Numerical Algorithms

, Volume 35, Issue 1, pp 97–120

Scaling by Binormalization

  • Oren E. Livne
  • Gene H. Golub

DOI: 10.1023/B:NUMA.0000016606.32820.69

Cite this article as:
Livne, O.E. & Golub, G.H. Numerical Algorithms (2004) 35: 97. doi:10.1023/B:NUMA.0000016606.32820.69


We present an iterative algorithm (BIN) for scaling all the rows and columns of a real symmetric matrix to unit 2-norm. We study the theoretical convergence properties and its relation to optimal conditioning. Numerical experiments show that BIN requires 2–4 matrix–vector multiplications to obtain an adequate scaling, and in many cases significantly reduces the condition number, more than other scaling algorithms. We present generalizations to complex, nonsymmetric and rectangular matrices.

optimal scaling BIN Gauss–Seidel–Newton relaxation methods 

Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  • Oren E. Livne
    • 1
  • Gene H. Golub
    • 1
  1. 1.The Program for Scientific Computing and Computational Mathematics, Computer Science Department, Gates 2BStanford UniversityStanfordUSA