On orthogonal block elimination
We consider the block elimination problem {ital Q}matrix (A{sub 1} over A{sub 2}) = matrix (-C over 0) where, given a matrix {ital A} {element_of} {ital {bold R}}{sup m x k}, {ital A}{sub 11} {element_of} {ital {bold R}}{sup k x k}, we try to find a matrix {ital C} with {ital C}{sup T}{ital C} = {ital A}{sup T}{ital A} and an orthogonal matrix {ital Q} that eliminates {ital A}{sub 2}. Sun and Bischof recently showed that any orthogonal matrix can be represented in the so-called basis-kernel representation {ital Q} = {ital Q}(Y, S) = {ital I} - Y{ital ST}{sup T}. Applying this framework to the block elimination problem, we show that there is considerable freedom in solving the block elimination problem and that, depending on {ital A} and {ital C}, we can find {ital Y} {element_of} {ital {bold R}}{sup m x r}, where r is between rank({ital A}{sub 2})and {ital k}, to solve the block elimination problem. We then introduce the canonical basis {ital Y} = matrix ({ital A}{sub 1} + {ital C} over {ital A}{sub 2}) and the canonical kernel {ital S} = ({ital A}{sub 1} + {ital C}){sup {dagger}}{ital C}{sup -T}, which can be determined easily once {ital C} has been computed, and relate this view to previously suggested approaches for computing block orthogonal matrices. We also show that the condition of {ital S} has a fundamental impact on the numerical stability with which {ital Q} can be computed and prove that the well-known compact {ital WY} representation approach, employed, for example, in LAPACK, leads to a well-conditioned kernel. Lastly, we discuss the computational promise of the canonical basis and kernel, in particular in the sparse setting, and suggest pre- and postconditioning strategies to ensure that {ital S} can be computed reliably and is wellconditioned.
- Research Organization:
- Argonne National Lab., IL (United States). Mathematics and Computer Science Div.
- Sponsoring Organization:
- USDOE Office of Energy Research, Washington, DC (United States); Advanced Research Projects Agency, Washington, DC (United States)
- DOE Contract Number:
- W-31109-ENG-38
- OSTI ID:
- 438466
- Report Number(s):
- MCS-P-450-0794; ON: DE97002614; CNN: Contract DM28E04120
- Resource Relation:
- Other Information: PBD: [1996]
- Country of Publication:
- United States
- Language:
- English
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