linear_algebra_part1

linear_algebra_part1 - Numerical Linear Algebra The two...

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Unformatted text preview: Numerical Linear Algebra The two principal problems in linear algebra are: Linear system Given an n n matrix A and an n-vector vector b , deter- mine vectorx IR n such that Avectorx = vector b Eigenvalue problem Given an n n matrix A , find a scalar (an eigenvalue) and a nonzero vector vectorx (an eigenvector) such that Avectorx = vectorx Goals: to review some basic concepts from undergraduate linear algebra; to study the three basic decompositions of a matrix A = LU A = QR A = U V T and understand what they tell us; to understand when a linear system is uniquely solvable; to be able to calculate and manipulate norms of vectors and matrices; to understand what it means for a system to be ill-conditioned and the consequences resulting from this ill-conditioning; to be exposed to both direct and iterative methods for solving linear systems and to understand when it is advantageous to use one or the other; to understand the importance of eigenvalues and eigenvectors and how to calculate them; to be able to compare the work required for two algorithms for solving a linear system. Some Important Results from Undergraduate Linear Algebra We will briefly review some basic results from undergraduate linear algebra that we will need throughout our exploration of numerical linear algebra. If you are not familiar with these, then you should review a good undergraduate text such as G. Strangs Linear Algebra. Recommended graduate texts which review this material and contain the material we will cover on numerical linear algebra are: G.W. Stewart, Introduction to Matrix Computations (used for approximately $7 from Amazon) G. Golub and C. van Loan, Matrix Computations (used for approximately $30 from Amazon) An overall excellent reference for ACS I and II is Gilbert Strang, Computational Science and Engineering (about $90 new from Amazon) Vectors We will denote an n-vector as vectorx and its components as x i , i = 1 ,. .. ,n . We think of vectorx as a column vector and vectorx T as a row vector. To add two vectors, they must have the same length and then we add corre- sponding entries. To multiply a vector by a scalar , we multiply each entry by . For n-vectors vectorx,vector y and scalar vector c = vectorx + vector y where c i = x i + y i To take the dot or scalar or inner product of two n-vectors vectorx and vector y we form n summationdisplay i =1 x i y i so the dot product of two vectors is a scalar. We denote the scalar product as vectorx vector y or vectorx T vector y or ( vectorx,vector y ) If two vectors have complex entries then their inner product is given by vector x T vector y where * denotes the complex conjugate. The standard Euclidean length of a vector is bardbl vectorx bardbl 2 = ( vectorx T vectorx ) 1 / 2 (we will discuss this notation later) We also know that vectorx T vector y = bardbl vectorx bardbl 2 bardbl vector y bardbl 2 cos...
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linear_algebra_part1 - Numerical Linear Algebra The two...

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