C++ for Scientists - Technische Universität Dresden

282 CHAPTER 14. NUMERICAL EXERCISES
When measurements come in sequentially, i.e. at time steps t1, t2, . . ., we receive at time step
tj the jth row of A and the jth element of b. In the algorithms we now discuss, we have the
following
14.7.1 The least squares QR algorithm
A numerically stable method for solving (14.8) is based on the QR factorization. The QR
factorization of the m × n matrix A is
A = QR
with Q ∈ R m×n and unitary (Q T Q = I) and R ∈ R n×n upper triangular. If A has full rank,
the diagonal elements of R are non-zero. Suppose we have computed the solution for
where
�Akx − bk�2 min
�Akx − bk�2 = �QkRkx − bk�2
= �Rkx − Q T k bk�2
We have to solve an upper triangular linear system. We can develop a ‘sequential’ method for
this QR decomposition without storing Q, but we will not discuss this any further.
14.7.2 The least squares method via the normal equations
One method to achieve this are the normal equations. That is, multiply (14.7) on the left by
A T , then we obtain
A T Ax = A T b (14.9)
If the A has full column rank, the solution of x is unique and satisfies (14.8).
14.7.3 Least squares Kalman filtering
The Kalman filter is a method to solve the normal equations (14.9) in a step by step way, i.e.
the measurements come in time step at time step. The Kalman filter adapts the least squares
solution to the newly arrived data.
Suppose we have computed the least squares solution of
Akxk = bk
where Ak are the first k rows of A and bk the first k elements of b with k ≥ n. Then we want
to compute the least squares solution of
Since
Ak+1 =
� Ak
a T k+1
Ak+1xk+1 = bk+1 .
�
�
and bk+1 =
bk
f(tk+1)
�