AMS526: Numerical Analysis I (Numerical Linear Algebra)
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1 AMS526: Numerical Analysis I (Numerical Linear Algebra) Lecture 1: Course Overview; Matrix Multiplication Xiangmin Jiao Stony Brook University Xiangmin Jiao Numerical Analysis I 1 / 21
2 Outline 1 Course Overview 2 Matrix-Vector Multiplication 3 Matrix-Matrix Multiplication Xiangmin Jiao Numerical Analysis I 2 / 21
3 Course Description What is numerical linear algebra? Solving linear algebra problems using efficient algorithms on computers Topics: Direct and iterative methods for solving simultaneous linear equations. Matrix factorization, conditioning, stability, sparsity, and efficiency. Computation of eigenvalues and eigenvectors. Singular value decomposition Required Textbook (adopted in Fall 2012) David S. Watkins, Fundamentals of Matrix Computations, 3rd edition, Wiley 2010, ISBN Xiangmin Jiao Numerical Analysis I 3 / 21
4 Prerequisite Prerequisite/Co-requisite: AMS 510 (linear algebra portion) or equivalent undergraduate-level linear algebra course. Familiarity with following concepts is assumed: Vector spaces, Gaussian elimination, Gram-Schmidt orthogonalization, and eigenvalues/eigenvectors AMS 595 (co-requisite for students without programming experience in C) This MUST NOT be your first course in linear algebra, or you will get lost To review fundamental concepts of linear algebra, see textbook such as Gilbert Strang, Linear Algebra and Its Applications, 4th Edition, Brooks Cole, Xiangmin Jiao Numerical Analysis I 4 / 21
5 Why Learn Numerical Linear Algebra? Numerical linear algebra is foundation of scientific computations Many problems ultimately reduce to linear algebra concepts or algorithms, either analytical or computational Examples: Finite-element analysis, data fitting, PageRank (Google) Focus of this course: Fundamental concepts, algorithms, and programming Xiangmin Jiao Numerical Analysis I 5 / 21
6 Course Outline Gaussian elimination and its variants (2.5 weeks) Sensitivity of linear systems (2.5 weeks) Linear least squares problem (1.5 weeks) Singular value decomposition (1.5 weeks) Eigenvalues and eigenvectors (3 weeks) Iterative methods for linear systems (2 weeks) Course webpage: Note: Course schedule online is tentative and is subject to change. Xiangmin Jiao Numerical Analysis I 6 / 21
7 Some Notes of Textbook and Notation We will cover all chapters of third-edition of Watkins, Fundamentals of Matrix Computations Previously this course used Trefethen and Bau, Numerical Linear Algebra, SIAM, 1997, which is a very insightful book Why change to Fundamentals of Matrix Computations (FMC)? FMC takes more progressive approach, so easier for students Starts with linear systems instead of SVD; section on iterative methods is easier to follow Starts with real numbers and switch to complex numbers later Has more examples FMC has more coverage on sparse matrices, which are important FMC uses non-standard names sometimes, so we sometimes will use different terminology from textbook Some explanations in lectures will still come from Trefethen and Bau Xiangmin Jiao Numerical Analysis I 7 / 21
8 Course Policy Assignments (written or programming) Assignments are due in class one to two weeks after assigned You can discuss course materials and homework problems with others, but you must write your answers completely independently Do NOT copy solutions from any source. Do NOT share your solutions to others Exams and tests Grading All exams are closed-book However, one-page cheat sheet is allowed Assignments: 30% Two tests: 40% Final exam: 30% Xiangmin Jiao Numerical Analysis I 8 / 21
9 Outline 1 Course Overview 2 Matrix-Vector Multiplication 3 Matrix-Matrix Multiplication Xiangmin Jiao Numerical Analysis I 9 / 21
10 Definition Suppose A is n m, x is size m, and b is size n a 11 a a 1m x 1 a 21 a a 2m A =......, x = x 2., b = a n1 a n2... a nm x m All entries belong to R, the field of real numbers. Space of m-vectors is R m, and the space of n m matrices is R n m Matrix-vector product b = Ax is defined as m b i = a ij x j j=1 The map x Ax is linear, which means that for any x, y R m and any α R A(x + y) = Ax + Ay, A(αx) = αax. Xiangmin Jiao Numerical Analysis I 10 / 21 b 1 b 2. b n
11 Linear Combination Let a j denote jth column of matrix A Alternatively, matrix-vector product can be viewed as a 11 a 12 a 21 a 22 b = Ax = m x j a j = x 1 j=1. a n1 + x 2. a n2 i.e., b is a linear combination of column vectors of A Two different views of matrix-vector products: + + x m 1 b i = m j=1 a ijx j : A acts on x to produce b; scalar operations; 2 b = m j=1 x ja j : x acts on A to produce b; vector operations. If A is n m, Ax can be viewed as a mapping from R m to R n a 1m a 2m. a nm Xiangmin Jiao Numerical Analysis I 11 / 21
12 Pseudo-Code for Matrix-Vector Multiplication Pseudo-code for b = Ax for i = 1 to n b i 0 for j = 1 to m b i b i + a ij x j Pseudo-code are not real code, in that they do not run on any computer. They are human readable, but they should be straightforward to convert into real computer codes in any programming language (e.g., C, FORTRAN, MATLAB, etc.) There is not a standard convention for pseudo-code Xiangmin Jiao Numerical Analysis I 12 / 21
13 Flop Count It is important to assess the efficiency of algorithms. But how? We could implement different algorithms and do head-to-head comparison, but implementation details can affect true performance We could estimate cost of all operations, but it is very tedious Relatively simple and effective approach is to estimate amount of floating-point operations, or flops, and focus on asymptotic analysis as sizes of matrices approach infinity Idealization Count each operation +,,, /, and as one flop, and make no distinction of real and complex numbers. This estimation is very crude, since it omits data movement in memory, which is non-negligible on modern computer architectures Matrix-vector product requires about 2mn flops, because b i b i + a ij x j requires two flops Suppose m = n, it takes quadratic time in n (or O(n 2 )) Xiangmin Jiao Numerical Analysis I 13 / 21
14 Outline 1 Course Overview 2 Matrix-Vector Multiplication 3 Matrix-Matrix Multiplication Xiangmin Jiao Numerical Analysis I 14 / 21
15 Matrix-Matrix Multiplication If A is n m and X is m p, then B = AX is n p, with entries defined by m b ij = a ik x kj. k=1 Written in columns, we have m b j = Ax j = x kj a k. k=1 In other word, jth column of B is A times jth column of X Xiangmin Jiao Numerical Analysis I 15 / 21
16 Pseudo-Code for Matrix-Matrix Multiplication Pseudo-code for B = AX for i = 1 to n for j = 1 to p b ij 0 for k = 1 to m b ij b ij + a ik x kj Matrix-matrix multiplication requires about 2mnp flops, because b ij b ij + a ik x kj requires two flops Suppose m = n = p, it takes cubic time in n (or O(n 3 )) Xiangmin Jiao Numerical Analysis I 16 / 21
17 Special Cases Matrix-Matrix Multiplication Vector is a special case of a matrix Matrix-vector multiplication is a special case of matrix-matrix Multiplication Other special cases of vector-vector multiplication Inner product: u T v (where u R n and v R n ) Outer product: uv T (where u R n and v R m ) Xiangmin Jiao Numerical Analysis I 17 / 21
18 Block Matrices Matrix can be partitioned into blocks. For example, A R n m and X R m p can be partitioned into A = m 1 m 2 [ A11 A 12 A 21 A 22 Then B = AX can be written as p ] 1 p [ 2 n1 X11 X, X = 12 n 2 X 21 X 22 ] m1 m 2, p 1 p 2 [ B11 B 12 B 21 B 22 ] [ n1 A11 A = 12 n 2 A 21 A 22 ] [ X11 X 12 X 21 X 22 ] where B ij = A i1 X 1j + A i2 X 2j, i, j = 1, 2. More generally, the matrix can be decomposed into more than two block rows/columns Xiangmin Jiao Numerical Analysis I 18 / 21
19 Block Matrix Operators Suppose A has s 1 block rows and s 2 block columns, and X has s 2 block rows and s 3 block columns. Matrix-matrix multiplication can be rewritten using block matrix operations: Pseudo-code for B = AX using block-matrix operators for i = 1 to s 1 for j = 1 to s 3 B ij 0 for k = 1 to s 2 B ij B ij + A ik X kj Block matrix operations do not reduce flop counts, but they can improve performance because of improved cache performance and reduced data movement on modern computer architecture, by fitting matrix blocks into cache at once Xiangmin Jiao Numerical Analysis I 19 / 21
20 Fast Matrix-Matrix Multiplication Multiplying two n n matrices requires 2n 3 flops using inner-product-based algorithm Is this optimal? Xiangmin Jiao Numerical Analysis I 20 / 21
21 Fast Matrix-Matrix Multiplication Multiplying two n n matrices requires 2n 3 flops using inner-product-based algorithm Is this optimal? Strassen s method requires O(n s ) flops, where s = log It is recursive algorithm applied to matrices of size 2 k 2 k For matrices of sizes not of 2 k 2 k, fill missing rows and columns with zeros Asymptotically faster algorithm, due to D. Coppersmith and S. Winograd (1990), requires O(n ) flops Asymptotically fastest algorithm currently known, due to V. Williams (2011), requires O(n ) flops Xiangmin Jiao Numerical Analysis I 20 / 21
22 Matrix-Matrix Multiplication in Practice In practice, inner-product-based algorithm is almost always used Earlier authors estimated that Strassen s algorithm is faster for matrices with widths of n 100 for optimized implementations Compared to a highly optimized traditional multiplication on current architectures, Strassen s algorithm is not faster unless n 1000, and the benefit is marginal for matrix sizes of several thousand Coppersmith and Winograd s algorithm has a huge constant C in front of it, so it is never fast enough in practice for realistic problem sizes The asymptotically faster methods are less accurate than standard multiplication when using floating-point numbers (more prone to rounding errors and cancellation errors) Lower-complexity algorithms are sometimes used to prove theoretical time bounds of other algorithms For sparse matrices, efficient matrix-matrix multiplication should take advantage of sparsity (see SMMP for further reading) Xiangmin Jiao Numerical Analysis I 21 / 21
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