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where the pre x value (P#) is calculated to provide the function value. X# is the parameter value that is modi ed for each value in the series. The parameter change can be a number of different operations, including squaring, square rooting, multiplying by the power of a negative number, and so on. For the multiplication and division operations shown here, I will be shifting the parameter by 1 bit for each series element. The theory and mathematics of calculating the pre x and parameter for arithmetic series can be quite complex but it can be used in cases such as producing the pre x values for simple multiplication or division operations, as I am going to show in this section. To demonstrate the operations, I have created the multiply and divide macros that can be found in the muldiv.inc le in the Macros\ MulDiv folder. The two macros provide the multiplication and division functions using MPLAB assembler capabilities that I haven t explained yet (although I do in Chap. 10). To avoid confusion, I will explain how the macros work from the perspective of a high level language before presenting the actual code. To further help explain how the macros work, I will present them from the perspective of implementing the function in straight PIC microcontroller assembler. Multiplication (and division) can be represented by a number of different methods. When you were taught basic arithmetic, multiplication was repeated addition. If you had to program it, you would use the high level code

Product = 0; for (i = 0; i < Multiplier; i++ ) Product = Product + Multiplicand;

This method works very well but will take a differing amount of time based on the multiplier (i.e., eight times something takes four times longer than two times something). This is not a problem for single-digit multiplication, but when multiplication gets more complex, the operations become signi cantly longer, which can have a negative impact on operation of the application code. Ideally, a multiplication method (or algorithm) that does not have such extreme ranges should be used. As you would expect, this is where the arithmetic series is involved. As you know, every number consists of constants multiplied by exponents of the number s base. For example, 123 decimal is actually

The energy balanc*e must also be cbloscd. If more liquid is boiled than condenses, there will be an increase in vapor invent,ory which will cause the pressure to rise. l rcssure csont,rol is very important in order to maintain equilibrium.

This also works for binary numbers and is used to convert constants between numbering systems. 123 decimal is 1111011 binary (0x07B). This can be represented like 123 decimal above as

In this binary sequence, I also have included any digits that are zero (which is 4 in the case of 123) because they will be used when multiplying two numbers together. This binary sequence can be used as the pre x of a multiplication arithmetic series if each value is used to add the multiplicand that has been shifted up by the number of the bit. The shifted-up multiplicand can be thought of as the parameter of the series. This series can be written out as

A * B = ((A ((A ((A ((A & (1 << & (1 << 1)) & (1 << 2)) & (1 << 7)) 0)) != 0)*(B << 0) + != 0)*(B << 1) + != 0)*(B << 2) + + != 0)*(B << 7)

This series can be converted to high level code very easily:

int Multiply(int A, int B) // Multiply two eight bit values // together and return the result { int Product = 0; int i; for (i = 0; i < 8; i++) { if ((A & 1) != 0) Product = Product + B; // Repeat for each bit // // // // // Bit of Multiplier is Set, Add Multiplicand to Product Shift down the Multiplier Shift up the Multiplicand

Each person in your audience is attending for his or her own reasons. Those who want to be there are already motivated to listen, and those who would rather be anywhere else need to be encouraged to tune in to you. Listeners in the latter group need to know the WIIFM factor (What s In It For Me ). Engage them as soon as possible. Let them know that you have something to offer them right up front. Similarly, hostile audiences should feel that you understand their position, even though it might differ from your own.

A = A >> 1; B = B << 1; } return Product;

This function will loop only eight times, and each time will shift up the multiplicand, which is the B << 2 term in this series, and shift down the multiplier, which is the equivalent of the (A & (1 << 0)) != 0 term in this series. This term is 100 percent mathematically correct; if the pre x result is not equal to zero, then the shifted term will be added to the Product.