'Upper triangular' diaporamas de présentation

Chapter 3 Linear Algebra Review and Elementary Differential Equations

Chapter 3 Linear Algebra Review and Elementary Differential Equations To review some important concept of linear algebra and differential equations that will be used subsequently. Outline. Vector spaces and subspaces Independence, basis and dimension Norm and inner product

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Chapter 3 Determinants

Chapter 3 Determinants. 3.1 The Determinant of a Matrix 3.2 Determinant and Elementary Operations 3.3 Properties of Determinants 3.4 Application of Determinants. Elementary Linear Algebra ????????

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Maria Ugryumova

Direct Solution Techniques in Spectral Methods. Maria Ugryumova. CASA Seminar, 13 December 2007. Outline. 1. Introduction. 2. Ad-hoc Direct Methods. 3. The matrix diagonalization techniques. 4. Direct methods. 5. Conclusions. 2 / 30. 1. Introduction. Steady and unsteady problems;.

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The Islamic University of Gaza Faculty of Engineering Civil Engineering Department

The Islamic University of Gaza Faculty of Engineering Civil Engineering Department Numerical Analysis ECIV 3306 Chapter 9. Gauss Elimination. Introduction. Roots of a single equation:. A general set of equations: - n equations, - n unknowns. Linear Algebraic Equations.

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ECIV 301

ECIV 301. Programming & Graphics Numerical Methods for Engineers Lecture 18 LU Decomposition and Matrix Inversion. EXAMPLE. Eliminate Column 1. PIVOTS. Eliminate Column 1. Eliminate Column 2. PIVOTS. { b }. Eliminate Column 2. Upper Triangular Matrix. Modified RHS. [ U ].

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Faster least squares approximation

Faster least squares approximation. Tamas Sarlos Yahoo! Research IPAM 23 Oct 2007 http://www.ilab.sztaki.hu/~stamas Joint work with P. Drineas, M. Mahoney, and S. Muthukrishnan. Outline. Randomized methods for Least squares approximation Lp regression Regularized regression

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Lecture 5 Determinants

Lecture 5 Determinants. Last Time Elementary Matrices Determinant of a Matrix Evaluation of Determinant Reading Assignment: Secs 2.4, 3.1, 3.2 of Textbook. Elementary Linear Algebra R. Larsen et al. (5 Edition) TKUEE ??? -NTUEE SCC_10_2007. Lecture 5: Determinants. Today

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STAR-P: High Productivity Parallel Computing

STAR-P: High Productivity Parallel Computing. David Cheng, Ron Choy, Alan Edelman Massachusetts Institute of Technology John R. Gilbert and Viral Shah University of California at Santa Barbara. Graph Algorithms and Sparse Matrix Land. Birth of Interactive Supercomputing.

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QR decomposition: A = Q R Q is orthonormal R is upper triangular To find QR decomposition:

QR decomposition: A = Q R Q is orthonormal R is upper triangular To find QR decomposition: 1.) Q: Use Gram-Schmidt to find orthonormal basis for column space of A 2.) Let R = Q T A. Find the QR decomposition of A = . 4 3 .

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Introduction & Systems of Linear Equations

Introduction & Systems of Linear Equations. Paul Heckbert Computer Science Department Carnegie Mellon University. State of the Art in Scientific Computing. aerospace: simulate subsonic & supersonic air flow around full aircraft, no wind tunnel

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Fine-Grain Communication

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Chapter 8 Numerical Technique

Linear Algebra. Chapter 8 Numerical Technique. ???? ????? ???. 8.1 Gaussian Elimination. Definition A matrix is in echelon form if Any rows consisting entirely of zeros are grouped at the bottom of the matrix.

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Chapter 1

Chapter 1. Section 5. Elementary Matrices. An elementary matrix of type I is a matrix obtained by interchanging two rows of I. Elementary Matrices. R ow 1 and Row 2 are switched!. C olumn 1 and Column 2 are switched!. Elementary Matrices.

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Advanced dynamic models

Advanced dynamic models. Martin Ellison University of Warwick and CEPR Bank of England, December 2005. Impulses. Propagation. Fluctuations. More complex models. Frisch-Slutsky paradigm. Impulses. Can add extra shocks to the model. Shocks may be correlated. Propagation.

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Chapter 2 Determinant

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Linear Least Squares

Linear Least Squares. Paul Heckbert Computer Science Department Carnegie Mellon University. Orthogonal and Hermitian Matrix. A square matrix Q is orthogonal iff Q -1 = Q T Q T Q = QQ T = I its rows are orthonormal ? its columns are orthonormal

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7.7B-Minors and Cofactors

7.7B-Minors and Cofactors. Minors of entry 1. Eliminate the row i and column j the entry is in 2. Find determinant of resulting matrix 2x2: 3x3: diagonal method OR expand cofactors Larger square matrix: expand cofactors Cofactor,

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Chapter 4

Chapter 4. Section 4.5 Eigenvectors and Eigenspaces. Eigenspaces and Geometric Multiplicity If A is a matrix and is an eigenvalue of A then we can define the following two concepts. The eigenspace of (denoted ) is the null space of the matrix .

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LU - Factorizations

LU - Factorizations. Matrix Factorization into Triangular Matrices. Triangular Matrices A n ? n square matrix L =( l ij ) is called Lower Triangular if l ij = 0 if i < j (i.e. all entries above the diagonal are zero).

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