Ans 5 - v3 see page 3

# Ans 5 - v3 see page 3 - f08 Least-squares and Eigenvalue...

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NAG C Library Chapter Introduction f08 – Least-squares and Eigenvalue Problems (LAPACK) Contents 1 Scope of the Chapter ........................................ 3 2 Background to the Problems .................................. 3 2.1 Linear Least-squares Problems ................................ 3 2.2 Orthogonal Factorizations and Least-squares Problems . ................ 4 2.2.1 QR factorization . ...................................... 4 2.2.2 LQ 5 2.2.3 QR factorization with column pivoting . . . . . . ................... 5 2.3 The Singular Value Decomposition . ............................ 6 2.4 The Singular Value Decomposition and Least-squares Problems . ......... 6 2.5 Symmetric Eigenvalue Problems . .............................. 6 2.6 Generalized Symmetric-De±nite Eigenvalue Problems . 7 2.7 Packed Storage for Symmetric Matrices . ......................... 8 2.8 Band Matrices . .......................................... 8 2.9 Nonsymmetric Eigenvalue Problems . 9 2.10 Generalized Nonsymmetric Eigenvalue Problem . .................... 9 2.11 The Sylvester Equation . .................................... 11 2.12 Error and Perturbation Bounds and Condition Numbers . ............... 2.12.1 Least-squares problems . 12 2.12.2 The singular value decomposition . ........................... 2.12.3 The symmetric eigenproblem . 13 2.12.4 The generalized symmetric-de±nite eigenproblem . .................. 14 2.12.5 The nonsymmetric eigenproblem . 15 2.12.6 Balancing and condition for the nonsymmetric eigenproblem . . . . . . ...... 2.12.7 The generalized nonsymmetric eigenvalue problem . ................. 16 2.12.8 Balancing the generalized eigenvalue problem . . ................... 2.13 Block Algorithms . 3 Recommendations on Choice and Use of Available Functions ........ 17 3.1 Available Functions . ....................................... 3.1.1 Orthogonal factorizations. ................................. 3.1.2 Singular value problems . 18 3.1.3 Symmetric eigenvalue problems ............................. 3.1.4 Generalized symmetric-de±nite eigenvalue problems . . . . . . . . . . . ...... 20 3.1.5 Nonsymmetric eigenvalue problems . . . . . . . . ................... 21 3.1.6 Generalized nonsymmetric eigenvalue problems . ................... 22 3.1.7 Sylvester’s equation . 23 3.2 NAG Names and LAPACK Names . 3.3 Matrix Storage Schemes 24 3.3.1 Conventional storage . ................................... 3.3.2 Packed storage . 25 f08 – Least-squares and Eigenvalue Problems (LAPACK) Introduction – f08 [NP3645/7] f08.1

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3.3.3 Band storage . ........................................ 26 3.3.4 Tridiagonal and bidiagonal matrices . . . . . . . . ................... 28 3.3.5 Real diagonal elements of complex matrices . . . ................... 3.3.6 Representation of orthogonal or unitary matrices ................... 3.4 Parameter Conventions ..................................... 3.4.1 Option parameters . 3.4.2 Problem dimensions. .................................... 29 4 Decision Tree .............................................. 4.1 General purpose functions (eigenvalues and eigenvectors) . ............. 4.2 General purpose functions (singular value decomposition) . 38 5 Index .................................................... 6 Functions Withdrawn or Scheduled for Withdrawal ............... 41 7 References ................................................ Introduction – f08 NAG C Library Manual f08.2 [NP3645/7]
1 Scope of the Chapter This chapter provides functions for the solution of linear least-squares problems, eigenvalue problems and singular value problems, as well as associated computations. It provides functions for: – solution of linear least-squares problems – solution of symmetric eigenvalue problems – solution of nonsymmetric eigenvalue problems – solution of singular value problems – solution of generalized symmetric-deFnite eigenvalue problems – matrix factorizations associated with the above problems – estimating condition numbers of eigenvalue and eigenvector problems – estimating the numerical rank of a matrix – solution of the Sylvester matrix equation ±unctions are provided for both real and complex data.

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## This note was uploaded on 01/14/2011 for the course ECE 210a taught by Professor Chandrasekara during the Fall '08 term at UCSB.

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Ans 5 - v3 see page 3 - f08 Least-squares and Eigenvalue...

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