the norms of a vector X = [1,12, ...,I" and matrix A = (a5) are respectively defined as ||X|| = E and ||A| - max laisl- j=1,2, i=1 Alternatively we can define

Advanced Engineering Mathematics
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Chapter2: Second-order Linear Odes
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the norms of a vector
X = [1, 12, ...,In"
and matrix
A = (a;5)
are respectively defined as
||X || =
Ei and ||A|| =
max laisl.
%3D
j=1,2,.n
i=1
Alternatively we can define
||X || = and ||A||
max
j=1,2,
i=0
i=1
The latter definition means that the norm of a matrix A is defined as the maximum of the sums
obtained by addition of the absolute values of the entries in each column of A.
Now consider the system
i1 = trı + 2tr2 + t
iz = (sin t)r1 +-r2 + 2r3 +e
i3 = 2(cos t)r1 + 2(sin t)r2 ++e.
Assume there exist solutions X and Y on the interval (1,4) with
X (2) = [3, 1.9, 4" and Y (2) = [3.2, 2.1, 3.9|".
Use alternative definitions given above and Gronwall's inequality to estimate the error
||X (t) – Y (t)|| for 1<t<4.
Transcribed Image Text:the norms of a vector X = [1, 12, ...,In" and matrix A = (a;5) are respectively defined as ||X || = Ei and ||A|| = max laisl. %3D j=1,2,.n i=1 Alternatively we can define ||X || = and ||A|| max j=1,2, i=0 i=1 The latter definition means that the norm of a matrix A is defined as the maximum of the sums obtained by addition of the absolute values of the entries in each column of A. Now consider the system i1 = trı + 2tr2 + t iz = (sin t)r1 +-r2 + 2r3 +e i3 = 2(cos t)r1 + 2(sin t)r2 ++e. Assume there exist solutions X and Y on the interval (1,4) with X (2) = [3, 1.9, 4" and Y (2) = [3.2, 2.1, 3.9|". Use alternative definitions given above and Gronwall's inequality to estimate the error ||X (t) – Y (t)|| for 1<t<4.
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