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Square or Rectangular Matrices

Factorizations and Decompositions. Square or Rectangular Matrices. QR factorization. Reduced QR factorization. SVD. Reduced SVD. Factorizations and Decompositions. Square Matrices. LU factorization. Cholesky factorization. Schur Factorization. Arnoldi iteration. Lanczos iteration.

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Square or Rectangular Matrices

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  1. Factorizations and Decompositions Square or Rectangular Matrices QR factorization Reduced QR factorization SVD Reduced SVD

  2. Factorizations and Decompositions Square Matrices LU factorization Cholesky factorization Schur Factorization Arnoldi iteration Lanczos iteration Eigenvalue decomposition

  3. Core Problem Three fundamental problems are considered 1) Square system of linear equations 2) Least Square Problem: m>n 3) Eigenvalue Problem (2 hw4)

  4. Core Problem Three fundamental problems are considered 1) Square system of linear equations Direct Methods Iterative Methods LU factorization Cholesky factorization stationary splitting Jacobi Gauss-Seidel SOR Krylov CG (SPD) MINRES (Symm) GMRES (general) Lanczos Arnoldi

  5. Core Problem Three fundamental problems are considered 2) Least Square Problem: m>n Normal Equation QR-factorization SVD

  6. Core Problem Three fundamental problems are considered 3) Eigenvalue Problem Schur Factorization QR-algorithm PHASE-I PHASE-II Divide and Conqur

  7. Applications Data Fitting Least square Problem Image processing SVD linear BVP (PDE or ODE) FDM, FEM, FVM Non-linear

  8. Applications Non-linear system of equations: Newton Raphson Method: Single variable jacobian In Newton Raphson Method, each iteration requires solving one linear system of equations

  9. Norms and Stability Vector p-norm Matrix p-norm Frobeniusnorm: Condition number

  10. Norms and Stability A well-conditioned Problem Is one with the property that all small changes in initial data lead to only small changes in the solution An ill-conditioned Problem Is one with the property that some small changes in initial data lead to a large change in the solution Frobeniusnorm: Condition number

  11. MATLAB LU factorization: lu, luinc Cholesky factorization: chol, cholinc Eigenvalue decomposition: eig QR factorization: qr Singula Value Decompsition: svd Schur Factorization: schur Arnoldi, Lanczos: hess Conjugate Gradient: pcg Minimal Residuals: minres Generalized Minimal Residuals: gmres Solving linear system: A\b Least Square Problem: lsqr

  12. References an in-depth, up-to-date view of practical algorithms for solving large-scale linear systems of equations Trefethen: encyclopedia Focuses on the use of iterative methods for solving large sparse systems of linear equations PDE in Fluid and its iterative

  13. References

  14. software library for numerical linear algebra LAPACK LINPACKNAG (The Numerical Algorithms Group)IMSLBLAS www.netlib.org

  15. software library for numerical linear algebra http://www.netlib.org/utk/people/JackDongarra/la-sw.html

  16. software library for numerical linear algebra FORTRAN c Parallel distributed methods Iterative methods for sparse and structured linear systems

  17. JOURNALS Springer Numerische Mathematik Computing Foundations of Computational Mathematics Numerical Algorithms Journal of Scientific Computing Free journals Electronic Transactions on Numerical Analysis (ETNA) Electronic Journal of Linear Algebra (ELA) Taylor & Francis 0.589: International Journal on Computer Mathematics Acta Numerica AMS Mathematics of Computation SIAM SIAM Journal on Scientific Computing (SISC) SIAM Journal on Matrix Analysis and Applications (SIMAX) SIAM Journal on Numerical Analysis (SINUM) SIAM Review (SIREV) Elsevier Linear Algebra and its Applications (LAA) Journal of Computational and Applied Mathematics (JCAM) Applied Numerical Mathematics (APNUM) Applied Mathematics and Computation (AMC) Mathematical and Computer Modeling (MCM) Mathematics and Computers in Simulation (MCM) Computers and Mathematics with Applications Wiley InterScience Numerical Linear Algebra with Applications (NLAA) Oxford University Press IMA Journal of Numerical Analysis Journal of Computational Mathematics Computational Methods in Applied Mathematics Linear and Multilinear Algebra (LAMA) NA-Digest

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