<i>Principles of Numerical Analysis</i>

“…It can be veri ed by direct calculation that is a J;unitary matrix, i.e., J = J = J : xlvii When xJx < 0, we use the same expression for but with g = x + e n;1 (A. 21) and a complex number that satis es j j 2 = ;xJx and x n;1 is real.…”

Section: A3 Fast Givens Transformationsmentioning

confidence: 99%

“…The unitary transformation j is highly nonunique and can be computed in many ways, the simplest ones being to construct it as a sequence of elementary (Givens or plane) rotations nulling one entry at a time in the pre-array, or as a sequence of elementary (Householder) re ections nulling out a block o f e n tries in each r o w. We refer to 21,5,7] and the Appendix for more details. 3 The numerical advantages of the square-root transformations arise from the length preserving properties of unitary transformations, and from the fact that the dynamic range of the entries in P 1=2 j is roughly the square-root of the dynamic range of those in P j .…”

Section: Square-root Arraysmentioning

confidence: 99%

Array Algorithms for H 2 and H ∞ Estimation

Hassibi¹,

Kailath²,

Sayed³

1999

Applied and Computational Control, Signals, and Circuits

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Currently, the preferred method for implementing H 2 estimation algorithms is what is called the array form, and includes two main families: square-root array algorithms which a r e t ypically more stable than conventional ones, and fast array algorithms which, when the system is time-invariant, typically o er an order of magnitude reduction in the computational e ort. Using our recent o b s e r v ation that H 1 ltering coincides with Kalman ltering in Krein space, in this paper we d e v elop array algorithms for H 1 ltering. These can be regarded as natural generalizations of their H 2 counterparts, and involve propagating the inde nite square-roots of the quantities of interest. The H 1 square-root and fast array algorithms bothhave the interesting feature that one does not need to explicitly check for the positivity conditions required for the existence of H 1 lters. These conditions are built into the algorithms themselves so that an H 1 estimator of the desired level exists if, and only if, the algorithms can be executed. However, since H 1 square-root algorithms predominantly use Junitary transformations, rather than the unitary transformations required in the H 2 case, further investigation is needed to determine the numerical behaviour of such algorithms.

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“…The three-term recurrence relation was used by 19th-century mathematicians to derive quadrature formulas (Szego, 1939(Szego, /1959. Householder (1953) mentioned its numerical applications and attributed such uses to C. Lanczos. Some program packages make use of the method and attribute it to Forsythe (1957).…”

mentioning

confidence: 99%

Orthogonal-polynomial programs: Good vs. better

Emerson

1982

Behavior Research Methods & Instrumentation

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Principles of Numerical Analysis

Cited by 82 publications

References 0 publications

A method of approximating the zeros of functions by quadratic formulas.

A method of approximating the zeros of functions by quadratic formulas.

Array Algorithms for H 2 and H ∞ Estimation

Orthogonal-polynomial programs: Good vs. better

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