerical stability of an algorithm cancellation171Sources of error in numerical computationExample: evaluate a function f : R R at
UCLA - EE - 103
EE103 (Fall 2008-09)Sources of error in numerical computationExample: evaluate a function f : R R at a given x (e.g., f (x) = sin x) sources of error in the result: x is not exactly known17. Problem conditioning and stability of algorithms t
UCLA - EE - 103
EE103 (Fall 2008-09)18. Ordinary dierential equations initial value problem examples forward and backward Euler method181Initial value problemFirst-order ordinary dierential equation (ODE) dx(t) = f (x(t), t), dt t usually represents time
UCLA - EE - 103
EE103 (Fall 2008-09)Initial value problemFirst-order ordinary dierential equation (ODE) dx(t) = f (x(t), t), dt x(0) = x018. Ordinary dierential equations initial value problem examples forward and backward Euler method t usually represent
UCLA - IPAM - 5
Institute for Pure and Applied Mathematics University of California, Los Angeles presentsMultiscale Geometry and Analysis in High Dimensions Workshop V: Math Analysis and Multiscale Geometric AnalysisNovember 15-19, 2004Members of the Organizing
UCLA - AAS - 116
A Truly American Experience By Jacqueline PonI would not be where I am today without the courage and hard work of my maternal and paternal grandparents. I was raised in an upper- middle class neighborhood in San Francisco and never felt like I was
UCLA - AAS - 116
My Struggles with Self and Society By Charito ViloriaMy name is Charito Viloria, and I am a 20-year-old first generation Pilipino-American, whose mother is a Registered Nurse and father is retired Navy. Born and raised in San Diego, California, Ive
UCLA - AAS - 116
Discoveries Terrible and Magnificent By Ali WongI was born in San Francisco April 19, 1982. While I lived in a neighborhood and went to school composed of predominantly wealthy white communities, I spent all of my summers and Friday nights at Donal
UCLA - AAS - 116
Try and Unite By Hanna KimMy name is Hanna Kim, and I had been a transfer student at Los Angeles City College. I wrote a grievance letter to LACC administrators because of an incident that occurred in my Asian American Studies class Monday night, 2
UCLA - AAS - 116
Learning from the Residents of Boston Chinatown By Katie LiHaaah? Tai dai sang ah! yelled an elderly Chinese woman wearing a set of earphones with the volume obviously turned up too loud. I rushed over to her, adjusted her walkman- looking machine,
UCLA - AAS - 116
My Life Is Connected to the Lives of My Ancestors By Karyn OkadaAs a fourth generation Japanese American, my life has intersected history in numerous ways. In this, I am speaking not only of my personal history, but also the histories of my ancesto
UCLA - AAS - 116
Walking the Same Streets as Carlos Bulosan By Marc LorestoThis essay will connect my life to history to the history of Filipino immigration to the United States and the racism experienced by them as well as other Asian minorities. I was born in Lo
UCLA - AAS - 116
Making a Difference in the World By Alejandro LopezThe journey began when my parents, my oldest brother and my two sisters left the state of Guanajuato in the country of Mexico in 1975. Like many others who have migrated, my father (who was a const
UCLA - AAS - 116
Stepping Stones towards a Successful Future By Sarah Marie P. MamarilI have always had assignments where I am to tell my whole life story, but this is the first time where I actually have to relate my life to the history that was happening around m
UCLA - AAS - 116
Kaoh 1 Immigration, Assimilation, and the Model Minority Myth By Christina KaohLyndon B. Johnsons signing of the Immigration Act of 1965 marked the shift in the demographics of America. According to Franklin Ngs The Taiwanese Americans, this act in
UCLA - AAS - 116
Why Did You Come to America? By Linda LamWhy did you come to America? I remember asking this question to my mother for a 4th grade project. My mother replied, Because of the Vietnam War, we decided to go to the United States. From her responses, I
UCLA - AAS - 116
How Do Race Relations in L.A. Koreatown Affect the Assi Workers Campaign? By Jacqueline PonIn class I thought about the possible effects of the Assi Market workers strike on the local community and came up with the question; how do race relations a
UCLA - AAS - 116
Mobilizing Students By Charito Viloria and Christine CorpusBefore we can discuss how students like us can mobilize support for the Assi workers, we must discuss the steps necessary in order to reach that stage of mobilizing. First, it is imperative
UCLA - AAS - 116
Why Care about Them? By Ali WongThe first day of Asian American Studies 116 was the first time I had ever heard about the ASSI boycott and KIWA as an organization. Since KIWA stands for Korean Immigrants Workers Advocate, specifically labeling thei
UCLA - AAS - 116
Life Lessons from Assi Workers By Hanna Kim and Katie LiThe initial goals and historical mission of Asian American Studies were to link students with the community. The students at San Francisco State who initiated the first Asian American Studies
UCLA - AAS - 116
How Does the Assi Workers Campaign Affect Race Relations in Koreatown? By Karyn OkadaAlthough immigrant worker struggles are rampant particularly in metropolitan cities such as Los Angeles there are many lessons that students and Asian Pacific Is
UCLA - AAS - 116
What Resources Do UCLA Students Have that Can Help Workers Campaigns for Justice? By Alejandro LopezStudent solidarity is important for various reasons in regards to the everyday struggles of immigrant workers. First and foremost, the economy would
UCLA - AAS - 116
Sarah MamarilHow and why is the role of a student important in alleviating the struggles of the immigrant workers?In the summer of 2001, 20 workers walked out of their jobs at the Assi Market. They had worked there for minimum wage and their hour
UCLA - AAS - 116
What Is the Significance of the Assi Workers Campaign for Asian American Students? By Christina KaohThree years ago, twenty Assi workers walked off their jobs, frustrated that the employer had reduced their hours from full, eight-hour workdays. Fro
UCLA - AAS - 116
What Resources Do Students Have That They Can Use to Support Workers Campaigns? By Linda Lam and Satomi ZeiglerStudent solidarity for immigrant workers struggles is important for several reasons. Students can provide time and resources that may not
UCLA - AAS - 116
Why Should Student s Support the Assi Workers Campaign for Justice? By Grace Chen and Delphina YuenAssi Market is one of seven large markets in Koreatown. Despite the success of Assi Market, over the past year an employer/employee conflict arose as
UCLA - AAS - 116
Let Freedom Ring! By Hanna KimWe can empower our community to empower themselves. Working with immigrant workers can humble students. We need to recognize their full humanity and understand that they have life experiences involving Asian American s
UCLA - AAS - 116
The Road to Mobilization By Jacqueline PonWhen trying to educate and mobilizing students, it is important to find methods that will appeal to the student population. The first step is education. Through a forum, students can listen and learn from g
UCLA - AAS - 116
Educating and Mobilizing By Katie LiStudent activists often have hard time mobilizing their peers. The pressures of class work, fear to get involved, and time constraints often deter people from wanting to take political action. In fact, being poli
UCLA - AAS - 116
Tools for Activists: Rethinking the Relationship between Awareness and ActionBy Glenn OmatsuAsk activists in the U.S. how they see the process of social change, and most will describe a linear relationship: i.e., social change occurs when people
UCLA - AAS - 116
How Can We Gain Support from Other Students for the Assi Workers Campaign? By Alejandro LopezIn order to gain the support of UCLA students towards the Assi worker struggle the process of educating and mobilizing needs to take effect. Consequently,
UCLA - AAS - 116
Experiencing Shared Leadership By Jacqueline PonWithin my committee, I have had some difficulties adjusting to the concept of shared leadership. I am used to having one leader organize and lead the rest of the group. So far, the shared leadership i
UCLA - AAS - 116
Less Dictator, More Responsibility By Ali WongMy friends and family share an inside joke about me. After I say anything controlling, theyll whisper dictator or dictadore. My sixth grade teacher once told me that I had Napoleonic syndrome. But despi
UCLA - AAS - 116
Kudos to Katie! By Hanna KimI am contributing to shared leadership by assuming shared responsibility for our campaign against corporate exploitation and the processes involved in making the rally and picket line both educational and fun. We as a gr
UCLA - AAS - 116
Teach Your Parents Well By Ali WongEvery culture perpetuates some sort of heroic icon. A person, place, group or thing can serve to represent that heroic icon, but the person tends to be the most powerful. For people of color in America, our heroic
UCLA - AAS - 116
Becoming an Active Agent for Social Change By Hanna KimI think that we can teach or demonstrate with the immigrant workers our solidarity with the cause against injustice. We, as UCLA students, can show them that we are not apathetic, inert, or com