Abstract
The matrix 1norm estimation algorithm used in LAPACK and various other software libraries and packages has proved to be a valuable tool. However, it has the limitations that it offers the user no control over the accuracy and reliability of the estimate and that it is based on level 2 BLAS operations. A block generalization of the 1norm power method underlying the estimator is derived here and developed into a practical algorithm applicable to both real and complex matrices. The algorithm works with n × t matrices, where t is a parameter. For t = 1 the original algorithm is recovered, but with two improvements (one for real matrices and one for complex matrices). The accuracy and reliability of the estimates generally increase with t and the computational kernels are level 3 BLAS operations for t > 1. The last t  1 columns of the starting matrix are randomly chosen, giving the algorithm a statistical flavor. As a byproduct of our investigations we identify a matrix for which the 1norm power method takes the maximum number of iterations. As an application of the new estimator we show how it can be used to efficiently approximate 1norm pseudospectra.
Original language  English 

Pages (fromto)  11851201 
Number of pages  16 
Journal  SIAM Journal on Matrix Analysis and Applications 
Volume  21 
Issue number  4 
Publication status  Published  Mar 2000 
Keywords
 1norm pseudospectrum
 Condition number estimation
 LAPACK
 Level 3 BLAS
 Matrix 1norm
 Matrix condition number
 Matrix norm estimation
 pnorm power method
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Mathematical Software for Computing Matrix Functions
Nicholas Higham (Participant), Francoise Tisseur (Participant) & Philip Davies (Participant)
Impact: Economic, Technological