Notizen
Gliederung
Birth of CG-Method
Prof. Dr. Urs Hochstrasser
Disposition
1. Presentation of the CG-METHOD
2. Applications of the CG-METHOD
3. Biography Stiefel
4. Biography Hestenes
5. Birth of the CG-METHOD at the Institut für Angewandte Mathematik  IAM, ETHZ
6. Birth of the CG-METHOD at the Institute for Numerical Analysis INA, UCLA
7. Final remarks
System of linear  equations
A. x + b = 0 e.g.
A symmetric, positive definite and
sparse, n by n matrix
Solution found by an iterative procedure:
      first guess x0 →  A x0 + b = r0 ≠  0
Relaxation Methods:
   change xi so that ri  →  0
Manuscript of Stiefel‘s 1949 course on numerical methods
            Cg-method:
select pi  such that
ri* rk = 0 i ≠ k orthogonal and
pi*A pk= 0 for all i,k   i ≠ k
pi conjugate gradients
Advantages of cg method
Theoretically, the solution is found in a finite number of steps, and, if stopped early, may already give a useful approximate solution
It is a strategical procedure taking into account in each step the information obtained in all the previous steps
At each step, the value of the error function f(x) is diminished. So also is the distance of the estimate xi from the solution x
The method allows to take advantage of parallel computing
Applications of cg method
Solving very large systems of linear equations with sparse coefficient matrix A (up to billions of unknowns) occuring e.g. in :
Civil engineering
Nuclear reactor theory
Aircraft engineering
Geodetics
Operations research
Curriculum Vitae Prof. Eduard Stiefel
Etching
by  
Stiefel‘s Father
Fields of activities of Prof. Stiefel
1. Topology
2. Group theory and representation of groups
3. Numerical linear algebra
4. Numerical methods in approximation theory
5. Analytical methods in mechanics, especially celestial mechanics
15 Stiefel giving a lecture
Curriculum Vitae Prof. Magnus R. Hestenes
Fields of activities of Prof. Hestenes
Variational problems
Problems of control theory
conjugate direction methods
18 Hestenes at INA 1952
Computation for the Grande Dixence Dam
Deformation of an airplane wing
21 Prof. Rutishauser (1918-1970) and his wife
Title page of my dissertation
Title of article by Hestenes and Stiefel
24 Lanczos at INA 1952
25 Forsythe and Lanczos at INA 1952
26 Prof. and Mrs. Stiefel at their motel in Westwood, LA 1951
27 Tea under palmtrees
28 Stiefel and his PhD-Candidate
29 The reason for our first contact
Final Remarks
After 50 years the cg-method is still very much alive, being widely used and further developed
Its ingenious inventors deserve our admiration and gratitude not only for creating it but also for their other outstanding achievements in academic teaching and research. They have essentially contributed to give the Computer Sciences a key role in today's scientific endeavors
A.N. Krylov August 3, 1863 - October 26, 1945
Maritime Engineer and Applied Mathematician