Homework 6 Solutions
|
|
- Gilbert Fleming
- 6 years ago
- Views:
Transcription
1 Homeork 6 Solutions Igor Yanovsky (Math 151B TA) Section 114, Problem 1: For the boundary-value problem y (y ) y + log x, 1 x, y(1) 0, y() log, (1) rite the nonlinear system and formulas for Neton s method We divide [1, ] into N + 1 subintervals hose endpoints are x i 1 + ih, for i 0, 1,, N + 1, and consider the discretization of the boundary-value problem in (1): y (x i ) (y(x i ) ) y(x i ) + log x i () Replacing y (x i ) and y (x i ) by appropriate centered difference formulas, equation () becomes: ( ) y(x i+1 ) y(x i ) + y(x i 1 ) h h y(xi+1 ) y(x i 1 ) 1 y(4) (ξ i ) h h 6 y (η i ) y(x i ) + log x i, for some ξ i and η i in the interval (x i 1, x i+1 ) The difference method results hen the error terms are deleted and the boundary conditions are employed: and 0 0, N+1 log, i+1 i + i 1 h ( ) i+1 i 1 i + log x i 0, (3) h for each i 1,,, N Multiplying (3) by h, e obtain ( ) i+1 i 1 i+1 + i i 1 h i + h log x i 0, hich can be ritten as: or ( ) i+1 i 1 i+1 + i i 1 h i + h log x i 0, i 1 + i i+1 1 4( i 1 i 1 i+1 + i+1) h i + h log x i 0 1
2 Thus, the N N nonlinear system is: ( ) h 1 + h log x 1 0, ( ) h + h log x 0, ( 4 + 4) h 3 + h log x 3 0, N + N 1 N 1 4( N N N + N) h N 1 + h log x N 1 0, N 1 + N log 1 4( N 1 N 1 log + (log ) ) h N + h log x N 0, here e designate the left-hand side of the first equation as F 1 ( 1,, N ), the second equation as F ( 1,, N ),, the last equation as F N ( 1,, N ) Also, e designate F (F 1,, F N ) T and ( 1,, N ) T We use Neton s method for nonlinear systems to approximate the solution to the system F ( ) 0 above A sequence of iterates (k) ( (k) 1, (k),, (k) N )T is generated that converges to the solution of this system The Jacobian matrix J for this system is F 1 F 1 F 1 F N J( 1,, N ) F 1 F F 3 F N 1 1 F N 1 F N 1 F N F N 1 3 F N 3 F N F N 1 N F N N h h N + 1 N h N 1 N h We can no use the Neton s method for nonlinear systems (k) (k 1) J 1( (k 1)) F ( (k 1) )
3 Section 71, Problem 1: Find x and x for the folloing vectors: a) x (3, 4, 0, 3 )T ; c) x (sin k, cos k, k ) T for a fixed positive integer k The L and L norms for the vector x (x 1, x,, x n ) T are defined by x max x i, 1in { } 1 x x i a) For x (3, 4, 0, 3 )T : { x max 3, 4, 0, 3 } 4, ( 3 ) x 3 + ( 4) c) For x (sin k, cos k, k ) T, k is a positive integer : x max { sin k, cos k, k } k, x sin k + cos k + ( k ) k Section 71, Problem (a): Verify that the function 1, defined on R n by x 1 x i, is a norm on R n (i) For all x R n, x 1 x i 0 (ii) If x 0, then x 1 x i 0 0 If x 1 0, e have n x i 0, and thus, x 0 (iii) For all α R and x R n, αx 1 αx i α x i α x i α x 1 (iii) For all x, y R n, x + y 1 x i + y i Thus, 1 is a norm on R n ( xi + y i ) 3 x i + y i x 1 + y 1
4 Section 71, Problem (c): Prove that for all x R n, x 1 x Let x (x 1, x,, x n ) T, and note that ( x1 + x + + x n ) x 1 + x + + x n, or or ( x i ) ( x i x i, x i, hich means that x 1 x Section 71, Problem 4(c): Find for the folloing matrix: 1 0 A Since We have A max 1in a ij a 1j a 11 + a 1 + a , a j a 1 + a + a , a 3j a 31 + a 3 + a , e have A max{3, 4, 3} 4 Section 71, Problem 7: Sho by example that, defined by A does not define a matrix norm max a ij, 1i,jn A function [ ] is a matrix [ norm ] only if it satisfies [ definition ] 78 on page Consider A and B Then, AB We have A , B 1, and AB, and thus, AB A B, hich contradicts one of the conditions for being a norm 4
5 Section 71, Problem 9(a): The Frobenius norm (hich is not a natural norm) is defined for an n n matrix A by ( A F a ij Sho that F is a matrix norm For all n n matrices A and B and all real numbers α, e have: (i) ( A F a ij 0 (ii) (iii) ( A F αa F a ij 0 if and only if A is a 0 matrix α a ij αa F α A F (iv) Here, e ill use Cauchy-Scharz Inequality: A + B F ( ( a ij + b ij ( a ij + b ij ) α a ij α ( a ij + a ij b ij + b ij ) a ij + ( a ij + a ij b ij + a ij ( + ( A F + B F ) ( x i y i a ij ( A + B F A F + B F b ij ) a ij α A F x i ( y i b ij (Cauchy-Scharz) b ij + (v) Note that n k1 a n 1kb k1 k1 a n 1kb k k1 a n kb k1 k1 a kb k AB n k1 a n nkb k1 k1 a nkb k n k1 a nkb k,n 1 5 b ij n k1 a 1kb kn n k1 a kb kn n k1 a nkb kn
6 AB ( F a ik b kj a ik b kj ) (Cauchy-Scharz) ( k1 k1 ( a ik b kj ) k1 k1 a ij )( AB F A F B F b ij ) A F B F CONTINUE TO THE NEXT PAGE 6
7 Section 71, Problem 9(c): For any matrix A, sho that A A F n 1/ A The definitions of F and norms are: ( A F a ij A max x 1 Ax Note, that Ax is a vector: n a 1jx j n Ax a jx j n a njx j Thus, e have ( ( ) Ax a ij x j ❶ We first sho that A A F For vector x, such that x 1, e have ( ) Ax a ij x j (Cauchy-Scharz) ( ( ( ( ( ( A F a ij a ij a ij a ij a ij ( )( ) x ) 1 x j x j ) We shoed that, Ax A F for all x, such that x 1 Thus, max Ax A F, or A A F x 1 ❷ We no sho that A F n 1/ A Let x i 1 n for all 1 i n Then, ) ) A max x 1 Ax ( ) a ij x j a ij 1 n 1 n a ij 1 n A F Thus, A F n 1/ A 7
8 Section 73, Problem (c): Find the first to iterations of the Jacobi method for the folloing linear system, using x (0) 0: 4x 1 + x x 3 + x 4, x 1 + 4x x 3 x 4 1, x 1 x + 5x 3 + x 4 0, x 1 x + x 3 + 3x 4 1 The linear system Ax b given by E 1 : 4x 1 + x x 3 + x 4, E : x 1 + 4x x 3 x 4 1, E 3 : x 1 x + 5x 3 + x 4 0, E 4 : x 1 x + x 3 + 3x 4 1 has the unique solution x ( 07534, , 0808, ) To convert Ax b to the form x T x + c, solve equation E 1 for x 1, E for x, E 3 for x 3, E 4 for x 4, to obtain x x x x 4 1, x 1 4 x x x 4 1 4, x x x 1 5 x 4, x x x 1 3 x Then Ax b can be ritten in the form x T x + c, ith T and c For initial approximation, e let x (0) (0, 0, 0, 0) T Then is given by x(0) x(0) x(0) , 1 4 x( x(0) x(0) , x( x(0) 1 5 x(0) 4 0, x( x(0) 1 3 x(0) /3 The next iterate, x (), is given by x () x(1) x(1) x(1) , x () 1 4 x( x(1) x(1) , x () x( x(1) 1 5 x(1) , x () x( x(1) 1 3 x(1)
9 Section 73, Problem 4(c): Find the first to iterations of the Gauss-Seidel method for the folloing linear system, using x (0) 0: 4x 1 + x x 3 + x 4, x 1 + 4x x 3 x 4 1, x 1 x + 5x 3 + x 4 0, x 1 x + x 3 + 3x 4 1 In section 73, Problem (c), e used Jacobi method to solve the linear system above The folloing equations ere used: x(k 1) x(k 1) x(k 1) 4 1, 1 4 x(k x(k 1) x(k 1) 4 1 4, x(k x(k 1) 1 5 x(k 1) 4, x(k x(k 1) 1 3 x(k 1) Hoever, since for i > 1, 1,, x(k) i 1 have already been computed, these are probably better approximations to the actual solutions x 1,, x i 1 than x (k 1) 1,, x (k 1) i 1 Hence, Gauss-Seidel uses the most recently available approximations to x 1,, x i 1 in a calculation of the next iterate: x(k 1) x(k 1) x(k 1) 4 1, 1 4 x(k x(k 1) x(k 1) 4 1 4, x(k x(k) 1 5 x(k 1) 4, x(k x(k) 1 3 x(k) For initial approximation, e let x (0) (0, 0, 0, 0) T Then is given by x(0) x(0) x(0) , 1 4 x( x(0) x(0) , x( x(1) 1 5 x(0) 4 015, x( x(1) 1 3 x(1) The next iterate, x (), is given by x () x(1) x(1) x(1) , x () 1 4 x( x(1) x(1) , x () x( x() 1 5 x(1) 4 05, x () x( x() 1 3 x() Comparing x () to the exact solution x ( 07534, , 0808, ), e see that Gauss-Seidel method gave more accurate results than Jacobi method 9
Problem 1. Possible Solution. Let f be the 2π-periodic functions defined by f(x) = cos ( )
Problem Let f be the π-periodic functions defined by f() = cos ( ) hen [ π, π]. Make a draing of the function f for the interval [ 3π, 3π], and compute the Fourier series of f. Use the result to compute
More informationMath 5630: Iterative Methods for Systems of Equations Hung Phan, UMass Lowell March 22, 2018
1 Linear Systems Math 5630: Iterative Methods for Systems of Equations Hung Phan, UMass Lowell March, 018 Consider the system 4x y + z = 7 4x 8y + z = 1 x + y + 5z = 15. We then obtain x = 1 4 (7 + y z)
More informationInner products and Norms. Inner product of 2 vectors. Inner product of 2 vectors x and y in R n : x 1 y 1 + x 2 y x n y n in R n
Inner products and Norms Inner product of 2 vectors Inner product of 2 vectors x and y in R n : x 1 y 1 + x 2 y 2 + + x n y n in R n Notation: (x, y) or y T x For complex vectors (x, y) = x 1 ȳ 1 + x 2
More informationSolving Linear Systems
Solving Linear Systems Iterative Solutions Methods Philippe B. Laval KSU Fall 207 Philippe B. Laval (KSU) Linear Systems Fall 207 / 2 Introduction We continue looking how to solve linear systems of the
More information3 - Vector Spaces Definition vector space linear space u, v,
3 - Vector Spaces Vectors in R and R 3 are essentially matrices. They can be vieed either as column vectors (matrices of size and 3, respectively) or ro vectors ( and 3 matrices). The addition and scalar
More informationLinear Algebraic Equations
Linear Algebraic Equations Linear Equations: a + a + a + a +... + a = c 11 1 12 2 13 3 14 4 1n n 1 a + a + a + a +... + a = c 21 2 2 23 3 24 4 2n n 2 a + a + a + a +... + a = c 31 1 32 2 33 3 34 4 3n n
More informationLecture Note 7: Iterative methods for solving linear systems. Xiaoqun Zhang Shanghai Jiao Tong University
Lecture Note 7: Iterative methods for solving linear systems Xiaoqun Zhang Shanghai Jiao Tong University Last updated: December 24, 2014 1.1 Review on linear algebra Norms of vectors and matrices vector
More informationChapter 7 Iterative Techniques in Matrix Algebra
Chapter 7 Iterative Techniques in Matrix Algebra Per-Olof Persson persson@berkeley.edu Department of Mathematics University of California, Berkeley Math 128B Numerical Analysis Vector Norms Definition
More informationCOURSE Numerical methods for solving linear systems. Practical solving of many problems eventually leads to solving linear systems.
COURSE 9 4 Numerical methods for solving linear systems Practical solving of many problems eventually leads to solving linear systems Classification of the methods: - direct methods - with low number of
More informationJACOBI S ITERATION METHOD
ITERATION METHODS These are methods which compute a sequence of progressively accurate iterates to approximate the solution of Ax = b. We need such methods for solving many large linear systems. Sometimes
More informationReview of Basic Concepts in Linear Algebra
Review of Basic Concepts in Linear Algebra Grady B Wright Department of Mathematics Boise State University September 7, 2017 Math 565 Linear Algebra Review September 7, 2017 1 / 40 Numerical Linear Algebra
More informationDEN: Linear algebra numerical view (GEM: Gauss elimination method for reducing a full rank matrix to upper-triangular
form) Given: matrix C = (c i,j ) n,m i,j=1 ODE and num math: Linear algebra (N) [lectures] c phabala 2016 DEN: Linear algebra numerical view (GEM: Gauss elimination method for reducing a full rank matrix
More informationLECTURE NOTES ELEMENTARY NUMERICAL METHODS. Eusebius Doedel
LECTURE NOTES on ELEMENTARY NUMERICAL METHODS Eusebius Doedel TABLE OF CONTENTS Vector and Matrix Norms 1 Banach Lemma 20 The Numerical Solution of Linear Systems 25 Gauss Elimination 25 Operation Count
More informationPhysically Based Rendering ( ) Geometry and Transformations
Phsicall Based Rendering (6.657) Geometr and Transformations 3D Point Specifies a location Origin 3D Point Specifies a location Represented b three coordinates Infinitel small class Point3D { public: Coordinate
More informationMath/Phys/Engr 428, Math 529/Phys 528 Numerical Methods - Summer Homework 3 Due: Tuesday, July 3, 2018
Math/Phys/Engr 428, Math 529/Phys 528 Numerical Methods - Summer 28. (Vector and Matrix Norms) Homework 3 Due: Tuesday, July 3, 28 Show that the l vector norm satisfies the three properties (a) x for x
More informationOptimal Preconditioners for Interval Gauss Seidel Methods
Optimal Preconditioners for Interval Gauss Seidel Methods R B Kearfott and Xiaofa Shi Introduction Consider the following nonlinear system f 1 x 1, x 2,, x n ) F X) f n x 1, x 2,, x n ) where bounds x
More informationMAT 771 FUNCTIONAL ANALYSIS HOMEWORK 3. (1) Let V be the vector space of all bounded or unbounded sequences of complex numbers.
MAT 771 FUNCTIONAL ANALYSIS HOMEWORK 3 (1) Let V be the vector space of all bounded or unbounded sequences of complex numbers. (a) Define d : V V + {0} by d(x, y) = 1 ξ j η j 2 j 1 + ξ j η j. Show that
More informationj=1 [We will show that the triangle inequality holds for each p-norm in Chapter 3 Section 6.] The 1-norm is A F = tr(a H A).
Math 344 Lecture #19 3.5 Normed Linear Spaces Definition 3.5.1. A seminorm on a vector space V over F is a map : V R that for all x, y V and for all α F satisfies (i) x 0 (positivity), (ii) αx = α x (scale
More informationTHE LINEAR BOUND FOR THE NATURAL WEIGHTED RESOLUTION OF THE HAAR SHIFT
THE INEAR BOUND FOR THE NATURA WEIGHTED RESOUTION OF THE HAAR SHIFT SANDRA POTT, MARIA CARMEN REGUERA, ERIC T SAWYER, AND BRETT D WIC 3 Abstract The Hilbert transform has a linear bound in the A characteristic
More informationThe Solution of Linear Systems AX = B
Chapter 2 The Solution of Linear Systems AX = B 21 Upper-triangular Linear Systems We will now develop the back-substitution algorithm, which is useful for solving a linear system of equations that has
More informationBucket handles and Solenoids Notes by Carl Eberhart, March 2004
Bucket handles and Solenoids Notes by Carl Eberhart, March 004 1. Introduction A continuum is a nonempty, compact, connected metric space. A nonempty compact connected subspace of a continuum X is called
More informationBasic Concepts in Linear Algebra
Basic Concepts in Linear Algebra Grady B Wright Department of Mathematics Boise State University February 2, 2015 Grady B Wright Linear Algebra Basics February 2, 2015 1 / 39 Numerical Linear Algebra Linear
More informationOn the approximation of real powers of sparse, infinite, bounded and Hermitian matrices
On the approximation of real poers of sparse, infinite, bounded and Hermitian matrices Roman Werpachoski Center for Theoretical Physics, Al. Lotnikó 32/46 02-668 Warszaa, Poland Abstract We describe a
More informationThe degree sequence of Fibonacci and Lucas cubes
The degree sequence of Fibonacci and Lucas cubes Sandi Klavžar Faculty of Mathematics and Physics University of Ljubljana, Slovenia and Faculty of Natural Sciences and Mathematics University of Maribor,
More informationThese notes give a quick summary of the part of the theory of autonomous ordinary differential equations relevant to modeling zombie epidemics.
NOTES ON AUTONOMOUS ORDINARY DIFFERENTIAL EQUATIONS MARCH 2017 These notes give a quick summary of the part of the theory of autonomous ordinary differential equations relevant to modeling zombie epidemics.
More informationParallel Scientific Computing
IV-1 Parallel Scientific Computing Matrix-vector multiplication. Matrix-matrix multiplication. Direct method for solving a linear equation. Gaussian Elimination. Iterative method for solving a linear equation.
More informationToday s class. Linear Algebraic Equations LU Decomposition. Numerical Methods, Fall 2011 Lecture 8. Prof. Jinbo Bi CSE, UConn
Today s class Linear Algebraic Equations LU Decomposition 1 Linear Algebraic Equations Gaussian Elimination works well for solving linear systems of the form: AX = B What if you have to solve the linear
More informationConjugate Gradient (CG) Method
Conjugate Gradient (CG) Method by K. Ozawa 1 Introduction In the series of this lecture, I will introduce the conjugate gradient method, which solves efficiently large scale sparse linear simultaneous
More information96 CHAPTER 4. HILBERT SPACES. Spaces of square integrable functions. Take a Cauchy sequence f n in L 2 so that. f n f m 1 (b a) f n f m 2.
96 CHAPTER 4. HILBERT SPACES 4.2 Hilbert Spaces Hilbert Space. An inner product space is called a Hilbert space if it is complete as a normed space. Examples. Spaces of sequences The space l 2 of square
More informationMatrices: 2.1 Operations with Matrices
Goals In this chapter and section we study matrix operations: Define matrix addition Define multiplication of matrix by a scalar, to be called scalar multiplication. Define multiplication of two matrices,
More informationLIMIT-POINT CRITERIA FOR THE MATRIX STURM-LIOUVILLE OPERATOR AND ITS POWERS. Irina N. Braeutigam
Opuscula Math. 37, no. 1 (2017), 5 19 http://dx.doi.org/10.7494/opmath.2017.37.1.5 Opuscula Mathematica LIMIT-POINT CRITERIA FOR THE MATRIX STURM-LIOUVILLE OPERATOR AND ITS POWERS Irina N. Braeutigam Communicated
More informationEnhancing Generalization Capability of SVM Classifiers with Feature Weight Adjustment
Enhancing Generalization Capability of SVM Classifiers ith Feature Weight Adjustment Xizhao Wang and Qiang He College of Mathematics and Computer Science, Hebei University, Baoding 07002, Hebei, China
More informationHomework If the inverse T 1 of a closed linear operator exists, show that T 1 is a closed linear operator.
Homework 3 1 If the inverse T 1 of a closed linear operator exists, show that T 1 is a closed linear operator Solution: Assuming that the inverse of T were defined, then we will have to have that D(T 1
More informationIterative Methods. Splitting Methods
Iterative Methods Splitting Methods 1 Direct Methods Solving Ax = b using direct methods. Gaussian elimination (using LU decomposition) Variants of LU, including Crout and Doolittle Other decomposition
More informationA turbulence closure based on the maximum entropy method
Advances in Fluid Mechanics IX 547 A turbulence closure based on the maximum entropy method R. W. Derksen Department of Mechanical and Manufacturing Engineering University of Manitoba Winnipeg Canada Abstract
More informationMath Introduction to Numerical Analysis - Class Notes. Fernando Guevara Vasquez. Version Date: January 17, 2012.
Math 5620 - Introduction to Numerical Analysis - Class Notes Fernando Guevara Vasquez Version 1990. Date: January 17, 2012. 3 Contents 1. Disclaimer 4 Chapter 1. Iterative methods for solving linear systems
More informationScientific Computing
Scientific Computing Direct solution methods Martin van Gijzen Delft University of Technology October 3, 2018 1 Program October 3 Matrix norms LU decomposition Basic algorithm Cost Stability Pivoting Pivoting
More informationSolving Linear Systems
Solving Linear Systems Iterative Solutions Methods Philippe B. Laval KSU Fall 2015 Philippe B. Laval (KSU) Linear Systems Fall 2015 1 / 12 Introduction We continue looking how to solve linear systems of
More informationREPRESENTATIONS FOR A SPECIAL SEQUENCE
REPRESENTATIONS FOR A SPECIAL SEQUENCE L. CARLITZ* RICHARD SCOVILLE Dyke University, Durham,!\!orth Carolina VERNERE.HOGGATTJR. San Jose State University, San Jose, California Consider the sequence defined
More information1. Nonlinear Equations. This lecture note excerpted parts from Michael Heath and Max Gunzburger. f(x) = 0
Numerical Analysis 1 1. Nonlinear Equations This lecture note excerpted parts from Michael Heath and Max Gunzburger. Given function f, we seek value x for which where f : D R n R n is nonlinear. f(x) =
More informationMAT 610: Numerical Linear Algebra. James V. Lambers
MAT 610: Numerical Linear Algebra James V Lambers January 16, 2017 2 Contents 1 Matrix Multiplication Problems 7 11 Introduction 7 111 Systems of Linear Equations 7 112 The Eigenvalue Problem 8 12 Basic
More information1 Effects of Regularization For this problem, you are required to implement everything by yourself and submit code.
This set is due pm, January 9 th, via Moodle. You are free to collaborate on all of the problems, subject to the collaboration policy stated in the syllabus. Please include any code ith your submission.
More informationAN ELEMENTARY PROOF OF THE SPECTRAL RADIUS FORMULA FOR MATRICES
AN ELEMENTARY PROOF OF THE SPECTRAL RADIUS FORMULA FOR MATRICES JOEL A. TROPP Abstract. We present an elementary proof that the spectral radius of a matrix A may be obtained using the formula ρ(a) lim
More informationMMSE Equalizer Design
MMSE Equalizer Design Phil Schniter March 6, 2008 [k] a[m] P a [k] g[k] m[k] h[k] + ṽ[k] q[k] y [k] P y[m] For a trivial channel (i.e., h[k] = δ[k]), e kno that the use of square-root raisedcosine (SRRC)
More informationLecture # 3 Orthogonal Matrices and Matrix Norms. We repeat the definition an orthogonal set and orthornormal set.
Lecture # 3 Orthogonal Matrices and Matrix Norms We repeat the definition an orthogonal set and orthornormal set. Definition A set of k vectors {u, u 2,..., u k }, where each u i R n, is said to be an
More informationCS 246 Review of Linear Algebra 01/17/19
1 Linear algebra In this section we will discuss vectors and matrices. We denote the (i, j)th entry of a matrix A as A ij, and the ith entry of a vector as v i. 1.1 Vectors and vector operations A vector
More informationCOURSE Iterative methods for solving linear systems
COURSE 0 4.3. Iterative methods for solving linear systems Because of round-off errors, direct methods become less efficient than iterative methods for large systems (>00 000 variables). An iterative scheme
More informationNumerical Linear Algebra
Numerical Linear Algebra The two principal problems in linear algebra are: Linear system Given an n n matrix A and an n-vector b, determine x IR n such that A x = b Eigenvalue problem Given an n n matrix
More informationHomework Set 2 Solutions
MATH 667-010 Introduction to Mathematical Finance Prof. D. A. Edards Due: Feb. 28, 2018 Homeork Set 2 Solutions 1. Consider the ruin problem. Suppose that a gambler starts ith ealth, and plays a game here
More informationAlgebra C Numerical Linear Algebra Sample Exam Problems
Algebra C Numerical Linear Algebra Sample Exam Problems Notation. Denote by V a finite-dimensional Hilbert space with inner product (, ) and corresponding norm. The abbreviation SPD is used for symmetric
More information4 Linear Algebra Review
4 Linear Algebra Review For this topic we quickly review many key aspects of linear algebra that will be necessary for the remainder of the course 41 Vectors and Matrices For the context of data analysis,
More informationand sinθ = cosb =, and we know a and b are acute angles, find cos( a+ b) Trigonometry Topics Accuplacer Review revised July 2016 sin.
Trigonometry Topics Accuplacer Revie revised July 0 You ill not be alloed to use a calculator on the Accuplacer Trigonometry test For more information, see the JCCC Testing Services ebsite at http://jcccedu/testing/
More informationIntroduction To Resonant. Circuits. Resonance in series & parallel RLC circuits
Introduction To esonant Circuits esonance in series & parallel C circuits Basic Electrical Engineering (EE-0) esonance In Electric Circuits Any passive electric circuit ill resonate if it has an inductor
More informationSeong Joo Kang. Let u be a smooth enough solution to a quasilinear hyperbolic mixed problem:
Comm. Korean Math. Soc. 16 2001, No. 2, pp. 225 233 THE ENERGY INEQUALITY OF A QUASILINEAR HYPERBOLIC MIXED PROBLEM Seong Joo Kang Abstract. In this paper, e establish the energy inequalities for second
More information1 Directional Derivatives and Differentiability
Wednesday, January 18, 2012 1 Directional Derivatives and Differentiability Let E R N, let f : E R and let x 0 E. Given a direction v R N, let L be the line through x 0 in the direction v, that is, L :=
More informationChapter 3. Linear and Nonlinear Systems
59 An expert is someone who knows some of the worst mistakes that can be made in his subject, and how to avoid them Werner Heisenberg (1901-1976) Chapter 3 Linear and Nonlinear Systems In this chapter
More informationChapter 3. Differentiable Mappings. 1. Differentiable Mappings
Chapter 3 Differentiable Mappings 1 Differentiable Mappings Let V and W be two linear spaces over IR A mapping L from V to W is called a linear mapping if L(u + v) = Lu + Lv for all u, v V and L(λv) =
More informationDifferential equations
Differential equations Math 7 Spring Practice problems for April Exam Problem Use the method of elimination to find the x-component of the general solution of x y = 6x 9x + y = x 6y 9y Soln: The system
More informationSolving Linear Systems of Equations
November 6, 2013 Introduction The type of problems that we have to solve are: Solve the system: A x = B, where a 11 a 1N a 12 a 2N A =.. a 1N a NN x = x 1 x 2. x N B = b 1 b 2. b N To find A 1 (inverse
More informationCh. 2 Math Preliminaries for Lossless Compression. Section 2.4 Coding
Ch. 2 Math Preliminaries for Lossless Compression Section 2.4 Coding Some General Considerations Definition: An Instantaneous Code maps each symbol into a codeord Notation: a i φ (a i ) Ex. : a 0 For Ex.
More informationA PROJECTED HESSIAN GAUSS-NEWTON ALGORITHM FOR SOLVING SYSTEMS OF NONLINEAR EQUATIONS AND INEQUALITIES
IJMMS 25:6 2001) 397 409 PII. S0161171201002290 http://ijmms.hindawi.com Hindawi Publishing Corp. A PROJECTED HESSIAN GAUSS-NEWTON ALGORITHM FOR SOLVING SYSTEMS OF NONLINEAR EQUATIONS AND INEQUALITIES
More informationIterative techniques in matrix algebra
Iterative techniques in matrix algebra Tsung-Ming Huang Department of Mathematics National Taiwan Normal University, Taiwan September 12, 2015 Outline 1 Norms of vectors and matrices 2 Eigenvalues and
More informationNumerical Analysis: Solutions of System of. Linear Equation. Natasha S. Sharma, PhD
Mathematical Question we are interested in answering numerically How to solve the following linear system for x Ax = b? where A is an n n invertible matrix and b is vector of length n. Notation: x denote
More information2. Linear algebra. matrices and vectors. linear equations. range and nullspace of matrices. function of vectors, gradient and Hessian
FE661 - Statistical Methods for Financial Engineering 2. Linear algebra Jitkomut Songsiri matrices and vectors linear equations range and nullspace of matrices function of vectors, gradient and Hessian
More informationA NEW EFFECTIVE PRECONDITIONED METHOD FOR L-MATRICES
Journal of Mathematical Sciences: Advances and Applications Volume, Number 2, 2008, Pages 3-322 A NEW EFFECTIVE PRECONDITIONED METHOD FOR L-MATRICES Department of Mathematics Taiyuan Normal University
More informationIntroduction - Motivation. Many phenomena (physical, chemical, biological, etc.) are model by differential equations. f f(x + h) f(x) (x) = lim
Introduction - Motivation Many phenomena (physical, chemical, biological, etc.) are model by differential equations. Recall the definition of the derivative of f(x) f f(x + h) f(x) (x) = lim. h 0 h Its
More informationPowerPoints organized by Dr. Michael R. Gustafson II, Duke University
Part 3 Chapter 10 LU Factorization PowerPoints organized by Dr. Michael R. Gustafson II, Duke University All images copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
More informationMATH 271 Summer 2016 Practice problem solutions Week 1
Part I MATH 271 Summer 2016 Practice problem solutions Week 1 For each of the following statements, determine whether the statement is true or false. Prove the true statements. For the false statement,
More informationMATH Topics in Applied Mathematics Lecture 12: Evaluation of determinants. Cross product.
MATH 311-504 Topics in Applied Mathematics Lecture 12: Evaluation of determinants. Cross product. Determinant is a scalar assigned to each square matrix. Notation. The determinant of a matrix A = (a ij
More informationDeterminants. Samy Tindel. Purdue University. Differential equations and linear algebra - MA 262
Determinants Samy Tindel Purdue University Differential equations and linear algebra - MA 262 Taken from Differential equations and linear algebra by Goode and Annin Samy T. Determinants Differential equations
More informationHere is an example of a block diagonal matrix with Jordan Blocks on the diagonal: J
Class Notes 4: THE SPECTRAL RADIUS, NORM CONVERGENCE AND SOR. Math 639d Due Date: Feb. 7 (updated: February 5, 2018) In the first part of this week s reading, we will prove Theorem 2 of the previous class.
More informationEcon 201: Problem Set 3 Answers
Econ 20: Problem Set 3 Ansers Instructor: Alexandre Sollaci T.A.: Ryan Hughes Winter 208 Question a) The firm s fixed cost is F C = a and variable costs are T V Cq) = 2 bq2. b) As seen in class, the optimal
More informationCS 323: Numerical Analysis and Computing
CS 323: Numerical Analysis and Computing MIDTERM #1 Instructions: This is an open notes exam, i.e., you are allowed to consult any textbook, your class notes, homeworks, or any of the handouts from us.
More information9. Iterative Methods for Large Linear Systems
EE507 - Computational Techniques for EE Jitkomut Songsiri 9. Iterative Methods for Large Linear Systems introduction splitting method Jacobi method Gauss-Seidel method successive overrelaxation (SOR) 9-1
More information7.5 Operations with Matrices. Copyright Cengage Learning. All rights reserved.
7.5 Operations with Matrices Copyright Cengage Learning. All rights reserved. What You Should Learn Decide whether two matrices are equal. Add and subtract matrices and multiply matrices by scalars. Multiply
More informationProcess Model Formulation and Solution, 3E4
Process Model Formulation and Solution, 3E4 Section B: Linear Algebraic Equations Instructor: Kevin Dunn dunnkg@mcmasterca Department of Chemical Engineering Course notes: Dr Benoît Chachuat 06 October
More informationComputational Methods. Systems of Linear Equations
Computational Methods Systems of Linear Equations Manfred Huber 2010 1 Systems of Equations Often a system model contains multiple variables (parameters) and contains multiple equations Multiple equations
More informationNotes on Linear Algebra and Matrix Theory
Massimo Franceschet featuring Enrico Bozzo Scalar product The scalar product (a.k.a. dot product or inner product) of two real vectors x = (x 1,..., x n ) and y = (y 1,..., y n ) is not a vector but a
More informationMax-Margin Ratio Machine
JMLR: Workshop and Conference Proceedings 25:1 13, 2012 Asian Conference on Machine Learning Max-Margin Ratio Machine Suicheng Gu and Yuhong Guo Department of Computer and Information Sciences Temple University,
More informationFIXED POINT ITERATIONS
FIXED POINT ITERATIONS MARKUS GRASMAIR 1. Fixed Point Iteration for Non-linear Equations Our goal is the solution of an equation (1) F (x) = 0, where F : R n R n is a continuous vector valued mapping in
More informationCHAPTER V MULTIPLE SCALES..? # w. 5?œ% 0 a?ß?ß%.?.? # %?œ!.>#.>
CHAPTER V MULTIPLE SCALES This chapter and the next concern initial value prolems of oscillatory type on long intervals of time. Until Chapter VII e ill study autonomous oscillatory second order initial
More informationJordan Journal of Mathematics and Statistics (JJMS) 5(3), 2012, pp A NEW ITERATIVE METHOD FOR SOLVING LINEAR SYSTEMS OF EQUATIONS
Jordan Journal of Mathematics and Statistics JJMS) 53), 2012, pp.169-184 A NEW ITERATIVE METHOD FOR SOLVING LINEAR SYSTEMS OF EQUATIONS ADEL H. AL-RABTAH Abstract. The Jacobi and Gauss-Seidel iterative
More informationSome definitions. Math 1080: Numerical Linear Algebra Chapter 5, Solving Ax = b by Optimization. A-inner product. Important facts
Some definitions Math 1080: Numerical Linear Algebra Chapter 5, Solving Ax = b by Optimization M. M. Sussman sussmanm@math.pitt.edu Office Hours: MW 1:45PM-2:45PM, Thack 622 A matrix A is SPD (Symmetric
More informationG1110 & 852G1 Numerical Linear Algebra
The University of Sussex Department of Mathematics G & 85G Numerical Linear Algebra Lecture Notes Autumn Term Kerstin Hesse (w aw S w a w w (w aw H(wa = (w aw + w Figure : Geometric explanation of the
More informationTheory of Iterative Methods
Based on Strang s Introduction to Applied Mathematics Theory of Iterative Methods The Iterative Idea To solve Ax = b, write Mx (k+1) = (M A)x (k) + b, k = 0, 1,,... Then the error e (k) x (k) x satisfies
More informationFirst, we review some important facts on the location of eigenvalues of matrices.
BLOCK NORMAL MATRICES AND GERSHGORIN-TYPE DISCS JAKUB KIERZKOWSKI AND ALICJA SMOKTUNOWICZ Abstract The block analogues of the theorems on inclusion regions for the eigenvalues of normal matrices are given
More informationNonlinear H control and the Hamilton-Jacobi-Isaacs equation
Proceedings of the 7th World Congress The International Federation of Automatic Control Seoul, Korea, July 6-, 8 Nonlinear H control and the Hamilton-Jacobi-Isaacs equation Henrique C. Ferreira Paulo H.
More informationy(x) = x w + ε(x), (1)
Linear regression We are ready to consider our first machine-learning problem: linear regression. Suppose that e are interested in the values of a function y(x): R d R, here x is a d-dimensional vector-valued
More informationFunctional Analysis Exercise Class
Functional Analysis Exercise Class Week: December 4 8 Deadline to hand in the homework: your exercise class on week January 5. Exercises with solutions ) Let H, K be Hilbert spaces, and A : H K be a linear
More informationLecture 8 January 30, 2014
MTH 995-3: Intro to CS and Big Data Spring 14 Inst. Mark Ien Lecture 8 January 3, 14 Scribe: Kishavan Bhola 1 Overvie In this lecture, e begin a probablistic method for approximating the Nearest Neighbor
More informationMath 304 (Spring 2010) - Lecture 2
Math 304 (Spring 010) - Lecture Emre Mengi Department of Mathematics Koç University emengi@ku.edu.tr Lecture - Floating Point Operation Count p.1/10 Efficiency of an algorithm is determined by the total
More informationA Simple Derivation of Newton-Cotes Formulas with Realistic Errors
Journal of Mathematics Research; Vol. 4, No. 5; 01 ISSN 1916-9795 E-ISSN 1916-9809 Published by Canadian Center of Science and Education A Simple Derivation of Neton-Cotes Formulas ith Realistic Errors
More informationCOS 511: Theoretical Machine Learning. Lecturer: Rob Schapire Lecture # 12 Scribe: Indraneel Mukherjee March 12, 2008
COS 511: Theoretical Machine Learning Lecturer: Rob Schapire Lecture # 12 Scribe: Indraneel Mukherjee March 12, 2008 In the previous lecture, e ere introduced to the SVM algorithm and its basic motivation
More informationNumerical solutions of nonlinear systems of equations
Numerical solutions of nonlinear systems of equations Tsung-Ming Huang Department of Mathematics National Taiwan Normal University, Taiwan E-mail: min@math.ntnu.edu.tw August 28, 2011 Outline 1 Fixed points
More informationImproving AOR Method for a Class of Two-by-Two Linear Systems
Alied Mathematics 2 2 236-24 doi:4236/am22226 Published Online February 2 (htt://scirporg/journal/am) Imroving AOR Method for a Class of To-by-To Linear Systems Abstract Cuixia Li Shiliang Wu 2 College
More information10.34: Numerical Methods Applied to Chemical Engineering. Lecture 7: Solutions of nonlinear equations Newton-Raphson method
10.34: Numerical Methods Applied to Chemical Engineering Lecture 7: Solutions of nonlinear equations Newton-Raphson method 1 Recap Singular value decomposition Iterative solutions to linear equations 2
More informationAn Improved Driving Scheme in an Electrophoretic Display
International Journal of Engineering and Technology Volume 3 No. 4, April, 2013 An Improved Driving Scheme in an Electrophoretic Display Pengfei Bai 1, Zichuan Yi 1, Guofu Zhou 1,2 1 Electronic Paper Displays
More informationIterative Methods for Ax=b
1 FUNDAMENTALS 1 Iterative Methods for Ax=b 1 Fundamentals consider the solution of the set of simultaneous equations Ax = b where A is a square matrix, n n and b is a right hand vector. We write the iterative
More informationThe smallest positive integer that is solution of a proportionally modular Diophantine inequality
The smallest positive integer that is solution of a proportionally modular Diophantine inequality P. Vasco Iberian meeting on numerical semigroups, Porto 2008 J. C. Rosales and P. Vasco, The smallest positive
More informationCHAPTER 3 THE COMMON FACTOR MODEL IN THE POPULATION. From Exploratory Factor Analysis Ledyard R Tucker and Robert C. MacCallum
CHAPTER 3 THE COMMON FACTOR MODEL IN THE POPULATION From Exploratory Factor Analysis Ledyard R Tucker and Robert C. MacCallum 1997 19 CHAPTER 3 THE COMMON FACTOR MODEL IN THE POPULATION 3.0. Introduction
More information