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Now that our ef cient scheme for polynomial multiplication is fully realized, let s hone in more closely on the core subroutine that makes it all possible, the fast Fourier transform The de nitive FFT algorithm The FFT takes as input a vector a = (a0 , , an 1 ) and a complex number whose powers 1, , 2 , , n 1 are the complex nth roots of unity It multiplies vector a by the n n matrix Mn ( ), which has (j, k)th entry (starting row- and column-count at zero) jk The potential for using divide-and-conquer in this matrix-vector multiplication becomes apparent when M s columns are segregated into evens and odds:

Figure 5-3 illustrates this process Because the multiplier is negative, we jump to the opposite side of the number reflector each time we multiply Then the result becomes a new multiplicand Because the

Start at -1 -4

Column 2k a0 a1 a2 a3 a4 an 1 Mn ( ) a Even columns Odd columns Column 2k a0 a2 an 2 a1 a3 an 1

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hate to put my life into any one single indicator, no matter what, he suggests To confirm an MACD bottom, Appel also looks at the New High New Low indicator A diminution of new lows sometimes takes place even when the market averages are dropping down, he says You begin to see more and more stocks holding support That s a pretty good indication that some significant reversal is going to take place in the stock market The opposite occurs at market tops, Appel says: As long as the number of new issues making new highs keeps expanding, you re getting pretty good upside momentum The number of new highs will decrease as the advance matures The number of stocks participating and making new highs diminishes before the market averages turn down Appel says you want to use different parameters to help recognize a gradual shift in the market momentum: If you are using MACD, I like to work with different time frames at the same time Then all of the market events taking place begin to confirm together But if you get a feeling the market is coming down and the short-term MACD begins to turn up, you want to see if it s a little blip or if you are getting changes in the intermediate MACD patterns The more confluence you have, the better the chances you are going in the right direction And likewise, he says, as the market goes up and begins to turn, the shorter-term MACD begins to weaken first Perhaps most important, don t make any sudden moves until you get confirmation from other indicators

1 and multiply by 2 over and over We jump back and forth across the number reflector, doubling our distance from it with each jump To avoid clutter, only the even-integer points are shown on the number line

absolute value of the multiplier is 2, we double our distance from the number reflector with each jump Expressed as equations, we have 1 ( 2) = 2 2 ( 2) = 4 4 ( 2) = 8 8 ( 2) = 16 16 ( 2) = 32 and so on, forever

In the second step, we have simpli ed entries in the bottom half of the matrix using n/2 = 1 and n = 1 Notice that the top left n/2 n/2 submatrix is M n/2 ( 2 ), as is the one on the bottom left And the top and bottom right submatrices are almost the same as M n/2 ( 2 ), but with their jth rows multiplied through by j and j , respectively Therefore the nal product is the vector 70

Although other people have made tremendous claims about MACD, Appel is humble about what it can do: Basically, it gives you a picture of how market momentum is going, he says But there is a difference between price direction and momentum In other words, if you re rising at a steep rate, you will see a lot of upside momentum But suppose at some point, as it always does, the rise doesn t continue at that angle but continues at a slower

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