SOME NOTES ON JAHN'S METHOD FOR THE
IMPROVEMENT OF APPROXIMATE LATENT
ROOTS AND VECTORS OF A SQUARE MATRIX
By A. R. COLLAR (The University, Bristol)
[Received 7 October 1947]
SUMMARY
A proof of Jahn's method is given in matrix notation and it is pointed out that
the latent roots are obtained with errors of the second order of small quantities
when the original vectors are accurate to first order only. The accuracy required
at various stages of the computation is discussed and a slight simplification for
subsequent approximations is indicated.
1. Introduction
IN many problems of mechanical vibration and of flutter, the accurate
determination of frequencies and modes of vibration is an important
matter. By the use of Rayleigh's principle accurate frequencies may be
found from relatively inaccurate modes; but when the modes are required
accurately considerable labour is often necessary. In (1)| Jahn has given
an interesting discussion of a method for the improvement of approximate
latent roots and the associated vectors of a square matrix, which is not
necessarily restricted to be the matrix of a dynamical system. It appears
to the writer that there are one or two interesting avenues of thought
arising from the paper which deserve further consideration.
2. Statement of Jahn's method
The method can be very simply stated in terms of matrices, and indeed
the matrix presentation is indicated in § 3 of the original paper. In brief,
the problem is, given a square matrix A, which for simplicity we assume
to have distinct latent roots.J to find as exactly as possible the modal
matrix x and the diagonal matrix A of the latent roots such that
Ax = xA,
(1)
starting from a given approximate matrix x0.
Jahn's method is expressible as follows. Evaluate
x^AX{] = Ax+o,
(2)
r
where A1 is the matrix of the diagonal elements of x^ Ax0 fcnd a is the
matrix of the remaining elements. Since x0 is approximately equal to x,
Ax will be approximately equal to A and the quantities arg will be small.
t See reference (1) at the end.
% It does not appear to tu essential to exclude complex roots, or to assume that A is
real.
146
A. R. COLLAR
Next, from a determine a matrix 6 such that, if Alr and Au are the rth
and sth (diagonal) elements of Ax,
so that the quantities brs are also small. When 6 has been found, an
improved modal matrix can be obtained from the equation
xx = *„(/+&).
(4)
For equation (3), in matrix form, is
a = bA1-A1b,
(5)
and substitution in (2) gives
XQ1AX0 = A j + f c A ^ A ^ .
(6)
Hence, from (4) and (6),
Ax,. = Axo(I+b)
=
xo(A1+bA1-A1b)(I+b)
= ar o (/+6)A 1 +a; o (6A 1 -A 1 6)6
= a;1A1+terms of order b2.
(7)
If the elements of a are of the first order of smallness, so also are those
of b, and we have thus derived matrices xlt Alt which satisfy (1) with
errors of the second order of smallness only.
3. Some comments
The presentation of § 2 shows that, if one of the standard methods is
used for the determination of x^1, the improved matrix xt can be obtained
without the evaluation of determinants. The product XQ1AX0 is found and
(3) used directly to find b, when xx is found directly from (4).
An interesting point arises from the result (7). It is that A t is correct
to the second order of small quantities, although it was derived from (2)
in which xQ is accurate only to the first order. The parallel with Rayleigh's
theorem will at> once spring to mind; but this result appears to be more
general since less restriction is placed on the nature of A. However, it
must be emphasized that x^1 must be found accurately, i.e. so that
X^XQ = / with errors of second order at most, or the result is no longer
true: see below.
No proof that Ax is correct to the second order other than that contained
in (7) is really necessary, but it may be helpful to give an alternative.
Let x = x o +e; then (1) yields
Ax0 = —
or
1
XQ AX0
= A+x^eA—x^^Ae.
(8)
SOME NOTES ON JAHN'S METHOD
147
On substitution for A on the right-hand side,
or
*olAxo = A+(pA—Ap)+Zo1e(pA—Ap),
(9)
1
where
2> = (xo+e)- ^.
Now the diagonal terms oi pA—Ap are all zero; thus the only terms
modifying A arise from the third factor in (9) which is of the order e2.
But if an accurate reciprocal of x0 is not found, it is readily shown that
an expression px A—Ap will appear in (9) where px and p, though small,
are not equal; and first-order errors will therefore appear in the approximation to A. If this happens, then there must, by (7), be first-order errors
in xx also, i.e. the iterative process proposed will only converge provided
appropriate accuracy is employed at each stage.
4. The second approximation
It is worth remarking that in a second approximation, which requires the
inversion of xv it is possible to make use of the known inverse of x0; this
is a simple numerical process involving only matrix summations and
multiplications. Thus, for example, since
xx = xo(I+b),
where xr is accurate to second order,
•-K"1
(io)
to any desired degree of accuracy. Alternatively, since x^1 as already
found initially is an approximate reciprocal of xv the accurate reciprocal
can be found by the successive approximation method of §4.11 of
Elementary Matrices (2), which converges with great rapidity.
5. A simple example
A - ['» »].
,.„
Then the exact solution of equation (1) may be written
rio 9] ri-5 —0-751
ri-5 _0-751
0-75] [16
0]
1 JL 0 -2\'
i.e.
(12)
1-5 —0-751
A f
A=
16
[0
0
J
1
- 2 JJ
(13)
148
SOME NOTES ON JAHN'S METHOD
Suppose the given approximate matrix x0 is
so that 'first-order' errors of the order of 30 per cent, have been introduced.
Then
and on multiplying out
_,.
J49
-51
Hence, taking the diagonal elements
so that the errors in the latent roots are relatively small. Also
and hence, with Ax—Ag = 18f = ^,
When constants of proportionality are removed from the columns this
gives approximately
-507 -0-7541
1 J'
in good agreement with the true answer.
REFERENCES
1. JAHN, 'Improvement of an approximate set of latent roots and modal columns
of a matrix by methods akin to those of classical perturbation theory':
See above, pp. 131-44.
2. FKAZER, DUNCAN, and COIXAK, Elementary Matrices (Cambridge Univ. Press,
1938).