Scientific Computing: An Introductory Survey


 Felix Singleton
 1 years ago
 Views:
Transcription
1 Scientific Computing: An Introductory Survey Chapter 3 Linear Least Squares Prof. Michael T. Heath Department of Computer Science University of Illinois at UrbanaChampaign Copyright c Reproduction permitted for noncommercial, educational use only. Michael T. Heath Scientific Computing 1 / 61
2 Outline Least Squares Data Fitting 1 Least Squares Data Fitting 2 3 Michael T. Heath Scientific Computing 2 / 61
3 Method of Least Squares Least Squares Data Fitting Measurement errors are inevitable in observational and experimental sciences Errors can be smoothed out by averaging over many cases, i.e., taking more measurements than are strictly necessary to determine parameters of system Resulting system is overdetermined, so usually there is no exact solution In effect, higher dimensional data are projected into lower dimensional space to suppress irrelevant detail Such projection is most conveniently accomplished by method of least squares Michael T. Heath Scientific Computing 3 / 61
4 Linear Least Squares Least Squares Data Fitting For linear problems, we obtain overdetermined linear system Ax = b, with m n matrix A, m > n System is better written Ax = b, since equality is usually not exactly satisfiable when m > n Least squares solution x minimizes squared Euclidean norm of residual vector r = b Ax, min r 2 x 2 = min b Ax 2 x 2 Michael T. Heath Scientific Computing 4 / 61
5 Data Fitting Least Squares Data Fitting Least Squares Data Fitting Given m data points (t i, y i ), find nvector x of parameters that gives best fit to model function f(t, x), min x m (y i f(t i, x)) 2 i=1 Problem is linear if function f is linear in components of x, f(t, x) = x 1 φ 1 (t) + x 2 φ 2 (t) + + x n φ n (t) where functions φ j depend only on t Problem can be written in matrix form as Ax = b, with a ij = φ j (t i ) and b i = y i Michael T. Heath Scientific Computing 5 / 61
6 Data Fitting Least Squares Data Fitting Least Squares Data Fitting Polynomial fitting f(t, x) = x 1 + x 2 t + x 3 t x n t n 1 is linear, since polynomial linear in coefficients, though nonlinear in independent variable t Fitting sum of exponentials f(t, x) = x 1 e x 2t + + x n 1 e xnt is example of nonlinear problem For now, we will consider only linear least squares problems Michael T. Heath Scientific Computing 6 / 61
7 Example: Data Fitting Least Squares Data Fitting Fitting quadratic polynomial to five data points gives linear least squares problem 1 t 1 t t 2 t 2 y 1 2 x 1 y 2 Ax = 1 t 3 t 2 3 x 2 = y 3 1 t 4 t 2 4 x 3 y 4 = b 1 t 5 t 2 5 y 5 Matrix whose columns (or rows) are successive powers of independent variable is called Vandermonde matrix Michael T. Heath Scientific Computing 7 / 61
8 Example, continued Least Squares Data Fitting For data t y overdetermined 5 3 linear system is x Ax = x 2 x 3 = = b 2.0 Solution, which we will see later how to compute, is x = [ ] T so approximating polynomial is p(t) = t + 1.4t 2 Michael T. Heath Scientific Computing 8 / 61
9 Example, continued Least Squares Data Fitting Resulting curve and original data points are shown in graph < interactive example > Michael T. Heath Scientific Computing 9 / 61
10 Existence and Uniqueness Existence and Uniqueness Orthogonality Conditioning Linear least squares problem Ax = b always has solution Solution is unique if, and only if, columns of A are linearly independent, i.e., rank(a) = n, where A is m n If rank(a) < n, then A is rankdeficient, and solution of linear least squares problem is not unique For now, we assume A has full column rank n Michael T. Heath Scientific Computing 10 / 61
11 Existence and Uniqueness Orthogonality Conditioning To minimize squared Euclidean norm of residual vector r 2 2 = r T r = (b Ax) T (b Ax) = b T b 2x T A T b + x T A T Ax take derivative with respect to x and set it to 0, 2A T Ax 2A T b = 0 which reduces to n n linear system of normal equations A T Ax = A T b Michael T. Heath Scientific Computing 11 / 61
12 Orthogonality Least Squares Data Fitting Existence and Uniqueness Orthogonality Conditioning Vectors v 1 and v 2 are orthogonal if their inner product is zero, v T 1 v 2 = 0 Space spanned by columns of m n matrix A, span(a) = {Ax : x R n }, is of dimension at most n If m > n, b generally does not lie in span(a), so there is no exact solution to Ax = b Vector y = Ax in span(a) closest to b in 2norm occurs when residual r = b Ax is orthogonal to span(a), 0 = A T r = A T (b Ax) again giving system of normal equations A T Ax = A T b Michael T. Heath Scientific Computing 12 / 61
13 Orthogonality, continued Existence and Uniqueness Orthogonality Conditioning Geometric relationships among b, r, and span(a) are shown in diagram Michael T. Heath Scientific Computing 13 / 61
14 Orthogonal Projectors Existence and Uniqueness Orthogonality Conditioning Matrix P is orthogonal projector if it is idempotent (P 2 = P ) and symmetric (P T = P ) Orthogonal projector onto orthogonal complement span(p ) is given by P = I P For any vector v, v = (P + (I P )) v = P v + P v For least squares problem Ax = b, if rank(a) = n, then P = A(A T A) 1 A T is orthogonal projector onto span(a), and b = P b + P b = Ax + (b Ax) = y + r Michael T. Heath Scientific Computing 14 / 61
15 Existence and Uniqueness Orthogonality Conditioning Pseudoinverse and Condition Number Nonsquare m n matrix A has no inverse in usual sense If rank(a) = n, pseudoinverse is defined by and condition number by A + = (A T A) 1 A T cond(a) = A 2 A + 2 By convention, cond(a) = if rank(a) < n Just as condition number of square matrix measures closeness to singularity, condition number of rectangular matrix measures closeness to rank deficiency Least squares solution of Ax = b is given by x = A + b Michael T. Heath Scientific Computing 15 / 61
16 Sensitivity and Conditioning Existence and Uniqueness Orthogonality Conditioning Sensitivity of least squares solution to Ax = b depends on b as well as A Define angle θ between b and y = Ax by cos(θ) = y 2 b 2 = Ax 2 b 2 Bound on perturbation x in solution x due to perturbation b in b is given by x 2 x 2 cond(a) 1 b 2 cos(θ) b 2 Michael T. Heath Scientific Computing 16 / 61
17 Existence and Uniqueness Orthogonality Conditioning Sensitivity and Conditioning, contnued Similarly, for perturbation E in matrix A, x 2 x 2 ( [cond(a)] 2 tan(θ) + cond(a) ) E 2 A 2 Condition number of least squares solution is about cond(a) if residual is small, but can be squared or arbitrarily worse for large residual Michael T. Heath Scientific Computing 17 / 61
18 Method If m n matrix A has rank n, then symmetric n n matrix A T A is positive definite, so its Cholesky factorization A T A = LL T can be used to obtain solution x to system of normal equations A T Ax = A T b which has same solution as linear least squares problem Ax = b Normal equations method involves transformations rectangular square triangular Michael T. Heath Scientific Computing 18 / 61
19 Example: Method For polynomial datafitting example given previously, normal equations method gives A T A = = , A T b = = Michael T. Heath Scientific Computing 19 / 61
20 Example, continued Cholesky factorization of symmetric positive definite matrix A T A gives A T A = = = LL T Solving lower triangular system Lz = A T b by forwardsubstitution gives z = [ ] T Solving upper triangular system L T x = z by backsubstitution gives x = [ ] T Michael T. Heath Scientific Computing 20 / 61
21 Shortcomings of Information can be lost in forming A T A and A T b For example, take 1 1 A = ɛ 0 0 ɛ where ɛ is positive number smaller than ɛ mach Then in floatingpoint arithmetic [ ] A T 1 + ɛ 2 1 A = ɛ 2 = which is singular [ ] Sensitivity of solution is also worsened, since cond(a T A) = [cond(a)] 2 Michael T. Heath Scientific Computing 21 / 61
22 Augmented System Method Definition of residual together with orthogonality requirement give (m + n) (m + n) augmented system [ ] [ ] [ ] I A r b A T = O x 0 Augmented system is not positive definite, is larger than original system, and requires storing two copies of A But it allows greater freedom in choosing pivots in computing LDL T or LU factorization Michael T. Heath Scientific Computing 22 / 61
23 Augmented System Method, continued Introducing scaling parameter α gives system [ ] [ ] [ ] αi A r/α b A T = O x 0 which allows control over relative weights of two subsystems in choosing pivots Reasonable rule of thumb is to take α = max a ij /1000 i,j Augmented system is sometimes useful, but is far from ideal in work and storage required Michael T. Heath Scientific Computing 23 / 61
24 Orthogonal Transformations We seek alternative method that avoids numerical difficulties of normal equations We need numerically robust transformation that produces easier problem without changing solution What kind of transformation leaves least squares solution unchanged? Square matrix Q is orthogonal if Q T Q = I Multiplication of vector by orthogonal matrix preserves Euclidean norm Qv 2 2 = (Qv) T Qv = v T Q T Qv = v T v = v 2 2 Thus, multiplying both sides of least squares problem by orthogonal matrix does not change its solution Michael T. Heath Scientific Computing 24 / 61
25 Triangular Least Squares Problems As with square linear systems, suitable target in simplifying least squares problems is triangular form Upper triangular overdetermined (m > n) least squares problem has form [ ] [ ] R x O b1 = where R is n n upper triangular and b is partitioned similarly b 2 Residual is r 2 2 = b 1 Rx b Michael T. Heath Scientific Computing 25 / 61
26 Triangular Least Squares Problems, continued We have no control over second term, b 2 2 2, but first term becomes zero if x satisfies n n triangular system Rx = b 1 which can be solved by backsubstitution Resulting x is least squares solution, and minimum sum of squares is r 2 2 = b So our strategy is to transform general least squares problem to triangular form using orthogonal transformation so that least squares solution is preserved Michael T. Heath Scientific Computing 26 / 61
27 QR Factorization Given m n matrix A, with m > n, we seek m m orthogonal matrix Q such that [ ] R A = Q O where R is n n and upper triangular Linear least squares problem Ax = b is then transformed into triangular least squares problem [ ] [ ] Q T R Ax = x O c1 = = Q T b which has same solution, since [ ] r 2 2 = b Ax 2 R 2 = b Q x 2 2 = Q T b O c 2 [ ] R x 2 2 O Michael T. Heath Scientific Computing 27 / 61
28 Orthogonal Bases If we partition m m orthogonal matrix Q = [Q 1 Q 2 ], where Q 1 is m n, then [ ] [ ] R R A = Q = [Q O 1 Q 2 ] = Q O 1 R is called reduced QR factorization of A Columns of Q 1 are orthonormal basis for span(a), and columns of Q 2 are orthonormal basis for span(a) Q 1 Q T 1 is orthogonal projector onto span(a) Solution to least squares problem Ax = b is given by solution to square system Q T 1 Ax = Rx = c 1 = Q T 1 b Michael T. Heath Scientific Computing 28 / 61
29 Computing QR Factorization To compute QR factorization of m n matrix A, with m > n, we annihilate subdiagonal entries of successive columns of A, eventually reaching upper triangular form Similar to LU factorization by Gaussian elimination, but use orthogonal transformations instead of elementary elimination matrices Possible methods include Householder transformations Givens rotations GramSchmidt orthogonalization Michael T. Heath Scientific Computing 29 / 61
30 Householder Transformations Householder transformation has form H = I 2 vvt v T v for nonzero vector v H is orthogonal and symmetric: H = H T = H 1 Given vector a, we want to choose v so that α Ha = = α = αe 1 Substituting into formula for H, we can take v = a αe 1 and α = ± a 2, with sign chosen to avoid cancellation Michael T. Heath Scientific Computing 30 / 61
31 Example: Householder Transformation If a = [ ] T, then we take α v = a αe 1 = 1 α 0 = where α = ± a 2 = ±3 Since a 1 is positive, we choose negative sign for α to avoid cancellation, so v = 1 0 = To confirm that transformation works, 2 Ha = a 2 vt a v T v v = = < interactive example > Michael T. Heath Scientific Computing 31 / 61
32 Householder QR Factorization To compute QR factorization of A, use Householder transformations to annihilate subdiagonal entries of each successive column Each Householder transformation is applied to entire matrix, but does not affect prior columns, so zeros are preserved In applying Householder transformation H to arbitrary vector u, ) ( ) Hu = (I 2 vvt v T u = u 2 vt u v v T v v which is much cheaper than general matrixvector multiplication and requires only vector v, not full matrix H Michael T. Heath Scientific Computing 32 / 61
33 Householder QR Factorization, continued Process just described produces factorization [ ] R H n H 1 A = O where R is n n and upper triangular [ ] R If Q = H 1 H n, then A = Q O To preserve solution of linear least squares problem, righthand side b is transformed by same sequence of Householder transformations [ ] R Then solve triangular least squares problem x O = Q T b Michael T. Heath Scientific Computing 33 / 61
34 Householder QR Factorization, continued For solving linear least squares problem, product Q of Householder transformations need not be formed explicitly R can be stored in upper triangle of array initially containing A Householder vectors v can be stored in (now zero) lower triangular portion of A (almost) Householder transformations most easily applied in this form anyway Michael T. Heath Scientific Computing 34 / 61
35 Example: Householder QR Factorization For polynomial datafitting example given previously, with A = , b = Householder vector v 1 for annihilating subdiagonal entries of first column of A is v 1 = = Michael T. Heath Scientific Computing 35 / 61
36 Example, continued Applying resulting Householder transformation H 1 yields transformed matrix and righthand side H 1 A = , H b = Householder vector v 2 for annihilating subdiagonal entries of second column of H 1 A is v 2 = = Michael T. Heath Scientific Computing 36 / 61
37 Example, continued Applying resulting Householder transformation H 2 yields H 2 H 1 A = , H H 1 b = Householder vector v 3 for annihilating subdiagonal entries of third column of H 2 H 1 A is v 3 = = Michael T. Heath Scientific Computing 37 / 61
38 Example, continued Applying resulting Householder transformation H 3 yields H 3 H 2 H 1 A = , H H 2 H 1 b = Now solve upper triangular system Rx = c 1 by backsubstitution to obtain x = [ ] T < interactive example > Michael T. Heath Scientific Computing 38 / 61
39 Givens Rotations Givens rotations introduce zeros one at a time Given vector [ a 1 ] T a 2, choose scalars c and s so that [ ] [ ] [ ] c s a1 α = s c 0 a 2 with c 2 + s 2 = 1, or equivalently, α = a a2 2 Previous equation can be rewritten [ ] [ ] [ ] a1 a 2 c α = a 2 a 1 s 0 Gaussian elimination yields triangular system [ ] [ ] [ ] a1 a 2 c α 0 a 1 a 2 2 /a = 1 s αa 2 /a 1 Michael T. Heath Scientific Computing 39 / 61
40 Givens Rotations, continued Backsubstitution then gives s = αa 2 a a2 2 and c = αa 1 a a2 2 Finally, c 2 + s 2 = 1, or α = a a2 2, implies c = a 1 a a 2 2 and s = a 2 a a 2 2 Michael T. Heath Scientific Computing 40 / 61
41 Example: Givens Rotation Let a = [ 4 3 ] T To annihilate second entry we compute cosine and sine c = a 1 a a 2 2 = 4 5 = 0.8 and s = a 2 a a 2 2 = 3 5 = 0.6 Rotation is then given by [ ] c s G = = s c [ ] To confirm that rotation works, [ ] [ ] Ga = = [ ] 5 0 Michael T. Heath Scientific Computing 41 / 61
42 Givens QR Factorization More generally, to annihilate selected component of vector in n dimensions, rotate target component with another component a 1 a 1 0 c 0 s 0 a 2 α s 0 c a 3 a 4 a 5 = a 3 0 a 5 By systematically annihilating successive entries, we can reduce matrix to upper triangular form using sequence of Givens rotations Each rotation is orthogonal, so their product is orthogonal, producing QR factorization Michael T. Heath Scientific Computing 42 / 61
43 Givens QR Factorization Straightforward implementation of Givens method requires about 50% more work than Householder method, and also requires more storage, since each rotation requires two numbers, c and s, to define it These disadvantages can be overcome, but requires more complicated implementation Givens can be advantageous for computing QR factorization when many entries of matrix are already zero, since those annihilations can then be skipped < interactive example > Michael T. Heath Scientific Computing 43 / 61
44 GramSchmidt Orthogonalization Given vectors a 1 and a 2, we seek orthonormal vectors q 1 and q 2 having same span This can be accomplished by subtracting from second vector its projection onto first vector and normalizing both resulting vectors, as shown in diagram < interactive example > Michael T. Heath Scientific Computing 44 / 61
45 GramSchmidt Orthogonalization Process can be extended to any number of vectors a 1,..., a k, orthogonalizing each successive vector against all preceding ones, giving classical GramSchmidt procedure for k = 1 to n q k = a k for j = 1 to k 1 r jk = q T j a k q k = q k r jk q j end r kk = q k 2 q k = q k /r kk end Resulting q k and r jk form reduced QR factorization of A Michael T. Heath Scientific Computing 45 / 61
46 Modified GramSchmidt Classical GramSchmidt procedure often suffers loss of orthogonality in finiteprecision Also, separate storage is required for A, Q, and R, since original a k are needed in inner loop, so q k cannot overwrite columns of A Both deficiencies are improved by modified GramSchmidt procedure, with each vector orthogonalized in turn against all subsequent vectors, so q k can overwrite a k Michael T. Heath Scientific Computing 46 / 61
47 Modified GramSchmidt QR Factorization Modified GramSchmidt algorithm for k = 1 to n r kk = a k 2 q k = a k /r kk for j = k + 1 to n r kj = q T k a j a j = a j r kj q k end end < interactive example > Michael T. Heath Scientific Computing 47 / 61
48 Rank Deficiency If rank(a) < n, then QR factorization still exists, but yields singular upper triangular factor R, and multiple vectors x give minimum residual norm Common practice selects minimum residual solution x having smallest norm Can be computed by QR factorization with column pivoting or by singular value decomposition () Rank of matrix is often not clear cut in practice, so relative tolerance is used to determine rank Michael T. Heath Scientific Computing 48 / 61
49 Example: Near Rank Deficiency Consider 3 2 matrix A = Computing QR factorization, [ ] R = R is extremely close to singular (exactly singular to 3digit accuracy of problem statement) If R is used to solve linear least squares problem, result is highly sensitive to perturbations in righthand side For practical purposes, rank(a) = 1 rather than 2, because columns are nearly linearly dependent Michael T. Heath Scientific Computing 49 / 61
50 QR with Column Pivoting Instead of processing columns in natural order, select for reduction at each stage column of remaining unreduced submatrix having maximum Euclidean norm If rank(a) = k < n, then after k steps, norms of remaining unreduced columns will be zero (or negligible in finiteprecision arithmetic) below row k Yields orthogonal factorization of form [ ] Q T R S AP = O O where R is k k, upper triangular, and nonsingular, and permutation matrix P performs column interchanges Michael T. Heath Scientific Computing 50 / 61
51 QR with Column Pivoting, continued Basic solution to least squares problem Ax = b can now be computed by solving triangular system Rz = c 1, where c 1 contains first k components of Q T b, and then taking [ ] z x = P 0 Minimumnorm solution can be computed, if desired, at expense of additional processing to annihilate S rank(a) is usually unknown, so rank is determined by monitoring norms of remaining unreduced columns and terminating factorization when maximum value falls below chosen tolerance < interactive example > Michael T. Heath Scientific Computing 51 / 61
52 Singular Value Decomposition Singular value decomposition () of m n matrix A has form A = UΣV T where U is m m orthogonal matrix, V is n n orthogonal matrix, and Σ is m n diagonal matrix, with { 0 for i j σ ij = σ i 0 for i = j Diagonal entries σ i, called singular values of A, are usually ordered so that σ 1 σ 2 σ n Columns u i of U and v i of V are called left and right singular vectors Michael T. Heath Scientific Computing 52 / 61
53 Example: Least Squares Data Fitting of A = is given by UΣV T = < interactive example > Michael T. Heath Scientific Computing 53 / 61
54 Applications of Minimum norm solution to Ax = b is given by x = σ i 0 u T i b v i σ i For illconditioned or rank deficient problems, small singular values can be omitted from summation to stabilize solution Euclidean matrix norm : A 2 = σ max Euclidean condition number of matrix : cond(a) = σ max σ min Rank of matrix : number of nonzero singular values Michael T. Heath Scientific Computing 54 / 61
55 Pseudoinverse Least Squares Data Fitting Define pseudoinverse of scalar σ to be 1/σ if σ 0, zero otherwise Define pseudoinverse of (possibly rectangular) diagonal matrix by transposing and taking scalar pseudoinverse of each entry Then pseudoinverse of general real m n matrix A is given by A + = V Σ + U T Pseudoinverse always exists whether or not matrix is square or has full rank If A is square and nonsingular, then A + = A 1 In all cases, minimumnorm solution to Ax = b is given by x = A + b Michael T. Heath Scientific Computing 55 / 61
56 Orthogonal Bases of matrix, A = UΣV T, provides orthogonal bases for subspaces relevant to A Columns of U corresponding to nonzero singular values form orthonormal basis for span(a) Remaining columns of U form orthonormal basis for orthogonal complement span(a) Columns of V corresponding to zero singular values form orthonormal basis for null space of A Remaining columns of V form orthonormal basis for orthogonal complement of null space of A Michael T. Heath Scientific Computing 56 / 61
57 LowerRank Matrix Approximation Another way to write is A = UΣV T = σ 1 E 1 + σ 2 E σ n E n with E i = u i v T i E i has rank 1 and can be stored using only m + n storage locations Product E i x can be computed using only m + n multiplications Condensed approximation to A is obtained by omitting from summation terms corresponding to small singular values Approximation using k largest singular values is closest matrix of rank k to A Approximation is useful in image processing, data compression, information retrieval, cryptography, etc. < interactive example > Michael T. Heath Scientific Computing 57 / 61
58 Total Least Squares Ordinary least squares is applicable when righthand side b is subject to random error but matrix A is known accurately When all data, including A, are subject to error, then total least squares is more appropriate Total least squares minimizes orthogonal distances, rather than vertical distances, between model and data Total least squares solution can be computed from of [A, b] Michael T. Heath Scientific Computing 58 / 61
59 Comparison of Methods Forming normal equations matrix A T A requires about n 2 m/2 multiplications, and solving resulting symmetric linear system requires about n 3 /6 multiplications Solving least squares problem using Householder QR factorization requires about mn 2 n 3 /3 multiplications If m n, both methods require about same amount of work If m n, Householder QR requires about twice as much work as normal equations Cost of is proportional to mn 2 + n 3, with proportionality constant ranging from 4 to 10, depending on algorithm used Michael T. Heath Scientific Computing 59 / 61
60 Comparison of Methods, continued Normal equations method produces solution whose relative error is proportional to [cond(a)] 2 Required Cholesky factorization can be expected to break down if cond(a) 1/ ɛ mach or worse Householder method produces solution whose relative error is proportional to cond(a) + r 2 [cond(a)] 2 which is best possible, since this is inherent sensitivity of solution to least squares problem Householder method can be expected to break down (in backsubstitution phase) only if cond(a) 1/ɛ mach or worse Michael T. Heath Scientific Computing 60 / 61
61 Comparison of Methods, continued Householder is more accurate and more broadly applicable than normal equations These advantages may not be worth additional cost, however, when problem is sufficiently well conditioned that normal equations provide sufficient accuracy For rankdeficient or nearly rankdeficient problems, Householder with column pivoting can produce useful solution when normal equations method fails outright is even more robust and reliable than Householder, but substantially more expensive Michael T. Heath Scientific Computing 61 / 61
Computational Methods. Least Squares Approximation/Optimization
Computational Methods Least Squares Approximation/Optimization Manfred Huber 2010 1 Least Squares Least squares methods are aimed at finding approximate solutions when no precise solution exists Find the
More informationNUMERICALLY EFFICIENT METHODS FOR SOLVING LEAST SQUARES PROBLEMS
NUMERICALLY EFFICIENT METHODS FOR SOLVING LEAST SQUARES PROBLEMS DO Q LEE Abstract. Computing the solution to Least Squares Problems is of great importance in a wide range of fields ranging from numerical
More informationCS3220 Lecture Notes: QR factorization and orthogonal transformations
CS3220 Lecture Notes: QR factorization and orthogonal transformations Steve Marschner Cornell University 11 March 2009 In this lecture I ll talk about orthogonal matrices and their properties, discuss
More informationSolving Sets of Equations. 150 B.C.E., 九章算術 Carl Friedrich Gauss,
Solving Sets of Equations 5 B.C.E., 九章算術 Carl Friedrich Gauss, 777855 GaussianJordan Elimination In GaussJordan elimination, matrix is reduced to diagonal rather than triangular form Row combinations
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a
More informationLinear Least Squares
Linear Least Squares Suppose we are given a set of data points {(x i,f i )}, i = 1,...,n. These could be measurements from an experiment or obtained simply by evaluating a function at some points. One
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +
More informationLinear Systems. Singular and Nonsingular Matrices. Find x 1, x 2, x 3 such that the following three equations hold:
Linear Systems Example: Find x, x, x such that the following three equations hold: x + x + x = 4x + x + x = x + x + x = 6 We can write this using matrixvector notation as 4 {{ A x x x {{ x = 6 {{ b General
More informationMATH 240 Fall, Chapter 1: Linear Equations and Matrices
MATH 240 Fall, 2007 Chapter Summaries for Kolman / Hill, Elementary Linear Algebra, 9th Ed. written by Prof. J. Beachy Sections 1.1 1.5, 2.1 2.3, 4.2 4.9, 3.1 3.5, 5.3 5.5, 6.1 6.3, 6.5, 7.1 7.3 DEFINITIONS
More informationLecture 5  Triangular Factorizations & Operation Counts
LU Factorization Lecture 5  Triangular Factorizations & Operation Counts We have seen that the process of GE essentially factors a matrix A into LU Now we want to see how this factorization allows us
More information9. Numerical linear algebra background
Convex Optimization Boyd & Vandenberghe 9. Numerical linear algebra background matrix structure and algorithm complexity solving linear equations with factored matrices LU, Cholesky, LDL T factorization
More informationMATH36001 Background Material 2015
MATH3600 Background Material 205 Matrix Algebra Matrices and Vectors An ordered array of mn elements a ij (i =,, m; j =,, n) written in the form a a 2 a n A = a 2 a 22 a 2n a m a m2 a mn is said to be
More informationLinear Systems COS 323
Linear Systems COS 323 Last time: Constrained Optimization Linear constrained optimization Linear programming (LP) Simplex method for LP General optimization With equality constraints: Lagrange multipliers
More informationUsing the Singular Value Decomposition
Using the Singular Value Decomposition Emmett J. Ientilucci Chester F. Carlson Center for Imaging Science Rochester Institute of Technology emmett@cis.rit.edu May 9, 003 Abstract This report introduces
More information1 Introduction to Matrices
1 Introduction to Matrices In this section, important definitions and results from matrix algebra that are useful in regression analysis are introduced. While all statements below regarding the columns
More informationLeast squares: the big idea
Week 5: Wednesday, Feb 22 Least squares: the big idea Least squares problems are a special sort of minimization. Suppose A R m n and m > n. In general, we will not be able to exactly solve overdetermined
More informationA matrix over a field F is a rectangular array of elements from F. The symbol
Chapter MATRICES Matrix arithmetic A matrix over a field F is a rectangular array of elements from F The symbol M m n (F) denotes the collection of all m n matrices over F Matrices will usually be denoted
More informationMAT Solving Linear Systems Using Matrices and Row Operations
MAT 171 8.5 Solving Linear Systems Using Matrices and Row Operations A. Introduction to Matrices Identifying the Size and Entries of a Matrix B. The Augmented Matrix of a System of Equations Forming Augmented
More information5. Orthogonal matrices
L Vandenberghe EE133A (Spring 2016) 5 Orthogonal matrices matrices with orthonormal columns orthogonal matrices tall matrices with orthonormal columns complex matrices with orthonormal columns 51 Orthonormal
More informationLinear Algebra Concepts
University of Central Florida School of Electrical Engineering Computer Science EGN3420  Engineering Analysis. Fall 2009  dcm Linear Algebra Concepts Vector Spaces To define the concept of a vector
More information9.3 Advanced Topics in Linear Algebra
548 93 Advanced Topics in Linear Algebra Diagonalization and Jordan s Theorem A system of differential equations x = Ax can be transformed to an uncoupled system y = diag(λ,, λ n y by a change of variables
More informationThe Classical Linear Regression Model
The Classical Linear Regression Model 1 September 2004 A. A brief review of some basic concepts associated with vector random variables Let y denote an n x l vector of random variables, i.e., y = (y 1,
More information4. Matrix inverses. left and right inverse. linear independence. nonsingular matrices. matrices with linearly independent columns
L. Vandenberghe EE133A (Spring 2016) 4. Matrix inverses left and right inverse linear independence nonsingular matrices matrices with linearly independent columns matrices with linearly independent rows
More information( % . This matrix consists of $ 4 5 " 5' the coefficients of the variables as they appear in the original system. The augmented 3 " 2 2 # 2 " 3 4&
Matrices define matrix We will use matrices to help us solve systems of equations. A matrix is a rectangular array of numbers enclosed in parentheses or brackets. In linear algebra, matrices are important
More informationInverses. Stephen Boyd. EE103 Stanford University. October 27, 2015
Inverses Stephen Boyd EE103 Stanford University October 27, 2015 Outline Left and right inverses Inverse Solving linear equations Examples Pseudoinverse Left and right inverses 2 Left inverses a number
More informationFactorization Theorems
Chapter 7 Factorization Theorems This chapter highlights a few of the many factorization theorems for matrices While some factorization results are relatively direct, others are iterative While some factorization
More informationLINEAR ALGEBRA. September 23, 2010
LINEAR ALGEBRA September 3, 00 Contents 0. LUdecomposition.................................... 0. Inverses and Transposes................................. 0.3 Column Spaces and NullSpaces.............................
More information7. LU factorization. factorsolve method. LU factorization. solving Ax = b with A nonsingular. the inverse of a nonsingular matrix
7. LU factorization EE103 (Fall 201112) factorsolve method LU factorization solving Ax = b with A nonsingular the inverse of a nonsingular matrix LU factorization algorithm effect of rounding error sparse
More informationRecall that two vectors in are perpendicular or orthogonal provided that their dot
Orthogonal Complements and Projections Recall that two vectors in are perpendicular or orthogonal provided that their dot product vanishes That is, if and only if Example 1 The vectors in are orthogonal
More informationNumerical Linear Algebra Chap. 4: Perturbation and Regularisation
Numerical Linear Algebra Chap. 4: Perturbation and Regularisation Heinrich Voss voss@tuharburg.de Hamburg University of Technology Institute of Numerical Simulation TUHH Heinrich Voss Numerical Linear
More informationLinear Dependence Tests
Linear Dependence Tests The book omits a few key tests for checking the linear dependence of vectors. These short notes discuss these tests, as well as the reasoning behind them. Our first test checks
More informationNumerical Methods I Solving Linear Systems: Sparse Matrices, Iterative Methods and NonSquare Systems
Numerical Methods I Solving Linear Systems: Sparse Matrices, Iterative Methods and NonSquare Systems Aleksandar Donev Courant Institute, NYU 1 donev@courant.nyu.edu 1 Course G63.2010.001 / G22.2420001,
More informationLinear Algebra Review. Vectors
Linear Algebra Review By Tim K. Marks UCSD Borrows heavily from: Jana Kosecka kosecka@cs.gmu.edu http://cs.gmu.edu/~kosecka/cs682.html Virginia de Sa Cogsci 8F Linear Algebra review UCSD Vectors The length
More informationMathematics Notes for Class 12 chapter 3. Matrices
1 P a g e Mathematics Notes for Class 12 chapter 3. Matrices A matrix is a rectangular arrangement of numbers (real or complex) which may be represented as matrix is enclosed by [ ] or ( ) or Compact form
More informationBasics Inversion and related concepts Random vectors Matrix calculus. Matrix algebra. Patrick Breheny. January 20
Matrix algebra January 20 Introduction Basics The mathematics of multiple regression revolves around ordering and keeping track of large arrays of numbers and solving systems of equations The mathematical
More informationPractical Numerical Training UKNum
Practical Numerical Training UKNum 7: Systems of linear equations C. Mordasini Max Planck Institute for Astronomy, Heidelberg Program: 1) Introduction 2) Gauss Elimination 3) Gauss with Pivoting 4) Determinants
More information2.5 Gaussian Elimination
page 150 150 CHAPTER 2 Matrices and Systems of Linear Equations 37 10 the linear algebra package of Maple, the three elementary 20 23 1 row operations are 12 1 swaprow(a,i,j): permute rows i and j 3 3
More informationDirect Methods for Solving Linear Systems. Linear Systems of Equations
Direct Methods for Solving Linear Systems Linear Systems of Equations Numerical Analysis (9th Edition) R L Burden & J D Faires Beamer Presentation Slides prepared by John Carroll Dublin City University
More informationSolution of Linear Systems
Chapter 3 Solution of Linear Systems In this chapter we study algorithms for possibly the most commonly occurring problem in scientific computing, the solution of linear systems of equations. We start
More informationInner Product Spaces and Orthogonality
Inner Product Spaces and Orthogonality week 34 Fall 2006 Dot product of R n The inner product or dot product of R n is a function, defined by u, v a b + a 2 b 2 + + a n b n for u a, a 2,, a n T, v b,
More informationTopic 1: Matrices and Systems of Linear Equations.
Topic 1: Matrices and Systems of Linear Equations Let us start with a review of some linear algebra concepts we have already learned, such as matrices, determinants, etc Also, we shall review the method
More informationLinear Algebra: Determinants, Inverses, Rank
D Linear Algebra: Determinants, Inverses, Rank D 1 Appendix D: LINEAR ALGEBRA: DETERMINANTS, INVERSES, RANK TABLE OF CONTENTS Page D.1. Introduction D 3 D.2. Determinants D 3 D.2.1. Some Properties of
More information1 VECTOR SPACES AND SUBSPACES
1 VECTOR SPACES AND SUBSPACES What is a vector? Many are familiar with the concept of a vector as: Something which has magnitude and direction. an ordered pair or triple. a description for quantities such
More informationAlgebra and Linear Algebra
Vectors Coordinate frames 2D implicit curves 2D parametric curves 3D surfaces Algebra and Linear Algebra Algebra: numbers and operations on numbers 2 + 3 = 5 3 7 = 21 Linear algebra: tuples, triples,...
More informationCHARACTERISTIC ROOTS AND VECTORS
CHARACTERISTIC ROOTS AND VECTORS 1 DEFINITION OF CHARACTERISTIC ROOTS AND VECTORS 11 Statement of the characteristic root problem Find values of a scalar λ for which there exist vectors x 0 satisfying
More informationApplied Linear Algebra I Review page 1
Applied Linear Algebra Review 1 I. Determinants A. Definition of a determinant 1. Using sum a. Permutations i. Sign of a permutation ii. Cycle 2. Uniqueness of the determinant function in terms of properties
More informationDirect Methods for Solving Linear Systems. Matrix Factorization
Direct Methods for Solving Linear Systems Matrix Factorization Numerical Analysis (9th Edition) R L Burden & J D Faires Beamer Presentation Slides prepared by John Carroll Dublin City University c 2011
More information= [a ij ] 2 3. Square matrix A square matrix is one that has equal number of rows and columns, that is n = m. Some examples of square matrices are
This document deals with the fundamentals of matrix algebra and is adapted from B.C. Kuo, Linear Networks and Systems, McGraw Hill, 1967. It is presented here for educational purposes. 1 Introduction In
More informationThe Second Undergraduate Level Course in Linear Algebra
The Second Undergraduate Level Course in Linear Algebra University of Massachusetts Dartmouth Joint Mathematics Meetings New Orleans, January 7, 11 Outline of Talk Linear Algebra Curriculum Study Group
More informationIntroduction to Matrix Algebra I
Appendix A Introduction to Matrix Algebra I Today we will begin the course with a discussion of matrix algebra. Why are we studying this? We will use matrix algebra to derive the linear regression model
More information13 MATH FACTS 101. 2 a = 1. 7. The elements of a vector have a graphical interpretation, which is particularly easy to see in two or three dimensions.
3 MATH FACTS 0 3 MATH FACTS 3. Vectors 3.. Definition We use the overhead arrow to denote a column vector, i.e., a linear segment with a direction. For example, in threespace, we write a vector in terms
More information6. Cholesky factorization
6. Cholesky factorization EE103 (Fall 201112) triangular matrices forward and backward substitution the Cholesky factorization solving Ax = b with A positive definite inverse of a positive definite matrix
More information3 Orthogonal Vectors and Matrices
3 Orthogonal Vectors and Matrices The linear algebra portion of this course focuses on three matrix factorizations: QR factorization, singular valued decomposition (SVD), and LU factorization The first
More informationMatrix Algebra LECTURE 1. Simultaneous Equations Consider a system of m linear equations in n unknowns: y 1 = a 11 x 1 + a 12 x 2 + +a 1n x n,
LECTURE 1 Matrix Algebra Simultaneous Equations Consider a system of m linear equations in n unknowns: y 1 a 11 x 1 + a 12 x 2 + +a 1n x n, (1) y 2 a 21 x 1 + a 22 x 2 + +a 2n x n, y m a m1 x 1 +a m2 x
More information7 Gaussian Elimination and LU Factorization
7 Gaussian Elimination and LU Factorization In this final section on matrix factorization methods for solving Ax = b we want to take a closer look at Gaussian elimination (probably the best known method
More informationThe Singular Value Decomposition in Symmetric (Löwdin) Orthogonalization and Data Compression
The Singular Value Decomposition in Symmetric (Löwdin) Orthogonalization and Data Compression The SVD is the most generally applicable of the orthogonaldiagonalorthogonal type matrix decompositions Every
More informationIterative Methods for Computing Eigenvalues and Eigenvectors
The Waterloo Mathematics Review 9 Iterative Methods for Computing Eigenvalues and Eigenvectors Maysum Panju University of Waterloo mhpanju@math.uwaterloo.ca Abstract: We examine some numerical iterative
More informationVector algebra Christian Miller CS Fall 2011
Vector algebra Christian Miller CS 354  Fall 2011 Vector algebra A system commonly used to describe space Vectors, linear operators, tensors, etc. Used to build classical physics and the vast majority
More information4. MATRICES Matrices
4. MATRICES 170 4. Matrices 4.1. Definitions. Definition 4.1.1. A matrix is a rectangular array of numbers. A matrix with m rows and n columns is said to have dimension m n and may be represented as follows:
More informationAdvanced Techniques for Mobile Robotics Compact Course on Linear Algebra. Wolfram Burgard, Cyrill Stachniss, Kai Arras, Maren Bennewitz
Advanced Techniques for Mobile Robotics Compact Course on Linear Algebra Wolfram Burgard, Cyrill Stachniss, Kai Arras, Maren Bennewitz Vectors Arrays of numbers Vectors represent a point in a n dimensional
More informationMatrix Norms. Tom Lyche. September 28, Centre of Mathematics for Applications, Department of Informatics, University of Oslo
Matrix Norms Tom Lyche Centre of Mathematics for Applications, Department of Informatics, University of Oslo September 28, 2009 Matrix Norms We consider matrix norms on (C m,n, C). All results holds for
More informationADVANCED LINEAR ALGEBRA FOR ENGINEERS WITH MATLAB. Sohail A. Dianat. Rochester Institute of Technology, New York, U.S.A. Eli S.
ADVANCED LINEAR ALGEBRA FOR ENGINEERS WITH MATLAB Sohail A. Dianat Rochester Institute of Technology, New York, U.S.A. Eli S. Saber Rochester Institute of Technology, New York, U.S.A. (g) CRC Press Taylor
More information8.3. Gauss elimination. Introduction. Prerequisites. Learning Outcomes
Gauss elimination 8.3 Introduction Engineers often need to solve large systems of linear equations; for example in determining the forces in a large framework or finding currents in a complicated electrical
More informationUNIT 2 MATRICES  I 2.0 INTRODUCTION. Structure
UNIT 2 MATRICES  I Matrices  I Structure 2.0 Introduction 2.1 Objectives 2.2 Matrices 2.3 Operation on Matrices 2.4 Invertible Matrices 2.5 Systems of Linear Equations 2.6 Answers to Check Your Progress
More informationNOTES on LINEAR ALGEBRA 1
School of Economics, Management and Statistics University of Bologna Academic Year 205/6 NOTES on LINEAR ALGEBRA for the students of Stats and Maths This is a modified version of the notes by Prof Laura
More informationSCHOOL OF MATHEMATICS MATHEMATICS FOR PART I ENGINEERING. Self Study Course
SCHOOL OF MATHEMATICS MATHEMATICS FOR PART I ENGINEERING Self Study Course MODULE 17 MATRICES II Module Topics 1. Inverse of matrix using cofactors 2. Sets of linear equations 3. Solution of sets of linear
More information8.3. Solution by Gauss Elimination. Introduction. Prerequisites. Learning Outcomes
Solution by Gauss Elimination 8.3 Introduction Engineers often need to solve large systems of linear equations; for example in determining the forces in a large framework or finding currents in a complicated
More informationNumerical Analysis Lecture Notes
Numerical Analysis Lecture Notes Peter J. Olver 4. Gaussian Elimination In this part, our focus will be on the most basic method for solving linear algebraic systems, known as Gaussian Elimination in honor
More informationLecture 7: Singular Value Decomposition
0368324801Algorithms in Data Mining Fall 2013 Lecturer: Edo Liberty Lecture 7: Singular Value Decomposition Warning: This note may contain typos and other inaccuracies which are usually discussed during
More informationSolving Linear Systems of Equations. Gerald Recktenwald Portland State University Mechanical Engineering Department gerry@me.pdx.
Solving Linear Systems of Equations Gerald Recktenwald Portland State University Mechanical Engineering Department gerry@me.pdx.edu These slides are a supplement to the book Numerical Methods with Matlab:
More informationDecember 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS
December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in twodimensional space (1) 2x y = 3 describes a line in twodimensional space The coefficients of x and y in the equation
More information1 Review of Least Squares Solutions to Overdetermined Systems
cs4: introduction to numerical analysis /9/0 Lecture 7: Rectangular Systems and Numerical Integration Instructor: Professor Amos Ron Scribes: Mark Cowlishaw, Nathanael Fillmore Review of Least Squares
More informationLecture 6. Inverse of Matrix
Lecture 6 Inverse of Matrix Recall that any linear system can be written as a matrix equation In one dimension case, ie, A is 1 1, then can be easily solved as A x b Ax b x b A 1 A b A 1 b provided that
More informationThinkwell s Homeschool Algebra 2 Course Lesson Plan: 34 weeks
Thinkwell s Homeschool Algebra 2 Course Lesson Plan: 34 weeks Welcome to Thinkwell s Homeschool Algebra 2! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson
More informationALGEBRAIC EIGENVALUE PROBLEM
ALGEBRAIC EIGENVALUE PROBLEM BY J. H. WILKINSON, M.A. (Cantab.), Sc.D. Technische Universes! Dsrmstedt FACHBEREICH (NFORMATiK BIBL1OTHEK Sachgebieto:. Standort: CLARENDON PRESS OXFORD 1965 Contents 1.
More informationNumerical Analysis Lecture Notes
Numerical Analysis Lecture Notes Peter J. Olver 6. Eigenvalues and Singular Values In this section, we collect together the basic facts about eigenvalues and eigenvectors. From a geometrical viewpoint,
More informationComputational Methods CMSC/AMSC/MAPL 460. Eigenvalues and Eigenvectors. Ramani Duraiswami, Dept. of Computer Science
Computational Methods CMSC/AMSC/MAPL 460 Eigenvalues and Eigenvectors Ramani Duraiswami, Dept. of Computer Science Eigen Values of a Matrix Definition: A N N matrix A has an eigenvector x (nonzero) with
More informationSolutions to Review Problems
Chapter 1 Solutions to Review Problems Chapter 1 Exercise 42 Which of the following equations are not linear and why: (a x 2 1 + 3x 2 2x 3 = 5. (b x 1 + x 1 x 2 + 2x 3 = 1. (c x 1 + 2 x 2 + x 3 = 5. (a
More informationMatrices, Determinants and Linear Systems
September 21, 2014 Matrices A matrix A m n is an array of numbers in rows and columns a 11 a 12 a 1n r 1 a 21 a 22 a 2n r 2....... a m1 a m2 a mn r m c 1 c 2 c n We say that the dimension of A is m n (we
More informationLinear Algebra: Matrices
B Linear Algebra: Matrices B 1 Appendix B: LINEAR ALGEBRA: MATRICES TABLE OF CONTENTS Page B.1. Matrices B 3 B.1.1. Concept................... B 3 B.1.2. Real and Complex Matrices............ B 3 B.1.3.
More informationElementary Matrices and The LU Factorization
lementary Matrices and The LU Factorization Definition: ny matrix obtained by performing a single elementary row operation (RO) on the identity (unit) matrix is called an elementary matrix. There are three
More informationWe call this set an ndimensional parallelogram (with one vertex 0). We also refer to the vectors x 1,..., x n as the edges of P.
Volumes of parallelograms 1 Chapter 8 Volumes of parallelograms In the present short chapter we are going to discuss the elementary geometrical objects which we call parallelograms. These are going to
More informationNumerical Methods Lecture 2 Simultaneous Equations
Numerical Methods Lecture 2 Simultaneous Equations Topics: matrix operations solving systems of equations Matrix operations: Mathcad is designed to be a tool for quick and easy manipulation of matrix forms
More informationSolving Systems of Linear Equations
LECTURE 5 Solving Systems of Linear Equations Recall that we introduced the notion of matrices as a way of standardizing the expression of systems of linear equations In today s lecture I shall show how
More informationDiagonal, Symmetric and Triangular Matrices
Contents 1 Diagonal, Symmetric Triangular Matrices 2 Diagonal Matrices 2.1 Products, Powers Inverses of Diagonal Matrices 2.1.1 Theorem (Powers of Matrices) 2.2 Multiplying Matrices on the Left Right by
More informationOrthogonal Projections
Orthogonal Projections and Reflections (with exercises) by D. Klain Version.. Corrections and comments are welcome! Orthogonal Projections Let X,..., X k be a family of linearly independent (column) vectors
More informationDETERMINANTS. b 2. x 2
DETERMINANTS 1 Systems of two equations in two unknowns A system of two equations in two unknowns has the form a 11 x 1 + a 12 x 2 = b 1 a 21 x 1 + a 22 x 2 = b 2 This can be written more concisely in
More informationChapter 4: Systems of Equations and Ineq. Lecture notes Math 1010
Section 4.1: Systems of Equations Systems of equations A system of equations consists of two or more equations involving two or more variables { ax + by = c dx + ey = f A solution of such a system is an
More informationMATH10212 Linear Algebra. Systems of Linear Equations. Definition. An ndimensional vector is a row or a column of n numbers (or letters): a 1.
MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0534405967. Systems of Linear Equations Definition. An ndimensional vector is a row or a column
More informationby the matrix A results in a vector which is a reflection of the given
Eigenvalues & Eigenvectors Example Suppose Then So, geometrically, multiplying a vector in by the matrix A results in a vector which is a reflection of the given vector about the yaxis We observe that
More information2.1: MATRIX OPERATIONS
.: MATRIX OPERATIONS What are diagonal entries and the main diagonal of a matrix? What is a diagonal matrix? When are matrices equal? Scalar Multiplication 45 Matrix Addition Theorem (pg 0) Let A, B, and
More informationLeastSquares Intersection of Lines
LeastSquares Intersection of Lines Johannes Traa  UIUC 2013 This writeup derives the leastsquares solution for the intersection of lines. In the general case, a set of lines will not intersect at a
More informationScientific Computing: An Introductory Survey
Scientific Computing: An Introductory Survey Chapter 10 Boundary Value Problems for Ordinary Differential Equations Prof. Michael T. Heath Department of Computer Science University of Illinois at UrbanaChampaign
More informationSystems of Linear Equations
Systems of Linear Equations Beifang Chen Systems of linear equations Linear systems 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 a, a,, a n and
More informationSystems of Linear Equations
A FIRST COURSE IN LINEAR ALGEBRA An Open Text by Ken Kuttler Systems of Linear Equations Lecture Notes by Karen Seyffarth Adapted by LYRYX SERVICE COURSE SOLUTION AttributionNonCommercialShareAlike (CC
More informationx = (x 1,..., x n ) x + y = (x 1 + y 1,..., x n + y n ) or the pointwise addition of functions (f + g)(t) = f(t) + g(t)
MODULE 1 Topics: Vectors space, subspace, span I. Vector spaces: General setting: We need V = a set: its elements will be the vectors x, y, f, u, etc. F = a scalar field: its elements are the numbers α,
More information6 Gaussian Elimination
G1BINM Introduction to Numerical Methods 6 1 6 Gaussian Elimination 61 Simultaneous linear equations Consider the system of linear equations a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 +
More informationSolving Linear Systems, Continued and The Inverse of a Matrix
, Continued and The of a Matrix Calculus III Summer 2013, Session II Monday, July 15, 2013 Agenda 1. The rank of a matrix 2. The inverse of a square matrix Gaussian Gaussian solves a linear system by reducing
More informationeissn: pissn: X Page 67
International Journal of Computer & Communication Engineering Research (IJCCER) Volume 2  Issue 2 March 2014 Performance Comparison of Gauss and GaussJordan Yadanar Mon 1, Lai Lai Win Kyi 2 1,2 Information
More informationHarvard College. Math 21b: Linear Algebra and Differential Equations Formula and Theorem Review
Harvard College Math 21b: Linear Algebra and Differential Equations Formula and Theorem Review Tommy MacWilliam, 13 tmacwilliam@college.harvard.edu May 5, 2010 1 Contents Table of Contents 4 1 Linear Equations
More information