2 Two-Point Boundary Value Problems
|
|
- Barnaby Moore
- 5 years ago
- Views:
Transcription
1 2 Two-Point Boundary Value Problems Another fundamental equation, in addition to the heat eq. and the wave eq., is Poisson s equation: n j=1 2 u x 2 j The unknown is the function u = u(x 1, x 2,..., x n ), where the x j s are spatial coordinates. In this chapter we study Poisson s eq. in one dimension (n = 1), so we get the two-point boundary value problem = f u (x) = f (x), x (, 1), u() = u(1) =. (2.1) We shall study numerical approximations and analytical properties. 1 / 21
2 2.1 Poisson s Equation in One Dimension We find an expression for the solution of (2.1), and obtain uniqueness. The fundamental theorem of calculus: u(x) = c 1 + x u (y)dy. So, using this for u (y), and assuming u is a solution of Poisson s eq. we get y y u (y) = c 2 + u (z)dz = c 2 f (z)dz so x y u(x) = c 1 + c 2 x ( f (z)dz)dy (2.5) Introduce F (y) = y f (z)dz, so by integration by parts x ( y f (z)dz)dy = x yf (y)dy = [yf (y)] x x yf (y)dy = xf (x) x yf (y)dy = x (x y)f (y)dy 2 / 21
3 So: u(x) = c 1 + c 2 x x (x y)f (y)dy and using the boundary conditions we get c 1 = and c 2 = 1 (1 y)f (y)dy. Since the constants are uniquely determined by the boundary conditions, the solution must be unique. And it is u(x) = x 1 (1 y)f (y)dy x Example 2.1 If f (x) = 1 (constant function), then u(x) = x 1 (1 y)dy x (x y)f (y)dy (2.7) (x y)dy = (1/2)x(1 x) 3 / 21
4 Introduce the Green s function: { y(1 x) if y x G(x, y) = x(1 y) if x y 1 Then we rewrite the Poisson solution from (2.7) u(x) = 1 G(x, y)f (y)dy. (2.9) The Green s function acts as an inverse, almost like Ax = b implies x = A 1 b when the matrix is invertible. Note that G is continuous, symmetric, nonnegative and piecewise linear if either x or y is fixed. Note also that the solution u is smoother than the data: u (m+2) = f (m), which follows from the Poisson eq. Theorem 2.1 For every f C([, 1]) there is a unique solution u C 2 ((, 1)) of the boundary value problem (2.1). Furthermore, the solution has the representation (2.9). 4 / 21
5 Using that G we get: Proposition 2.1 Assume that f C([, 1]) is a nonnegative function.then the corresponding solution u of (2.1) is also nonnegative. The next result is called the maximum principle and it says that the solution of the Poisson eq. is smaller than the given function f in the supremum norm: Proposition 2.2 Assume that f C([, 1]) and let u be the unique solution of (2.1). Then u (1/8) f Proof. This follows from (2.9) by a small calculation, using that 1 G(x, y)dy = (1/2)x(1 x) and the supremum of this is 1/8. 5 / 21
6 2.2 A Finite Difference Approximation Want to find approximate solution of the Poisson eq. by solving a linear algebraic system Ax = b. For a four-times continuously differentiable function g = g(x) we can use the Taylor series at x to get expressions for g(x + h) and g(x h), and show that g(x + h) 2g(x) + g(x h) h 2 = g (x) + E h (x) where the error term satisfies E h (x) M g h 2 /12 and M g = sup x g (4) (x). For instance, if g is a polynomial of degree at most 3, then the error is zero. Partition the unit interval: x j = jh (j =, 1,..., n + 1) where h = 1/(n + 1). The discrete problem is obtained by using variables v j that approximate the solution u at x j. We use the approximation for u (x) and get the linear (algebraic) equations v j 1 2v j + v j+1 h 2 = f (x j ) (j = 1,..., n), v = v n+1 =. 6 / 21
7 This may be written as Av = b where A = The right hand side column vector b is given by b = (b 1,..., b n ) (we identify n-tuples and column vectors, for typographical reasons) where b j = h 2 f (x j ) j = 1,..., n. We will later show that A is invertible. Example 2.3 With f (x) = (3x + x 2 )e x we can find the exact solution u(x) = x(1 x)e x. Figures in the book show good approximation of the discrete solution, and the error E h = seems to satisfy E h = O(h 2 ). max u(x j) v j j n+1 7 / 21
8 A is tridiagonal. How to solve tridiagonal linear systems? So more generally: α 1 γ β 2 α 2 γ.. 2. A = (2.18). β n 1 α n 1 γ n 1... β n α n and we consider Av = b. In the second-order difference matrix before we had α j = 2 and β j = γ j = 1 for each j. We use usual row-reduction: First, add β 2 /α 1 times the first row to the second row. In the resulting matrix we then get a in position (2, 1). Then we add a suitable multiple of the second row to the third row, so we get in position (3, 2). Continuing like this, adding a multiple of the ith row to the next, we end up with a row-equivalent matrix which is upper triangular. Of course, similar operations are made to the right hand side b, i.e., we work on the augmented matrix. 8 / 21
9 Then we find the solution v of Av = b by back substitution: determine v n from the last row (only one unknown), and use this value to find v n 1 from row n 1 etc. If we never get a coefficient in front of variable v k in equation k, this process can be completed. This holds if and only if A is invertible and we return to conditions that assure this. As a result (see TW) one gets the following algorithm Algorithm 2.1 δ 1 = α 1 c 1 = b 1 for k = 2, 3,..., n m k = β k /δ k 1 δ k = α k m k γ k 1 c k = b k m k c k 1 v n = c n /δ n for k = n 1, n 2,..., 1 v k = (c k γ k v k+1 )/δ k Note that the number of operations is O(n), in contrast to elimination for a full matrix which is O(n 3 ). 9 / 21
10 An n n matrix A = [a ij ] is called diagonally dominant if a ii j i a ij (i n). A slight variation is for our tridiagonal matrix A above which we call diagonally dominant if α 1 > γ 1, α k β k + γ k (k = 2, 3,..., n) where we define γ n = Lemma 2.1 Assume that the coefficient matrix A of the triangular system (2.19) is diagonal dominant and that β k for k = 2, 3,..., n. Then the variables δ k (k = 1, 2,..., n) determined by Algorithm 2.1 are well defined and nonzero. 1 / 21
11 Proposition 2.3 Assume that the coefficient matrix A of (2.19) satisfies the properties specified in Lemma 2.1 above. Then, the system has a unique solution which can be computed by Algorithm 2.1. Corollary 2.1 The system of equations defined by (2.14)-(2.15), has a unique solution that can be computed using Algorithm 2.1. The second-order difference matrix A has another important property, it is symmetric and positive definite. Recall: A symmetric n n matrix A ia called positive definite if v T Av > for all nonzero v R n. From linear algebra we know that this property holds if and only if all eigenvalues of A are positive. Another equivalent condition is that there exists an n k matrix B with linearly independent columns such that A = B T B. Every positive definite matrix A is invertible, because if Ax = O for some nonzero x, then x T Ax = x T O = ; a contradiction. 11 / 21
12 Proposition 2.4 Consider a tridiagonal system of the form (2.19) and assume that the corresponding coefficient matrix (2.18) is symmetric and positive definite. Then the system has a unique solution that can be computed by Algorithm 2.1. Proof: As just explained, A must be invertible. Moreover, the only problem that can occur in Algorithm 2.1 is that in some iteration k 1 we obtain that the entry in position (k, k) of the present matrix Ā is zero, i.e., that δ k =. But then the first k columns of Ā are linearly dependent. This is a contradiction, as A and Ā are row-equivalent, and therefore have the same rank, namely n. 12 / 21
13 2.3 Continuous and Discrete Solutions We shall show that almost all essential properties of the exact, or continuous, solution are somehow present in the approximate solution. Let L be the differential operator (Lu)(x) = u (x), f C([, 1]). So the two-point boundary value problem is: Find u C 2 ([, 1]) such that (Lu)(x) = f (x) for all x (, 1) (2.26) C 2 ([, 1]): functions that are twice continuously differentiable, and are zero at the boundaries. Let D h be the discrete functions defined at the grid points x j for j =,..., n + 1. We write v(x j ) or just v j. D h, is the subset of D h where the function is at the boundary points. Define the discrete operator which gives the finite difference approximation to the second derivative: (L h w)(x j ) = w(x j+1) 2w j + w j 1 h 2 13 / 21
14 The discrete problem: Find v C h, such that (L h v)(x j ) = f (x j ) for j = 1, 2,..., n (2.27) We use inner products (and norms) as follows. For continuous functions (as above): u, v = 1 u(x)v(x)dx. and for discrete functions u, v D h almost like the usual inner product for vectors: u, v h = h( u v + u n+1 v n+1 2 n + u j v j ) j=1 A matrix A is symmetric if A = A T, and this is equivalent to (Ax, y) = (x, Ay) for all x, y R n, where (, ) denotes the usual Euclidean inner product. 14 / 21
15 Lemma 2.2 The operator L given in (2.26) is symmetric in the sense that u, Lv = Lu, v for all u, v C 2 ((, 1)). Proof. If u, v C 2 ((, 1)), then by integration by parts and using that u() = v() = u(1) = v(1) = : Lu, v = 1 u vdx = u (x)v(x) u v dx = 1 u v dx = 1 uv dx = u, Lv By so-called summation by part we get the same property for the discrete operator: Lemma 2.3 The operator L given in (2.26) is symmetric in the sense that u, L h v h = L h u, v for all u, v D h,. 15 / 21
16 Lemma 2.4 The operators L and L h are positive definite in the following sense: (i) For any u C 2 ((, 1)) we have Lu, u and with equality only if u =. (ii) For any v D h, we have L h v, v and with equality only if v =. Proof. Follows from prev calculations: and also Lu, u = 1 (u (x)) 2 dx n L h u, u h = h 1 (v j+1 v j ) 2. j= 16 / 21
17 We showed before the uniqueness of solution of both the continuous and the discrete problem. Now we see that this also follows from the positive definiteness. Lemma 2.5 The solution u of (2.26) and the solution v of (2.27) are unique solutions of the continuous and the discrete problems, respectively. Proof. Assume Lu 1 = f and Lu 2 = f, and let e = u 1 u 2. Then Le = Lu 1 Lu 2 = f f =. So Le, e = and by positive definite property, this implies that e =, so u 1 = u 2. The discrete case is similar. 17 / 21
18 Remember that the exact solution of the Poisson solution may be written in terms of the Green s function u(x) = 1 G(x, y)f (y)dy. (2.9) For a given grid point x k = kh define a grid function G k D h, by G k (x j ) = G(x j, x k ). Using linearity of G(x, y) in x for x y, we get L h (G k )(x j ) = for j k. A computation shows that L h (G k )(x j ) = 1/h. So L h G k = (1/h)e k where e k (x j ) equal 1 if k = j, and it is otherwise. For f D h, define w D h, by n w = h f (x k )G k k=1 Then w is the unique solution of the discrete problem as n n L h w = h f (x k )(L h G k ) = f (x k )e k = f k=1 k=1 18 / 21
19 So we have v j = h n G(x j, x k )f (x k ) k=1 which is the discrete solution of (2.27). So, this is similar to the continuous solution. And since G k we get Proposition 2.5 Assume that f (x) > for all x [, 1], and let v D h, be the solution of (2.27). Then v(x j ) for all j = 1,..., n. Similarly, we also obtain a discrete maximum principle. Here we use the norm v h, = max v(x j). j=,...,n+1 Proposition 2.6 The solution v D h, of (2.27) satisfies v h, (1/8) f h, 19 / 21
20 Convergence of the discrete solution Definition 2.2 Let f C([, 1]), and let u C 2 ((, 1)) be the solution of (2.26). Then we define the discrete vector τ h, called the truncation error, by τ h (x j ) = (L h u)(x j ) f (x j ) for all j = 1,...,, n. We say that the finite difference scheme (2.27) is consistent with the differential equation (2.26) if lim τ h h, =. h Note that the error is obtained by applying u, not v, to the differential operator L h. 2 / 21
21 By using the Poisson equation, second order Taylor series and our maximum principle we can derive: Theorem 2.2 Assume that f C 2 ([, 1]) is given. Let u and v be the corresponding solutions of (2.26) and (2.27), respectively. Then u v h, f h So, we can get arbitrary small error, by choosing h small enough. 21 / 21
LECTURE 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 informationFinite difference method for elliptic problems: I
Finite difference method for elliptic problems: I Praveen. C praveen@math.tifrbng.res.in Tata Institute of Fundamental Research Center for Applicable Mathematics Bangalore 560065 http://math.tifrbng.res.in/~praveen
More informationLinear Algebra. Session 8
Linear Algebra. Session 8 Dr. Marco A Roque Sol 08/01/2017 Abstract Linear Algebra Range and kernel Let V, W be vector spaces and L : V W, be a linear mapping. Definition. The range (or image of L is the
More informationNumerical Solutions to Partial Differential Equations
Numerical Solutions to Partial Differential Equations Zhiping Li LMAM and School of Mathematical Sciences Peking University Numerical Methods for Partial Differential Equations Finite Difference Methods
More informationNORMS ON SPACE OF MATRICES
NORMS ON SPACE OF MATRICES. Operator Norms on Space of linear maps Let A be an n n real matrix and x 0 be a vector in R n. We would like to use the Picard iteration method to solve for the following system
More information1 Last time: least-squares problems
MATH Linear algebra (Fall 07) Lecture Last time: least-squares problems Definition. If A is an m n matrix and b R m, then a least-squares solution to the linear system Ax = b is a vector x R n such that
More informationOnline Exercises for Linear Algebra XM511
This document lists the online exercises for XM511. The section ( ) numbers refer to the textbook. TYPE I are True/False. Lecture 02 ( 1.1) Online Exercises for Linear Algebra XM511 1) The matrix [3 2
More informationMath113: Linear Algebra. Beifang Chen
Math3: Linear Algebra Beifang Chen Spring 26 Contents Systems of Linear Equations 3 Systems of Linear Equations 3 Linear Systems 3 2 Geometric Interpretation 3 3 Matrices of Linear Systems 4 4 Elementary
More informationOR MSc Maths Revision Course
OR MSc Maths Revision Course Tom Byrne School of Mathematics University of Edinburgh t.m.byrne@sms.ed.ac.uk 15 September 2017 General Information Today JCMB Lecture Theatre A, 09:30-12:30 Mathematics revision
More informationDS-GA 1002 Lecture notes 0 Fall Linear Algebra. These notes provide a review of basic concepts in linear algebra.
DS-GA 1002 Lecture notes 0 Fall 2016 Linear Algebra These notes provide a review of basic concepts in linear algebra. 1 Vector spaces You are no doubt familiar with vectors in R 2 or R 3, i.e. [ ] 1.1
More informationLecture Summaries for Linear Algebra M51A
These lecture summaries may also be viewed online by clicking the L icon at the top right of any lecture screen. Lecture Summaries for Linear Algebra M51A refers to the section in the textbook. Lecture
More informationLinear Algebra. Week 7
Linear Algebra. Week 7 Dr. Marco A Roque Sol 10 / 09 / 2018 If {v 1, v 2,, v n } is a basis for a vector space V, then any vector v V, has a unique representation v = x 1 v 1 + x 2 v 2 + + x n v n where
More informationMath 344 Lecture # Linear Systems
Math 344 Lecture #12 2.7 Linear Systems Through a choice of bases S and T for finite dimensional vector spaces V (with dimension n) and W (with dimension m), a linear equation L(v) = w becomes the linear
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 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 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 information1. General Vector Spaces
1.1. Vector space axioms. 1. General Vector Spaces Definition 1.1. Let V be a nonempty set of objects on which the operations of addition and scalar multiplication are defined. By addition we mean a rule
More informationLinear Algebra Massoud Malek
CSUEB Linear Algebra Massoud Malek Inner Product and Normed Space In all that follows, the n n identity matrix is denoted by I n, the n n zero matrix by Z n, and the zero vector by θ n An inner product
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 informationPreliminary/Qualifying Exam in Numerical Analysis (Math 502a) Spring 2012
Instructions Preliminary/Qualifying Exam in Numerical Analysis (Math 502a) Spring 2012 The exam consists of four problems, each having multiple parts. You should attempt to solve all four problems. 1.
More informationLinear Algebra Section 2.6 : LU Decomposition Section 2.7 : Permutations and transposes Wednesday, February 13th Math 301 Week #4
Linear Algebra Section. : LU Decomposition Section. : Permutations and transposes Wednesday, February 1th Math 01 Week # 1 The LU Decomposition We learned last time that we can factor a invertible matrix
More informationMTH 215: Introduction to Linear Algebra
MTH 215: Introduction to Linear Algebra Lecture 5 Jonathan A. Chávez Casillas 1 1 University of Rhode Island Department of Mathematics September 20, 2017 1 LU Factorization 2 3 4 Triangular Matrices Definition
More informationOHSx XM511 Linear Algebra: Solutions to Online True/False Exercises
This document gives the solutions to all of the online exercises for OHSx XM511. The section ( ) numbers refer to the textbook. TYPE I are True/False. Answers are in square brackets [. Lecture 02 ( 1.1)
More information1 Inner Product and Orthogonality
CSCI 4/Fall 6/Vora/GWU/Orthogonality and Norms Inner Product and Orthogonality Definition : The inner product of two vectors x and y, x x x =.., y =. x n y y... y n is denoted x, y : Note that n x, y =
More informationMath 240 Calculus III
The Calculus III Summer 2015, Session II Wednesday, July 8, 2015 Agenda 1. of the determinant 2. determinants 3. of determinants What is the determinant? Yesterday: Ax = b has a unique solution when A
More informationQuadratic forms. Here. Thus symmetric matrices are diagonalizable, and the diagonalization can be performed by means of an orthogonal matrix.
Quadratic forms 1. Symmetric matrices An n n matrix (a ij ) n ij=1 with entries on R is called symmetric if A T, that is, if a ij = a ji for all 1 i, j n. We denote by S n (R) the set of all n n symmetric
More informationApplied Linear Algebra
Applied Linear Algebra Gábor P. Nagy and Viktor Vígh University of Szeged Bolyai Institute Winter 2014 1 / 262 Table of contents I 1 Introduction, review Complex numbers Vectors and matrices Determinants
More informationTEST CODE: PMB SYLLABUS
TEST CODE: PMB SYLLABUS Convergence and divergence of sequence and series; Cauchy sequence and completeness; Bolzano-Weierstrass theorem; continuity, uniform continuity, differentiability; directional
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 informationChapter 1: Systems of Linear Equations
Chapter : Systems of Linear Equations February, 9 Systems of linear equations Linear systems Lecture A linear equation in variables x, x,, x n is an equation of the form a x + a x + + a n x n = b, where
More informationLecture 1 INF-MAT3350/ : Some Tridiagonal Matrix Problems
Lecture 1 INF-MAT3350/4350 2007: Some Tridiagonal Matrix Problems Tom Lyche University of Oslo Norway Lecture 1 INF-MAT3350/4350 2007: Some Tridiagonal Matrix Problems p.1/33 Plan for the day 1. Notation
More informationMATH 1120 (LINEAR ALGEBRA 1), FINAL EXAM FALL 2011 SOLUTIONS TO PRACTICE VERSION
MATH (LINEAR ALGEBRA ) FINAL EXAM FALL SOLUTIONS TO PRACTICE VERSION Problem (a) For each matrix below (i) find a basis for its column space (ii) find a basis for its row space (iii) determine whether
More informationx 3y 2z = 6 1.2) 2x 4y 3z = 8 3x + 6y + 8z = 5 x + 3y 2z + 5t = 4 1.5) 2x + 8y z + 9t = 9 3x + 5y 12z + 17t = 7
Linear Algebra and its Applications-Lab 1 1) Use Gaussian elimination to solve the following systems x 1 + x 2 2x 3 + 4x 4 = 5 1.1) 2x 1 + 2x 2 3x 3 + x 4 = 3 3x 1 + 3x 2 4x 3 2x 4 = 1 x + y + 2z = 4 1.4)
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 informationMath 405: Numerical Methods for Differential Equations 2016 W1 Topics 10: Matrix Eigenvalues and the Symmetric QR Algorithm
Math 405: Numerical Methods for Differential Equations 2016 W1 Topics 10: Matrix Eigenvalues and the Symmetric QR Algorithm References: Trefethen & Bau textbook Eigenvalue problem: given a matrix A, find
More informationThis can be accomplished by left matrix multiplication as follows: I
1 Numerical Linear Algebra 11 The LU Factorization Recall from linear algebra that Gaussian elimination is a method for solving linear systems of the form Ax = b, where A R m n and bran(a) In this method
More information12/1/2015 LINEAR ALGEBRA PRE-MID ASSIGNMENT ASSIGNED BY: PROF. SULEMAN SUBMITTED BY: M. REHAN ASGHAR BSSE 4 ROLL NO: 15126
12/1/2015 LINEAR ALGEBRA PRE-MID ASSIGNMENT ASSIGNED BY: PROF. SULEMAN SUBMITTED BY: M. REHAN ASGHAR Cramer s Rule Solving a physical system of linear equation by using Cramer s rule Cramer s rule is really
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 informationNumerical Linear Algebra Homework Assignment - Week 2
Numerical Linear Algebra Homework Assignment - Week 2 Đoàn Trần Nguyên Tùng Student ID: 1411352 8th October 2016 Exercise 2.1: Show that if a matrix A is both triangular and unitary, then it is diagonal.
More informationNumerical Solutions to Partial Differential Equations
Numerical Solutions to Partial Differential Equations Zhiping Li LMAM and School of Mathematical Sciences Peking University Discretization of Boundary Conditions Discretization of Boundary Conditions On
More informationReview of Linear Algebra
Review of Linear Algebra Definitions An m n (read "m by n") matrix, is a rectangular array of entries, where m is the number of rows and n the number of columns. 2 Definitions (Con t) A is square if m=
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 informationSolution to Homework 1
Solution to Homework Sec 2 (a) Yes It is condition (VS 3) (b) No If x, y are both zero vectors Then by condition (VS 3) x = x + y = y (c) No Let e be the zero vector We have e = 2e (d) No It will be false
More informationA matrix is a rectangular array of. objects arranged in rows and columns. The objects are called the entries. is called the size of the matrix, and
Section 5.5. Matrices and Vectors A matrix is a rectangular array of objects arranged in rows and columns. The objects are called the entries. A matrix with m rows and n columns is called an m n matrix.
More informationAn Alternative Proof of Primitivity of Indecomposable Nonnegative Matrices with a Positive Trace
An Alternative Proof of Primitivity of Indecomposable Nonnegative Matrices with a Positive Trace Takao Fujimoto Abstract. This research memorandum is aimed at presenting an alternative proof to a well
More informationScientific Computing: Dense Linear Systems
Scientific Computing: Dense Linear Systems Aleksandar Donev Courant Institute, NYU 1 donev@courant.nyu.edu 1 Course MATH-GA.2043 or CSCI-GA.2112, Spring 2012 February 9th, 2012 A. Donev (Courant Institute)
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 informationA matrix is a rectangular array of. objects arranged in rows and columns. The objects are called the entries. is called the size of the matrix, and
Section 5.5. Matrices and Vectors A matrix is a rectangular array of objects arranged in rows and columns. The objects are called the entries. A matrix with m rows and n columns is called an m n matrix.
More informationLecture 7. Econ August 18
Lecture 7 Econ 2001 2015 August 18 Lecture 7 Outline First, the theorem of the maximum, an amazing result about continuity in optimization problems. Then, we start linear algebra, mostly looking at familiar
More informationIMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET
IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET This is a (not quite comprehensive) list of definitions and theorems given in Math 1553. Pay particular attention to the ones in red. Study Tip For each
More informationBoundary Value Problems and Iterative Methods for Linear Systems
Boundary Value Problems and Iterative Methods for Linear Systems 1. Equilibrium Problems 1.1. Abstract setting We want to find a displacement u V. Here V is a complete vector space with a norm v V. In
More informationLINEAR ALGEBRA SUMMARY SHEET.
LINEAR ALGEBRA SUMMARY SHEET RADON ROSBOROUGH https://intuitiveexplanationscom/linear-algebra-summary-sheet/ This document is a concise collection of many of the important theorems of linear algebra, organized
More informationSolving Ax = b w/ different b s: LU-Factorization
Solving Ax = b w/ different b s: LU-Factorization Linear Algebra Josh Engwer TTU 14 September 2015 Josh Engwer (TTU) Solving Ax = b w/ different b s: LU-Factorization 14 September 2015 1 / 21 Elementary
More informationLU Factorization. A m x n matrix A admits an LU factorization if it can be written in the form of A = LU
LU Factorization A m n matri A admits an LU factorization if it can be written in the form of Where, A = LU L : is a m m lower triangular matri with s on the diagonal. The matri L is invertible and is
More informationMath Camp Lecture 4: Linear Algebra. Xiao Yu Wang. Aug 2010 MIT. Xiao Yu Wang (MIT) Math Camp /10 1 / 88
Math Camp 2010 Lecture 4: Linear Algebra Xiao Yu Wang MIT Aug 2010 Xiao Yu Wang (MIT) Math Camp 2010 08/10 1 / 88 Linear Algebra Game Plan Vector Spaces Linear Transformations and Matrices Determinant
More informationLecture Notes for Math 414: Linear Algebra II Fall 2015, Michigan State University
Lecture Notes for Fall 2015, Michigan State University Matthew Hirn December 11, 2015 Beginning of Lecture 1 1 Vector Spaces What is this course about? 1. Understanding the structural properties of a wide
More information2 Eigenvectors and Eigenvalues in abstract spaces.
MA322 Sathaye Notes on Eigenvalues Spring 27 Introduction In these notes, we start with the definition of eigenvectors in abstract vector spaces and follow with the more common definition of eigenvectors
More informationMath 61CM - Solutions to homework 6
Math 61CM - Solutions to homework 6 Cédric De Groote November 5 th, 2018 Problem 1: (i) Give an example of a metric space X such that not all Cauchy sequences in X are convergent. (ii) Let X be a metric
More information(v, w) = arccos( < v, w >
MA322 Sathaye Notes on Inner Products Notes on Chapter 6 Inner product. Given a real vector space V, an inner product is defined to be a bilinear map F : V V R such that the following holds: For all v
More information3. Vector spaces 3.1 Linear dependence and independence 3.2 Basis and dimension. 5. Extreme points and basic feasible solutions
A. LINEAR ALGEBRA. CONVEX SETS 1. Matrices and vectors 1.1 Matrix operations 1.2 The rank of a matrix 2. Systems of linear equations 2.1 Basic solutions 3. Vector spaces 3.1 Linear dependence and independence
More informationMTH 5102 Linear Algebra Practice Final Exam April 26, 2016
Name (Last name, First name): MTH 5 Linear Algebra Practice Final Exam April 6, 6 Exam Instructions: You have hours to complete the exam. There are a total of 9 problems. You must show your work and write
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 informationIterative Methods for Linear Systems
Iterative Methods for Linear Systems 1. Introduction: Direct solvers versus iterative solvers In many applications we have to solve a linear system Ax = b with A R n n and b R n given. If n is large the
More informationTravis Schedler. Thurs, Oct 27, 2011 (version: Thurs, Oct 27, 1:00 PM)
Lecture 13: Proof of existence of upper-triangular matrices for complex linear transformations; invariant subspaces and block upper-triangular matrices for real linear transformations (1) Travis Schedler
More informationMath 18, Linear Algebra, Lecture C00, Spring 2017 Review and Practice Problems for Final Exam
Math 8, Linear Algebra, Lecture C, Spring 7 Review and Practice Problems for Final Exam. The augmentedmatrix of a linear system has been transformed by row operations into 5 4 8. Determine if the system
More informationChapter 5 Eigenvalues and Eigenvectors
Chapter 5 Eigenvalues and Eigenvectors Outline 5.1 Eigenvalues and Eigenvectors 5.2 Diagonalization 5.3 Complex Vector Spaces 2 5.1 Eigenvalues and Eigenvectors Eigenvalue and Eigenvector If A is a n n
More informationEAD 115. Numerical Solution of Engineering and Scientific Problems. David M. Rocke Department of Applied Science
EAD 115 Numerical Solution of Engineering and Scientific Problems David M. Rocke Department of Applied Science Taylor s Theorem Can often approximate a function by a polynomial The error in the approximation
More informationGaussian Elimination and Back Substitution
Jim Lambers MAT 610 Summer Session 2009-10 Lecture 4 Notes These notes correspond to Sections 31 and 32 in the text Gaussian Elimination and Back Substitution The basic idea behind methods for solving
More informationMath 443 Differential Geometry Spring Handout 3: Bilinear and Quadratic Forms This handout should be read just before Chapter 4 of the textbook.
Math 443 Differential Geometry Spring 2013 Handout 3: Bilinear and Quadratic Forms This handout should be read just before Chapter 4 of the textbook. Endomorphisms of a Vector Space This handout discusses
More informationSecond Order Elliptic PDE
Second Order Elliptic PDE T. Muthukumar tmk@iitk.ac.in December 16, 2014 Contents 1 A Quick Introduction to PDE 1 2 Classification of Second Order PDE 3 3 Linear Second Order Elliptic Operators 4 4 Periodic
More informationChapter 1: Systems of Linear Equations and Matrices
: Systems of Linear Equations and Matrices Multiple Choice Questions. Which of the following equations is linear? (A) x + 3x 3 + 4x 4 3 = 5 (B) 3x x + x 3 = 5 (C) 5x + 5 x x 3 = x + cos (x ) + 4x 3 = 7.
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 informationSimple Examples on Rectangular Domains
84 Chapter 5 Simple Examples on Rectangular Domains In this chapter we consider simple elliptic boundary value problems in rectangular domains in R 2 or R 3 ; our prototype example is the Poisson equation
More informationLinear Algebra Review. Vectors
Linear Algebra Review 9/4/7 Linear Algebra Review By Tim K. Marks UCSD Borrows heavily from: Jana Kosecka http://cs.gmu.edu/~kosecka/cs682.html Virginia de Sa (UCSD) Cogsci 8F Linear Algebra review Vectors
More information5.7 Cramer's Rule 1. Using Determinants to Solve Systems Assumes the system of two equations in two unknowns
5.7 Cramer's Rule 1. Using Determinants to Solve Systems Assumes the system of two equations in two unknowns (1) possesses the solution and provided that.. The numerators and denominators are recognized
More informationLinear Analysis Lecture 16
Linear Analysis Lecture 16 The QR Factorization Recall the Gram-Schmidt orthogonalization process. Let V be an inner product space, and suppose a 1,..., a n V are linearly independent. Define q 1,...,
More informationWe denote the derivative at x by DF (x) = L. With respect to the standard bases of R n and R m, DF (x) is simply the matrix of partial derivatives,
The derivative Let O be an open subset of R n, and F : O R m a continuous function We say F is differentiable at a point x O, with derivative L, if L : R n R m is a linear transformation such that, for
More informationSolution of Linear Equations
Solution of Linear Equations (Com S 477/577 Notes) Yan-Bin Jia Sep 7, 07 We have discussed general methods for solving arbitrary equations, and looked at the special class of polynomial equations A subclass
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 informationFundamentals of Engineering Analysis (650163)
Philadelphia University Faculty of Engineering Communications and Electronics Engineering Fundamentals of Engineering Analysis (6563) Part Dr. Omar R Daoud Matrices: Introduction DEFINITION A matrix is
More informationAbsolute value equations
Linear Algebra and its Applications 419 (2006) 359 367 www.elsevier.com/locate/laa Absolute value equations O.L. Mangasarian, R.R. Meyer Computer Sciences Department, University of Wisconsin, 1210 West
More informationThe goal of this chapter is to study linear systems of ordinary differential equations: dt,..., dx ) T
1 1 Linear Systems The goal of this chapter is to study linear systems of ordinary differential equations: ẋ = Ax, x(0) = x 0, (1) where x R n, A is an n n matrix and ẋ = dx ( dt = dx1 dt,..., dx ) T n.
More informationLECTURE 6: VECTOR SPACES II (CHAPTER 3 IN THE BOOK)
LECTURE 6: VECTOR SPACES II (CHAPTER 3 IN THE BOOK) In this lecture, F is a fixed field. One can assume F = R or C. 1. More about the spanning set 1.1. Let S = { v 1, v n } be n vectors in V, we have defined
More informationProblem Set (T) If A is an m n matrix, B is an n p matrix and D is a p s matrix, then show
MTH 0: Linear Algebra Department of Mathematics and Statistics Indian Institute of Technology - Kanpur Problem Set Problems marked (T) are for discussions in Tutorial sessions (T) If A is an m n matrix,
More informationDeterminants. Beifang Chen
Determinants Beifang Chen 1 Motivation Determinant is a function that each square real matrix A is assigned a real number, denoted det A, satisfying certain properties If A is a 3 3 matrix, writing A [u,
More informationClass notes: Approximation
Class notes: Approximation Introduction Vector spaces, linear independence, subspace The goal of Numerical Analysis is to compute approximations We want to approximate eg numbers in R or C vectors in R
More informationEE5120 Linear Algebra: Tutorial 3, July-Dec
EE5120 Linear Algebra: Tutorial 3, July-Dec 2017-18 1. Let S 1 and S 2 be two subsets of a vector space V such that S 1 S 2. Say True/False for each of the following. If True, prove it. If False, justify
More informationAlgebra II. Paulius Drungilas and Jonas Jankauskas
Algebra II Paulius Drungilas and Jonas Jankauskas Contents 1. Quadratic forms 3 What is quadratic form? 3 Change of variables. 3 Equivalence of quadratic forms. 4 Canonical form. 4 Normal form. 7 Positive
More informationCHAPTER 3 Further properties of splines and B-splines
CHAPTER 3 Further properties of splines and B-splines In Chapter 2 we established some of the most elementary properties of B-splines. In this chapter our focus is on the question What kind of functions
More informationApril 26, Applied mathematics PhD candidate, physics MA UC Berkeley. Lecture 4/26/2013. Jed Duersch. Spd matrices. Cholesky decomposition
Applied mathematics PhD candidate, physics MA UC Berkeley April 26, 2013 UCB 1/19 Symmetric positive-definite I Definition A symmetric matrix A R n n is positive definite iff x T Ax > 0 holds x 0 R n.
More informationMATH 115A: SAMPLE FINAL SOLUTIONS
MATH A: SAMPLE FINAL SOLUTIONS JOE HUGHES. Let V be the set of all functions f : R R such that f( x) = f(x) for all x R. Show that V is a vector space over R under the usual addition and scalar multiplication
More informationWhat is it we are looking for in these algorithms? We want algorithms that are
Fundamentals. Preliminaries The first question we want to answer is: What is computational mathematics? One possible definition is: The study of algorithms for the solution of computational problems in
More informationMATH 304 Linear Algebra Lecture 10: Linear independence. Wronskian.
MATH 304 Linear Algebra Lecture 10: Linear independence. Wronskian. Spanning set Let S be a subset of a vector space V. Definition. The span of the set S is the smallest subspace W V that contains S. If
More informationSystem of Linear Equations
Chapter 7 - S&B Gaussian and Gauss-Jordan Elimination We will study systems of linear equations by describing techniques for solving such systems. The preferred solution technique- Gaussian elimination-
More informationMATH 3511 Lecture 1. Solving Linear Systems 1
MATH 3511 Lecture 1 Solving Linear Systems 1 Dmitriy Leykekhman Spring 2012 Goals Review of basic linear algebra Solution of simple linear systems Gaussian elimination D Leykekhman - MATH 3511 Introduction
More informationMatrix Algebra for Engineers Jeffrey R. Chasnov
Matrix Algebra for Engineers Jeffrey R. Chasnov The Hong Kong University of Science and Technology The Hong Kong University of Science and Technology Department of Mathematics Clear Water Bay, Kowloon
More informationMAT Linear Algebra Collection of sample exams
MAT 342 - Linear Algebra Collection of sample exams A-x. (0 pts Give the precise definition of the row echelon form. 2. ( 0 pts After performing row reductions on the augmented matrix for a certain system
More informationLinear Algebra Review
Chapter 1 Linear Algebra Review It is assumed that you have had a course in linear algebra, and are familiar with matrix multiplication, eigenvectors, etc. I will review some of these terms here, but quite
More informationChapter 4: Interpolation and Approximation. October 28, 2005
Chapter 4: Interpolation and Approximation October 28, 2005 Outline 1 2.4 Linear Interpolation 2 4.1 Lagrange Interpolation 3 4.2 Newton Interpolation and Divided Differences 4 4.3 Interpolation Error
More information2: LINEAR TRANSFORMATIONS AND MATRICES
2: LINEAR TRANSFORMATIONS AND MATRICES STEVEN HEILMAN Contents 1. Review 1 2. Linear Transformations 1 3. Null spaces, range, coordinate bases 2 4. Linear Transformations and Bases 4 5. Matrix Representation,
More information