By C. R. Weisbin (auth.), Jeffery Lewins, Martin Becker (eds.)
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Additional info for Advances in Nuclear Science and Technology: Volume 14 Sensitivity and Uncertainty Analysis of Reactor Performance Parameters
R. W. PEELLE 32 c. "Normalized" Uncertainty Matrices. Some nuclear parameters customarily are represented as the product of a mUltiplicity and a normalized spectrum, or as the product of an integral cross-section and an angular distribution expressed in other than a normalized Legendre expansion NM 1 + I n=l a p (11). n n Though no such uncertainty data are included in ENDF/B-V, such data will be required in a complete file. As a specific example, take the fission neutron source spectrum v X(E'), where J X(E')dE' = 1, and assume that X(E') is represented in histogram form with X.
Data evaluation for cross-section values and uncertainties involves objective combination of data. For reasons clarified below, a variety of least-squares techniques are becoming increasingly popular as more data become available and as workers recognize that this technique is not confined to curvefitting problems. The method has been available for nearly two centuries; the formulae are laid out below to make explicit the forms of the equations useful for uncertainty evaluation and to accent the approximations typically made.
Are . 1 rea I , uncertalnty matrlces are real and by detlnltlon, symmetric. Since the variance of any function r(y) must be non-negative, by Equation (8) the quadratic form r'V r' and the rna trix V are posi ti ve semide fini te . The V can be made diagonal by an orthogonal transformation r' = Ts with TT = TT = I the unit matrix such that D = tv T = (<5 •. A. ) is diagonal, with A. the non-negative eigenvalues OfJV1SO that Equation (8) t~kes the form (19): t v(r) '" r'Vr' '" sDs '" (Tr') D Tr' '" I j \ (10) ci r') ~ To apply these ideas, note that if a proposed uncertainty matrix were to have a negative eigenvalue A , then an apparent negative "variance" would be computed ~or a response r having a sensitivity vector r' proportional to the nth column of T.
Advances in Nuclear Science and Technology: Volume 14 Sensitivity and Uncertainty Analysis of Reactor Performance Parameters by C. R. Weisbin (auth.), Jeffery Lewins, Martin Becker (eds.)