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The matrix P = H R H is referred to as the information matrix Note that quality of the estimate of x (ie, P 1 ) increases as R decreases Note also that P does not depend on the actual data, but on the statistics of the measurement noise and the matrix H that relates the vector x to the noisy measurements Table 51 estimates the memory and computational requirements to implement eqns (527-528) Computational requirements are measured by counting oating point operations (FLOP s) The FLOP is a machine independent unit of measurement corresponding to one oating point multiplication and addition Note that both the computation and memory requirements grow linearly with the number of measurements m Example 51 Consider the problem of estimating the value of a scalar constant x from measurements corrupted with iid zero mean Gaussian white noise In this problem, each measurement is modeled as y k = x + k

and so on, forever The value keeps getting smaller and smaller, becoming half its former size every time you decrease the integer power by 1, but always remaining positive This sequence of values is said to converge In this case it approaches 0

1 2 2n

Now that we ve seen what happens when a number is raised to an integer power, let s find out what goes on when a number is raised to a power that is the reciprocal of an integer

.

Integer roots are reciprocal-of-integer powers Suppose we take some number or quantity a, and raise it to the power 1/p where p is a positive integer We write this as

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Table 51: Computational and memory requirements for the weighted least squares algorithm of eqns (527-528) assuming m n with x Rn , H Rm n , Y Rm , and diagonal R Rm m y where k N (0, 2 ) Therefore, after m measurements, Y = [ 1 , , ym ] Rm , H = [1, , 1] Rm and R = 2 Im According to eqn (528), Pm = (H R 1 H) 1 = 2 m (529)

a1/p We can surround the exponent with parentheses for clarity If we do that to the above expression, we get a(1/p) In this case, the parentheses are not technically necessary because the whole ratio is written as a superscript anyway When we take a reciprocal-of-integer power of a quantity, the result is often called a root If you have a number and raise it to the power 1/p, it is the same thing as taking the pth root of that number If p is a positive integer, then the pth root of a quantity is something we must multiply by itself p times in order to get that quantity

2 m

The square root If the general formulas above confuse you, it can help if we look at an example We know that

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