Here we consider a numerical procedure to interpolate on matrix Lie groups. By using the exponential map and its (1, 1) diagonal Padé approximant, piecewice interpolants may be derived. The approach ...based on the Padé map has the advantage that the computation of exponentials and logarithms of matrices are reduced. We show that the updating technique proposed by Enright in 1 may be applied when a dense output is required. The application to the numerical solution of a system ODEs on matrix group and to a classical interpolation problem are reported.
Arterial cross-clamping is a common strategy used in vascular surgery and its duration is an independent predictor of surgical outcomes. The impact of arterial cross-clamping on the viscoelastic ...properties of the arterial system and its underlying mechanisms still remain unclear. The aim of this study was to evaluate the effect of arterial cross-clamping on arterial stiffness. A cross-sectional, observational, before-after study was designed to enroll adult patients undergoing vascular surgery. The Augmentation Index normalized to 75 beats-per-minute (AIx@75) and Fractal Dimension (FD) –indirect indicators of arterial stiffness– were calculated from radial arterial pressure tracings during surgery. The arterial pressure tracings from 8 patients were analyzed. Overall data included 4 aortic and 11 iliofemoral interventions. In both aortic and iliofemoral interventions, after arterial clamping, median AIx@75 rose and FD dropped significantly; the opposite occurred after arterial unclamping. Spearman’s correlation suggests a strong significant negative correlation between median AIx@75 and FD during each hemodynamic state for aortic interventions. Our results are consistent at many levels:
a)
opposite events (i.e.
,
clamping and unclamping) produce changes in different directions,
b)
two different indicators (i.e.
,
AIx@75 and FD) suggest the same underlying phenomenon, and
c)
similar results are observed at different vascular locations (i.e., aortic and iliofemoral). Overall, our data consistently suggests an increase in arterial stiffness during clamping and a reduction during unclamping. Despite the large distance from the aortic or iliofemoral intervention sites, radial artery pressure monitoring is still able to detect consistently these vascular events.
This note deals with the numerical solution of the matrix differential system
Y′ =
B(
t,
Y),
Y,
Y(0) =
Y
0,
t ⩾ 0, where
Y
0 is a real constant symmetric matrix,
B maps symmetric into skew-symmetric ...matrices, and
B(
t,
Y),
Y is the Lie bracket commutator of
B(
t,
Y) and
Y, i.e.
B(
t,
Y),
Y =
B(
t,
Y)
Y −
YB(
t,
Y). The unique solution of (1) is isospectral, that is the matrix
Y(
t) preserves the eigenvalues of
Y
0 and is symmetric for all
t (see 1, 5). Isospectral methods exploit the Flaschka formulation of (1) in which
Y(
t) is written as
Y(
t) =
U(
t)
Y
0
U
T(
t), for
t ⩾ 0, where
U(
t) is the orthogonal solution of the differential system
U′ =
B(
t,
UY
0
U
T)
U,
U(0) =
I,
t ⩾ 0, (see 5). Here a numerical procedure based on the Cayley transform is proposed and compared with known isospectral methods.
In this paper we consider numerical methods for solving nonlinear equations on matrix Lie groups. Recently Owren and Welfert (Technical Report Numerics, No 3/1996, Norwegian University of Science and ...Technology, Trondheim, Norway, 1996) have proposed a method where the original nonlinear equation
F(
Y)=0 is transformed into a nonlinear equation on the Lie algebra of the group, thus Newton-type methods may be applied which require the evaluation of exponentials of matrices. Here the previous transformation will be performed by the Cayley approximant of the exponential map. This approach has the advantage that no exponentials of matrices are needed. The numerical tests reported in the last section seem to show that our approach is less expensive and provides a larger convergence region than the method of Owren and Welfert.
This paper deals with the numerical solution of the Lax system
L′ =
B(
L),
L,
L(0) =
L
0 (∗), where
L
0 is a constant symmetric matrix,
B(·) maps symmetric matrices into skew-symmetric matrices, and ...
B(
L),
L is the commutator of
B(
L) and
L. Here two different procedures, based on the approach recently proposed by Calvo, Iserles and Zanna (the MGLRK methods), are suggested. Such an approach is a computational form for the Flaschka formulation of (∗). Our numerical procedures consist in solving (∗) by a Runge-Kutta method, then, a single step of a Gauss-Legendre Runge-Kutta (GLRK) method may be applied to the Flaschka formulation of (∗). In the first procedure we compute the approximation of the Lax system by a continuous explicit RK method, instead, the second procedure computes the approximation of the Lax system by a GLRK method (the same method used for the Flaschka system). The computational costs have been derived and compared with the ones of the MGLRK methods. Finally, several numerical tests and computational comparisons will be shown.
In this paper we consider numerical methods for computing functions of matrices being Hamiltonian and skew-symmetric. Analytic functions of this kind of matrices (i.e., exponential and rational ...functions) appear in the numerical solutions of ortho-symplectic matrix differential systems when geometric integrators are involved. The main idea underlying the presented techniques is to exploit the special block structure of a Hamiltonian and skew-symmetric matrix to gain a cheaper computation of the functions. First, we will consider an approach based on the numerical solution of structured linear systems and then another one based on the Schur decomposition of the matrix. Splitting techniques are also considered in order to reduce the computational cost. Several numerical tests and comparison examples are shown.
The authors analyze the problem of solving tridiagonal linear systems on parallel computers. A wide class of efficient parallel solvers is derived by considering different parallel factorizations of ...partitioned matrices. These solvers have a minimum requirement of data transmission. In fact, communication is only needed for solving a "reduced system," whose dimension depends on the number of parallel processors used. Moreover, for a given partitioned tridiagonal matrix, the reduced system (which is again tridiagonal) is the same, and represents the only sequential part of the corresponding parallel solver. Three examples are discussed in more detail; one of them derives a very efficient parallel method based on the cyclic reduction algorithm.
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