1 / 27

Introduction

HKOI Intermediate Training. High Precision Arithmetic. Introduction. Overview. Introduction :: Overview. HKOI Intermediate Training :: High Precision Arithmetic. Why HPA (High Precision Arithmetic)?. LongInt (32-bit signed) : -2147483648 .. 2147483647

lyris
Download Presentation

Introduction

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. HKOI Intermediate Training High Precision Arithmetic Introduction Overview

  2. Introduction :: Overview HKOI Intermediate Training :: High Precision Arithmetic Why HPA (High Precision Arithmetic)? • LongInt (32-bit signed) : -2147483648 .. 2147483647 • Cardinal (32-bit unsigned): 0 .. 4294967295 • Int64 (64-bit signed) : -9223372036854775808 .. 9223372036854775807 • - QWord (64-bit unsigned): 0 .. 18446744073709551615 • Sometimes, we need to deal with really large numbers (10000 digits~) • Often integrated with other problems • Don’t Fear

  3. Introduction :: Overview HKOI Intermediate Training :: High Precision Arithmetic HPA Operations • Addition, Subtraction (negative addition) • Multiplication, Division • Arbitrary base arithmetic • 123(8) + 456(8) = 601(8) • 45A(H) + C9F(H) = 10F9(H) • Usually only with integers

  4. Introduction :: Overview HKOI Intermediate Training :: High Precision Arithmetic HPA Extensions • Exponent (^), Modulus (mod) • Factorial (!) • Composed of basic HPA operations • Recommended : Write your standard HPA functions • in a modular manner • - Recommended : Do not modify the operands Procedure HPAadd(A as HPAtype; B as HPAtype; var C as HPAtype); Function HPAadd(A as HPAtype; B as HPAtype) : HPAtype;{may not work!}

  5. HKOI Intermediate Training High Precision Arithmetic Introduction Representation

  6. Introduction :: Representation HKOI Intermediate Training :: High Precision Arithmetic Common Representations • Array of byte • Type HPAtype = Array[1..1024] of byte; • Advantage : Can apply the integer operations directly • String / ANSIstring • Type HPAtype = ANSIstring; • Advantage : Simple Input Strength of each one is the weakness of the other!

  7. Introduction :: Representation HKOI Intermediate Training :: High Precision Arithmetic Variation in Representations • Zero based or One-based? • Type HPAtype = Array[1..1024] of byte; • Type HPAtype = Array[0..1023] of byte; • Zero based : Corresponds to our concept of digit-value • However, it is not applicable to Strings • Can pack multiple digits into the same cell • Byte/ShortInt - 2 digits • SmallInt - 4 digits • Longint/Cardinal - 9 digits • Int64 - 18 digits • QWord - 19 digits (!) • Advantage : Reduces looping overhead, saves memory

  8. Introduction :: Representation HKOI Intermediate Training :: High Precision Arithmetic Input Considerations • Highly related to your representation • Arrays : Usually the lowest cell stores • the least significant digit (LSD) Array [ 8 7 6 5 4 3 2 1 ] 00 1 3 1 0 7 2 • Strings : If you use readln() directly, • the lowest cell stores the MSD! • You may have to reverse the string first, • in order to make the HPA functions easy • to understand / debug String [ 1 2 3 4 5 6 7 8 ] 1 3 1 0 7 2 0 0

  9. Introduction :: Representation HKOI Intermediate Training :: High Precision Arithmetic Output Considerations • Arrays : When to stop? • Solution 1 : Loop from the end (MSD) back • to the start, and check for the • first non-zero digit • Problem : Can be very time-consuming • Solution 2 : Keep a “size” variable • Problem : Your number can expand or shrink during operations • Increase size whenever needed • Use Solution 1, but loop from size back to the start to check Array [ 8 7 6 5 4 3 2 1 ] 00 1 3 1 0 7 2

  10. Introduction :: Representation HKOI Intermediate Training :: High Precision Arithmetic Output Considerations • Strings : Simple? No. • You have to take care of the length! • ShortStrings : s[0] := chr(size); • - ANSIstring : setlength(s,size); • Since ANSIstring is a pointer type and is dynamic, you must • set the length before accessing the individual cells Var S : ANSIstring; S[5] := ‘x’; SetLength(S,5); Writeln(S); { RUNTIME ERROR } Var S : ANSIstring; SetLength(S,5); S[5] := ‘x’; Writeln(S); { WORKS FINE }

  11. HKOI Intermediate Training High Precision Arithmetic Basic Operations Addition

  12. Basic Operations :: Addition HKOI Intermediate Training :: High Precision Arithmetic How do we do addition? • For normal numeric types, there are hard wired • circuits for + - * / • For HPA types, there is no standard way of doing so, • so you have to implement your own “addition algorithm” • - RETURN TO THE PRIMARY SCHOOL!

  13. Basic Operations :: Addition HKOI Intermediate Training :: High Precision Arithmetic Addition Algorithm • Assumed : 2 non-negative integers • Add the digits one by one • Keep a “carry” • Think about the exact process now • Extra Problem : When to stop? [ 4 3 2 1 ] 1 2 3 +) 4 5 ---------- 1 6 8 [ 4 3 2 1 ] 3 9 +) 8 5 ---------- 1 2 4

  14. Basic Operations :: Addition HKOI Intermediate Training :: High Precision Arithmetic Addition Algorithm • Lazy method: • Carry := 0; • For I := 1 to Max(SizeA,SizeB)+5 Do Begin • C[I] := A[I] + B[I] + Carry; • Carry := C[I] mod 10; • C[I] := C[I] div 10; • If (C[I] <> 0) and (I > SizeC) SizeC := I; • End; [ 4 3 2 1 ] 1 2 3 +) 4 5 ---------- 1 6 8 [ 4 3 2 1 ] 3 9 +) 8 5 ---------- 1 2 4 • Does it always work? • - Think about the growth rate of Carry

  15. Basic Operations :: Addition HKOI Intermediate Training :: High Precision Arithmetic Negative Numbers • In the previous section, we assumed that • the integers to be added are both non-negative • If negative values are allowed, the simple • add-and-carry approach will not succeed, • moreover, even the representation has problems! • More cases : • Negative + Negative = Negative • Smaller Positive + Bigger Negative = Negative • Bigger Positive + Smaller Negative = Positive [ 4 3 2 1 ] 3 9 +) - 8 5 ---------- ?????

  16. Basic Operations :: Addition HKOI Intermediate Training :: High Precision Arithmetic Negative Representation • You can use a flag (e.g. 1 or –1) • Negative + Negative : • Do it the normal way, but mark the output as Negative • Positive + Negative : • Check the absolute sizes, and do a subtraction • Then set a correct flag in the output [ 4 3 2 1 ] 3 9 +) - 8 5 ---------- ?????

  17. HKOI Intermediate Training High Precision Arithmetic Basic Operations Subtraction

  18. Basic Operations :: Subtraction HKOI Intermediate Training :: High Precision Arithmetic Subtraction Algorithm • Assumed : 2 non-negative integers • Subtract instead of adding the digits • Borrow when necessary • When to stop : You should have figured it out • New problem : The size may shrink [ 4 3 2 1 ] 3 9 -) 8 5 ---------- - 4 6

  19. Basic Operations :: Subtraction HKOI Intermediate Training :: High Precision Arithmetic Subtraction Algorithm • Pseudocode • GreatThan(A,B) :- • Returns TRUE if A is greater than B • (It checks the negative flag of A and B) • If (GreaterThan(A,B)) Then • C->Flag := +1; • C->Number := NonNegativeSubtract(A->Number,B->Number); • Else • C->Flag := -1; • C->Number := NonNegativeSubtract(B->Number,A->Number); • End If [ 4 3 2 1 ] 3 9 -) 8 5 ---------- - 4 6

  20. HKOI Intermediate Training High Precision Arithmetic Basic Operations Multiplication

  21. Basic Operations :: Multiplication HKOI Intermediate Training :: High Precision Arithmetic Multiplication Algorithm • Positive/Negative is no longer a problem • How many digits will the output have? • Each output digit is a summation of multiplied values • In the example : 7 = 1*5 + 2*9 + 5*9/10 (carry) • How many terms can there be? • General Strategy : Use 2 loops [ 4 3 2 1 ] 1 9 *) 2 5 ---------- 4 7 6

  22. Basic Operations :: Multiplication HKOI Intermediate Training :: High Precision Arithmetic Forward Multiplication • Loop through each (A[I],B[J]) combination • Where does A[I]*B[J] -> C[K] go? • During the multiplication process, you can • accumulate up the values without worrying • about the size of integers (why?) • Finally, make the number stored in each cell • single-digited [ 4 3 2 1 ] 1 9 *) 2 5 ---------- 4 7 6

  23. Basic Operations :: Multiplication HKOI Intermediate Training :: High Precision Arithmetic Backward Multiplication • Loop through each (C[K],A[I]) combination • What is the corresponding B[J]? • This algorithm can be adapted such that every • number calculated can be printed instantly (how?) [ 4 3 2 1 ] 1 9 *) 2 5 ---------- 4 7 6

  24. HKOI Intermediate Training High Precision Arithmetic Basic Operations Division

  25. Basic Operations :: Division HKOI Intermediate Training :: High Precision Arithmetic Division Algorithm • Positive/Negative is also not a problem • This is the most difficult operation • Align the MSD of the operands • Repeatly subtract the divider instead of “finding the • correct multiple” • When shifting the divider, do it blindly instead of • finding the first non-zero digit. This will simplify • the work and reduce the chance of bug

  26. HKOI Intermediate Training High Precision Arithmetic Extended Operations Others

  27. Basic Operations :: Multiplication HKOI Intermediate Training :: High Precision Arithmetic H-L Precision Multiplication • Sometimes, you need to multiply a large number • with a ordinary sized one • Example : Factorial • Use integer / longint for each cell • Do the multiplication directly for each digit, • and finally convert it into single-digited

More Related