This document contains summaries of problems and solutions from applied mathematics. 1) The first problem involves finding the asymptotic expansion of Euler's constant. The solution uses known formulas to derive the expansion to higher orders. 2) The second problem involves finding the least squares estimate of a rotation matrix that aligns two sets of points. The solution uses properties of orthogonal and positive semidefinite matrices to show the optimal rotation matrix maximizes a trace function. 3) The third problem is a third order differential equation arising in physics. The solution identifies power series solutions in terms of special functions and evaluates associated integrals and asymptotic expressions.

1. 384 PROBLEMS AND SOLUTIONS
where
p(x) _xe
n!
Solution by J. H. VAN LINT (Technological University, Eindhoven, Nel,!e,
lands).
We use the known formula for Euler’s constant
1 Sn r=l
r-1
log n -Jr ’t 2(n -t- 1)
zc 0
Note that
(2) e
nih’-0
--(n+2)
It now follows from (1) and (2) that
e_ z log n
O(z-).
’-t-logx-- Ei(--x) --.(1 e-z)--(e
-1-- x) + O(x-’)
1
log x + O(x-).
To obtain more terms of the asymptotic expansion, we just use more terms in the
expansion (1).
Problem 65-1, A Least Squares Estimate of Satellite Attitude, by Gnncn WAIBa
(IBM--Federal Systems Division).
Given two sets of n points {v, v, ...,v}, and {v*, v*, ...,v*}, where
n ->_ 2, find the rotation matrix M (i.e., the orthogonal matrix with determinant
-1) which brings the first set into the best least squares coincidence with the
second. That is, find M which minimizes
This problem has arisen in the estimation of the attitude of a satellite by using
direction cosines {v*} of objects as observed in a satellite fixed frame of reference
and direction cosines Ivy} of the same objects in a known frame of reference. M
is then a least squares estimate of the rotation matrix which carries the known
frame of reference into the satellite fixed frame of reference.
Solution by J. L. FhRRELL and J. C. STUELPNAGEL (Westinghouse Defense
and Space Center).
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2. PROBLEMS AND SOLU’IIONS 385
Let k denote the dimension of the column vectors vl, ...,vn, vl ...,vn
nd let V and V* denote the two/c )< n matrices obtained by juxtaposing Vl,
* * respectively.
"’’Vn and Vl ""vn
For any orthogonal matrix M, define Q(M) as the sum of squares to be
minimized, so
Q(M) v3.* Mv3.
3"=1
tr(V*- MV)r(V* MV),
where tr denotes the trace function and a superscript T denotes transposition.
Q(M) may be rewritten as
Q(M) tr (V*r
VrMr)(V* MV) tr V*rV* + tr VrV- 2 tr VrMrV*.
Since the first two terms are independent of M, Q(M) is minimized by maximiz-
ing F(M) tr VrMrV*, which may be written as
F(M) tr MrV*Vr.
It is a well-known fact that an arbitrary real square matrix A can be written
as a product UP, where U is orthogonal and P is symmetric and positive semi-
definite. Furthermore, if A is nonsingular, U is uniquely defined and P is positive
definite. If A is singular, U is not unique, but it may be taken to have deter-
minant +1. (The corresponding statement of the first result above for complex:
A may be fotmd in [1, 2.8] and the result for real A follows from it.)
Applying this result to A V*Vr, we have F(M) tr MrUP. Since P is
symmetric, there is an orthogonal matrix N such that NPNr
is a diagond
matrix D, whose diagonal elements dl, ..., dk are arranged in decreasing order.
All di are nonnegative, since P is positive semidefinite. Now, letting X NMr
UNr, we obtain
F(M) tr MrUNrDN tr NMrUNrD tr XD dx
i=l
Since F(M) is a linear function of the nonnegative numbers dl, .-., dk, its
maximum is attained when the diagonal elements of X attain their maximum
values. Because X is an orthogonal matrix, all elements of X are between -1
and 1, so F(M) is maximized when xi 1, x 0, i j.
Because det M is required to be +1, detX det (NMrUNr) (detN)
det M det U det U. If det U 1, then it is required that det X 1,
and it is not hard to see that
is solution (since dl >= d >= >_- d). Letting X0 be the matrix which maxi-
mizesF(M) (Xo IorXo=(I-1
_0) accordingasdetU= -lor-1)
1
Xo NMorUN,or M0 UNrXorN is a rotation matrix which minimizes the
sum of squares Q(M). If V*V is nonsingular, it is the unique rotation matrix
which does so.
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3. 386 PROBLEMS AND SOLUTIONS
REFERENCE
[1] M. MRcus, Basic Theorems in Matrix Theory, Nat. Bureau of Stds., Appl. Math.
Series No. 57, 1960.
R. H. WESSEn (Hughes Aircraft Company) in his solution points out that if
det A 0, then V*Vr
A UP,
V (Ar)-(ArA)1, P (ArA)1,
where (ArA) is the symmetric square root of ArA with positive eigeavalues,
and, hence, for det A > 0,
M0 (VV*r)-(VV*rV*V)1/.
J. R. VnLMN (Hughes Aircraft Company) iu his solution demonstrates that
in the case det A < 0, M0 U(I 2G) where G is any one-dimensional projection
satisfying GE G, where E is the eigenspace of the smallest eigeavalue of P,
hence Farrel aad Stuelpnagle’s solution in this case is unique if the smallest
eigenvalue of P has multiplicity one.
J. E. BROCK (U. S. Naval Postgraduate School) solved the problem for
det V*Vr
>-_ 0 by differentiating
a -tr [VrM-V* V*rMV]
with respect to each of the 9 elemeats of M aud setting the results equal to 0. The
resulting equutions turn out to be
MrAMr
At,
which implies that MrA is symmetric, (MrA)(MrA) ArA, MrA is any
symmetric square root (ArA)x of ArA, and M (Ar)-i(ArA).He then
gives an example in which the actual residual sum of squares is minimized by
tking the positive definite symmetric square root.
Also solved by R. DnSZnDINS (Goddard Spce Flight Center) nd the pro-
poser.
Problem 65-2, A Third Order Differential Equation, by DONALD E. AMOS (Uni-
versity of Missouri).
The differential equation
[D
-t2D
-3t]y 0
arises in a problem describing the motion of a particle in a magnetic field.
(1) Identify the power series solutions in terms of special functions,
(2) evaluate the associated integral
fotxy(x) dx,
und
(3) find asymptotic expressions for lurge in (1) and (2).
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