// The ShortDivide() algorithm works like dividing a polynomial which
// looks like:
// (ax3 + bx2 + cx + d) / N = (ax3 + bx2 + cx + d) * (1/N)
// The 1/N distributes over the polynomial:
// (ax3 * (1/N)) + (bx2 * (1/N)) + (cx * (1/N)) + (d * (1/N))
// (ax3/N) + (bx2/N) + (cx/N) + (d/N)
// The algorithm goes from left to right and reduces that polynomial
// expression. So it starts with Quotient being a copy of ToDivide
// and then it reduces Quotient from left to right.
private bool ShortDivide( Integer ToDivide,
Integer DivideBy,
Integer Quotient,
Integer Remainder )
{
Quotient.Copy( ToDivide );
// DivideBy has an Index of zero:
ulong DivideByU = DivideBy.GetD( 0 );
ulong RemainderU = 0;
// Get the first one set up.
if( DivideByU > Quotient.GetD( Quotient.GetIndex()) )
{
Quotient.SetD( Quotient.GetIndex(), 0 );
}
else
{
ulong OneDigit = Quotient.GetD( Quotient.GetIndex() );
Quotient.SetD( Quotient.GetIndex(), OneDigit / DivideByU );
RemainderU = OneDigit % DivideByU;
ToDivide.SetD( ToDivide.GetIndex(), RemainderU );
}
// Now do the rest.
for( int Count = Quotient.GetIndex(); Count >= 1; Count-- )
{
ulong TwoDigits = ToDivide.GetD( Count );
TwoDigits <<= 32;
TwoDigits |= ToDivide.GetD( Count - 1 );
Quotient.SetD( Count - 1, TwoDigits / DivideByU );
RemainderU = TwoDigits % DivideByU;
ToDivide.SetD( Count, 0 );
ToDivide.SetD( Count - 1, RemainderU ); // What's left to divide.
}
// Set the index for the quotient.
// The quotient would have to be at least 1 here,
// so it will find where to set the index.
for( int Count = Quotient.GetIndex(); Count >= 0; Count-- )
{
if( Quotient.GetD( Count ) != 0 )
{
Quotient.SetIndex( Count );
break;
}
}
Remainder.SetD( 0, RemainderU );
Remainder.SetIndex( 0 );
if( RemainderU == 0 )
return true;
else
return false;
}
internal void DoSquare( Integer ToSquare )
{
if( ToSquare.GetIndex() == 0 )
{
ToSquare.Square0();
return;
}
if( ToSquare.GetIndex() == 1 )
{
ToSquare.Square1();
return;
}
if( ToSquare.GetIndex() == 2 )
{
ToSquare.Square2();
return;
}
// Now Index is at least 3:
int DoubleIndex = ToSquare.GetIndex() << 1;
if( DoubleIndex >= Integer.DigitArraySize )
{
throw( new Exception( "Square() overflowed." ));
}
for( int Row = 0; Row <= ToSquare.GetIndex(); Row++ )
{
if( ToSquare.GetD( Row ) == 0 )
{
for( int Column = 0; Column <= ToSquare.GetIndex(); Column++ )
M[Column + Row, Row] = 0;
}
else
{
for( int Column = 0; Column <= ToSquare.GetIndex(); Column++ )
M[Column + Row, Row] = ToSquare.GetD( Row ) * ToSquare.GetD( Column );
}
}
// Add the columns up with a carry.
ToSquare.SetD( 0, M[0, 0] & 0xFFFFFFFF );
ulong Carry = M[0, 0] >> 32;
for( int Column = 1; Column <= DoubleIndex; Column++ )
{
ulong TotalLeft = 0;
ulong TotalRight = 0;
for( int Row = 0; Row <= Column; Row++ )
{
if( Row > ToSquare.GetIndex() )
break;
if( Column > (ToSquare.GetIndex() + Row) )
continue;
TotalRight += M[Column, Row] & 0xFFFFFFFF;
TotalLeft += M[Column, Row] >> 32;
}
TotalRight += Carry;
ToSquare.SetD( Column, TotalRight & 0xFFFFFFFF );
Carry = TotalRight >> 32;
Carry += TotalLeft;
}
ToSquare.SetIndex( DoubleIndex );
if( Carry != 0 )
{
ToSquare.SetIndex( ToSquare.GetIndex() + 1 );
if( ToSquare.GetIndex() >= Integer.DigitArraySize )
throw( new Exception( "Square() overflow." ));
ToSquare.SetD( ToSquare.GetIndex(), Carry );
}
}
internal void SubtractULong( Integer Result, ulong ToSub )
{
if( Result.IsULong())
{
ulong ResultU = Result.GetAsULong();
if( ToSub > ResultU )
throw( new Exception( "SubULong() (IsULong() and (ToSub > Result)." ));
ResultU = ResultU - ToSub;
Result.SetD( 0, ResultU & 0xFFFFFFFF );
Result.SetD( 1, ResultU >> 32 );
if( Result.GetD( 1 ) == 0 )
Result.SetIndex( 0 );
else
Result.SetIndex( 1 );
return;
}
// If it got this far then Index is at least 2.
SignedD[0] = (long)Result.GetD( 0 ) - (long)(ToSub & 0xFFFFFFFF);
SignedD[1] = (long)Result.GetD( 1 ) - (long)(ToSub >> 32);
if( (SignedD[0] >= 0) && (SignedD[1] >= 0) )
{
// No need to reorganize it.
Result.SetD( 0, (ulong)SignedD[0] );
Result.SetD( 1, (ulong)SignedD[1] );
return;
}
for( int Count = 2; Count <= Result.GetIndex(); Count++ )
SignedD[Count] = (long)Result.GetD( Count );
for( int Count = 0; Count < Result.GetIndex(); Count++ )
{
if( SignedD[Count] < 0 )
{
SignedD[Count] += (long)0xFFFFFFFF + 1;
SignedD[Count + 1]--;
}
}
if( SignedD[Result.GetIndex()] < 0 )
throw( new Exception( "SubULong() SignedD[Index] < 0." ));
for( int Count = 0; Count <= Result.GetIndex(); Count++ )
Result.SetD( Count, (ulong)SignedD[Count] );
for( int Count = Result.GetIndex(); Count >= 0; Count-- )
{
if( Result.GetD( Count ) != 0 )
{
Result.SetIndex( Count );
return;
}
}
// If this was zero it wouldn't find a nonzero
// digit to set the Index to and it would end up down here.
Result.SetIndex( 0 );
}
// Finding the square root of a number is similar to division since
// it is a search algorithm. The TestSqrtBits method shown next is
// very much like TestDivideBits(). It works the same as
// FindULSqrRoot(), but on a bigger scale.
/*
private void TestSqrtBits( int TestIndex, Integer Square, Integer SqrRoot )
{
Integer Test1 = new Integer();
uint BitTest = 0x80000000;
for( int BitCount = 31; BitCount >= 0; BitCount-- )
{
Test1.Copy( SqrRoot );
Test1.D[TestIndex] |= BitTest;
Test1.Square();
if( !Square.ParamIsGreater( Test1 ) )
SqrRoot.D[TestIndex] |= BitTest; // Use the bit.
BitTest >>= 1;
}
}
*/
// In the SquareRoot() method SqrRoot.Index is half of Square.Index.
// Compare this to the Square() method where the Carry might or
// might not increment the index to an odd number. (So if the Index
// was 5 its square root would have an Index of 5 / 2 = 2.)
// The SquareRoot1() method uses FindULSqrRoot() either to find the
// whole answer, if it's a small number, or it uses it to find the
// top part. Then from there it goes on to a bit by bit search
// with TestSqrtBits().
public bool SquareRoot( Integer Square, Integer SqrRoot )
{
ulong ToMatch;
if( Square.IsULong() )
{
ToMatch = Square.GetAsULong();
SqrRoot.SetD( 0, FindULSqrRoot( ToMatch ));
SqrRoot.SetIndex( 0 );
if( (SqrRoot.GetD(0 ) * SqrRoot.GetD( 0 )) == ToMatch )
return true;
else
return false;
}
int TestIndex = Square.GetIndex() >> 1; // LgSquare.Index / 2;
SqrRoot.SetDigitAndClear( TestIndex, 1 );
// if( (TestIndex * 2) > (LgSquare.Index - 1) )
if( (TestIndex << 1) > (Square.GetIndex() - 1) )
{
ToMatch = Square.GetD( Square.GetIndex());
}
else
{
// LgSquare.Index is at least 2 here.
ToMatch = Square.GetD( Square.GetIndex()) << 32;
ToMatch |= Square.GetD( Square.GetIndex() - 1 );
}
SqrRoot.SetD( TestIndex, FindULSqrRoot( ToMatch ));
TestIndex--;
while( true )
{
// TestSqrtBits( TestIndex, LgSquare, LgSqrRoot );
SearchSqrtXPart( TestIndex, Square, SqrRoot );
if( TestIndex == 0 )
break;
TestIndex--;
}
// Avoid squaring the whole thing to see if it's an exact square root:
if( ((SqrRoot.GetD( 0 ) * SqrRoot.GetD( 0 )) & 0xFFFFFFFF) != Square.GetD( 0 ))
return false;
TestForSquareRoot.Copy( SqrRoot );
DoSquare( TestForSquareRoot );
if( Square.IsEqual( TestForSquareRoot ))
return true;
else
return false;
}
// This is a variation on ShortDivide that returns the remainder.
// Also, DivideBy is a ulong.
internal ulong ShortDivideRem( Integer ToDivideOriginal,
ulong DivideByU,
Integer Quotient )
{
if( ToDivideOriginal.IsULong())
{
ulong ToDiv = ToDivideOriginal.GetAsULong();
ulong Q = ToDiv / DivideByU;
Quotient.SetFromULong( Q );
return ToDiv % DivideByU;
}
ToDivide.Copy( ToDivideOriginal );
Quotient.Copy( ToDivide );
ulong RemainderU = 0;
if( DivideByU > Quotient.GetD( Quotient.GetIndex() ))
{
Quotient.SetD( Quotient.GetIndex(), 0 );
}
else
{
ulong OneDigit = Quotient.GetD( Quotient.GetIndex() );
Quotient.SetD( Quotient.GetIndex(), OneDigit / DivideByU );
RemainderU = OneDigit % DivideByU;
ToDivide.SetD( ToDivide.GetIndex(), RemainderU );
}
for( int Count = Quotient.GetIndex(); Count >= 1; Count-- )
{
ulong TwoDigits = ToDivide.GetD( Count );
TwoDigits <<= 32;
TwoDigits |= ToDivide.GetD( Count - 1 );
Quotient.SetD( Count - 1, TwoDigits / DivideByU );
RemainderU = TwoDigits % DivideByU;
ToDivide.SetD( Count, 0 );
ToDivide.SetD( Count - 1, RemainderU );
}
for( int Count = Quotient.GetIndex(); Count >= 0; Count-- )
{
if( Quotient.GetD( Count ) != 0 )
{
Quotient.SetIndex( Count );
break;
}
}
return RemainderU;
}
internal void SubtractPositive( Integer Result, Integer ToSub )
{
if( ToSub.IsULong() )
{
SubtractULong( Result, ToSub.GetAsULong());
return;
}
if( ToSub.GetIndex() > Result.GetIndex() )
throw( new Exception( "In Subtract() ToSub.Index > Index." ));
for( int Count = 0; Count <= ToSub.GetIndex(); Count++ )
SignedD[Count] = (long)Result.GetD( Count ) - (long)ToSub.GetD( Count );
for( int Count = ToSub.GetIndex() + 1; Count <= Result.GetIndex(); Count++ )
SignedD[Count] = (long)Result.GetD( Count );
for( int Count = 0; Count < Result.GetIndex(); Count++ )
{
if( SignedD[Count] < 0 )
{
SignedD[Count] += (long)0xFFFFFFFF + 1;
SignedD[Count + 1]--;
}
}
if( SignedD[Result.GetIndex()] < 0 )
throw( new Exception( "Subtract() SignedD[Index] < 0." ));
for( int Count = 0; Count <= Result.GetIndex(); Count++ )
Result.SetD( Count, (ulong)SignedD[Count] );
for( int Count = Result.GetIndex(); Count >= 0; Count-- )
{
if( Result.GetD( Count ) != 0 )
{
Result.SetIndex( Count );
return;
}
}
// If it never found a non-zero digit it would get down to here.
Result.SetIndex( 0 );
}
internal int MultiplyUIntFromCopy( Integer Result, Integer FromCopy, ulong ToMul )
{
int FromCopyIndex = FromCopy.GetIndex();
// The compiler knows that FromCopyIndex doesn't change here so
// it can do its range checking on the for-loop before it starts.
Result.SetIndex( FromCopyIndex );
for( int Column = 0; Column <= FromCopyIndex; Column++ )
Scratch[Column] = ToMul * FromCopy.GetD( Column );
// Add these up with a carry.
Result.SetD( 0, Scratch[0] & 0xFFFFFFFF );
ulong Carry = Scratch[0] >> 32;
for( int Column = 1; Column <= FromCopyIndex; Column++ )
{
ulong Total = Scratch[Column] + Carry;
Result.SetD( Column, Total & 0xFFFFFFFF );
Carry = Total >> 32;
}
if( Carry != 0 )
{
Result.IncrementIndex(); // This might throw an exception if it overflows.
Result.SetD( FromCopyIndex + 1, Carry );
}
return Result.GetIndex();
}
// This is another optimization. This is used when the top digit
// is 1 and all of the other digits are zero.
// This is effectively just a shift-left operation.
internal void MultiplyTopOne( Integer Result, Integer ToMul )
{
// try
// {
int TotalIndex = Result.GetIndex() + ToMul.GetIndex();
if( TotalIndex >= Integer.DigitArraySize )
throw( new Exception( "MultiplyTopOne() overflow." ));
for( int Column = 0; Column <= ToMul.GetIndex(); Column++ )
Result.SetD( Column + Result.GetIndex(), ToMul.GetD( Column ));
for( int Column = 0; Column < Result.GetIndex(); Column++ )
Result.SetD( Column, 0 );
// No Carrys need to be done.
Result.SetIndex( TotalIndex );
/*
}
catch( Exception ) // Except )
{
// "Bug in MultiplyTopOne: " + Except.Message
}
*/
}
// This is an optimization for multiplying when only the top digit
// of a number has been set and all of the other digits are zero.
internal void MultiplyTop( Integer Result, Integer ToMul )
{
// try
// {
int TotalIndex = Result.GetIndex() + ToMul.GetIndex();
if( TotalIndex >= Integer.DigitArraySize )
throw( new Exception( "MultiplyTop() overflow." ));
// Just like Multiply() except that all the other rows are zero:
for( int Column = 0; Column <= ToMul.GetIndex(); Column++ )
M[Column + Result.GetIndex(), Result.GetIndex()] = Result.GetD( Result.GetIndex() ) * ToMul.GetD( Column );
for( int Column = 0; Column < Result.GetIndex(); Column++ )
Result.SetD( Column, 0 );
ulong Carry = 0;
for( int Column = 0; Column <= ToMul.GetIndex(); Column++ )
{
ulong Total = M[Column + Result.GetIndex(), Result.GetIndex()] + Carry;
Result.SetD( Column + Result.GetIndex(), Total & 0xFFFFFFFF );
Carry = Total >> 32;
}
Result.SetIndex( TotalIndex );
if( Carry != 0 )
{
Result.SetIndex( Result.GetIndex() + 1 );
if( Result.GetIndex() >= Integer.DigitArraySize )
throw( new Exception( "MultiplyTop() overflow." ));
Result.SetD( Result.GetIndex(), Carry );
}
/*
}
catch( Exception ) // Except )
{
// "Bug in MultiplyTop: " + Except.Message
}
*/
}
// See also: http://en.wikipedia.org/wiki/Karatsuba_algorithm
internal void Multiply( Integer Result, Integer ToMul )
{
// try
// {
if( Result.IsZero())
return;
if( ToMul.IsULong())
{
MultiplyULong( Result, ToMul.GetAsULong());
SetMultiplySign( Result, ToMul );
return;
}
// It could never get here if ToMul is zero because GetIsULong()
// would be true for zero.
// if( ToMul.IsZero())
int TotalIndex = Result.GetIndex() + ToMul.GetIndex();
if( TotalIndex >= Integer.DigitArraySize )
throw( new Exception( "Multiply() overflow." ));
for( int Row = 0; Row <= ToMul.GetIndex(); Row++ )
{
if( ToMul.GetD( Row ) == 0 )
{
for( int Column = 0; Column <= Result.GetIndex(); Column++ )
M[Column + Row, Row] = 0;
}
else
{
for( int Column = 0; Column <= Result.GetIndex(); Column++ )
M[Column + Row, Row] = ToMul.GetD( Row ) * Result.GetD( Column );
}
}
// Add the columns up with a carry.
Result.SetD( 0, M[0, 0] & 0xFFFFFFFF );
ulong Carry = M[0, 0] >> 32;
for( int Column = 1; Column <= TotalIndex; Column++ )
{
ulong TotalLeft = 0;
ulong TotalRight = 0;
for( int Row = 0; Row <= ToMul.GetIndex(); Row++ )
{
if( Row > Column )
break;
if( Column > (Result.GetIndex() + Row) )
continue;
// Split the ulongs into right and left sides
// so that they don't overflow.
TotalRight += M[Column, Row] & 0xFFFFFFFF;
TotalLeft += M[Column, Row] >> 32;
}
TotalRight += Carry;
Result.SetD( Column, TotalRight & 0xFFFFFFFF );
Carry = TotalRight >> 32;
Carry += TotalLeft;
}
Result.SetIndex( TotalIndex );
if( Carry != 0 )
{
Result.IncrementIndex(); // This can throw an exception if it overflowed the index.
Result.SetD( Result.GetIndex(), Carry );
}
SetMultiplySign( Result, ToMul );
}
private void AddUpAccumulateArray( Integer Result, int HowManyToAdd, int BiggestIndex )
{
try
{
for( int Count = 0; Count <= (BiggestIndex + 1); Count++ )
Result.SetD( Count, 0 );
Result.SetIndex( BiggestIndex );
for( int Count = 0; Count < HowManyToAdd; Count++ )
{
int HowManyDigits = AccumulateArray[Count].GetIndex() + 1;
for( int CountDigits = 0; CountDigits < HowManyDigits; CountDigits++ )
{
ulong Sum = AccumulateArray[Count].GetD( CountDigits ) + Result.GetD( CountDigits );
Result.SetD( CountDigits, Sum );
}
}
// This is like ax + by + cz + ... = Result.
// You know what a, b, c... are.
// But you don't know what x, y, and z are.
// So how do you reverse this and get x, y and z?
// Is this reversible?
Result.OrganizeDigits();
}
catch( Exception Except )
{
throw( new Exception( "Exception in AddUpAccumulateArray(): " + Except.Message ));
}
}