Discussion on the biginteger. SQRT Method

We know that. NET Framework 4 already has the system. numerics. biginteger structure. However, this biginteger structure does not have the SQRT method. Let's write one by ourselves:

 01:  Using System; 02:  Using System. numerics; 03:  04:  Namespace Skyiv 05: { 06:    Public static class  Extensions07: { 08:      Public static  Biginteger SQRT ( This  Biginteger X) 09: { 10:        If (X. Sign <0) Throw new  Argumentoutofrangeexception ( "X" , "Must be greater than or equal to zero" );11:        Biginteger Low, high; 12: Getlowandhigh (X, Out Low, Out High ); 13:        VaR Mid = low; 14:        VaR CMP = 0; 15:        While (Low. compareto (high) <= 0) 16: {17: Mid = (low + high)/2; 18: CMP = (mid * mid). compareto (X ); 19:          If (CMP <0) Low = Mid + 1; 20:          Else if (CMP> 0) High = Mid + (-1 ); 21:          Else return Mid; 22: } 23:        If (CMP> 0) Mid --;24:        Return Mid; 25: } 26:  27:      Static void Getlowandhigh ( Biginteger X, Out  Biginteger Low, Out  Biginteger High) 28: { 29:       VaR N = x. tobytearray (). length; 30:        If (N <2) 31: { 32: Low = 0; 33: High = X; 34:          Return ; 35: } 36:        VaR BS = New byte [N/2 + 1];37:        VaR K = BS. Length-2; 38:        If (N % 2 = 0) 39: { 40: BS [k] = 0x0b; 41: Low = New  Biginteger (BS ); 42: BS [k] = 0xb6; 43: High = New Biginteger (BS ); 44: } 45:        Else46: { 47: BS [k] = 0xb5; 48: Low = New  Biginteger (BS ); 49: BS [k] = 0x51; 50: BS [k + 1] = 0x0b;51: High = New  Biginteger (BS ); 52: } 53: } 54: } 55: }

AboveProgramThe getlowandhigh method of rows 27th to 53 is used to obtain the lower limit and upper limit of the square root of a specified large integer. Then, the second-point search is used in the SQRT method of rows 8th to 25 to find the square root of the request.

The biginteger. tobytearray method converts biginteger to a byte array. For biginteger greater than zero, if the length of the converted byte array is fixed, the minimum value is in the format of 00-80-00-00-00, and the maximum value is in the form of 7f-ff-ff-ff-ff, it is exactly one smaller than the minimum value of the next group. See the following table:

N	X	SQRT (X)
2	00-80	0b
3	00-80-00	00-b5
4	00-80-00-00	0b-50
5	00-80-00-00-00	00-b5-04
6	00-80-00-00-00-00	0b-50-4f
7	00-80-00-00-00-00-00-00	00-b5-04-f3
8	00-80-00-00-00-00-00-00-00	0b-50-4f-33
9	00-80-00-00-00-00-00-00-00-00	00-b5-04-f3-33

As shown in the preceding table, if X is converted to a byte array and the length is N, the length of SQRT (X) is n/2 + 1 after being converted to a byte array.

If n is an even number (lines 40th to 43 in the program ):

• The minimum value is greater than 00-0b-00-00 -...
• The maximum value is less than 00-b6-00-00 -...

If n is an odd number (lines 47th to 51 in the program ):

• The minimum value is greater than 00-b5-00-00 -...
• The maximum value is less than 0b-51-00-00 -...

The. NET Framework 2.0 does not contain biginteger. I used to write biginteger in C # and implemented the SQRT method. See the following document:

• Timus 1153. supercomputer
• About biginteger
• Let's talk about biginteger-using Fast Fourier Transformation
• Bigarithmetic-static class that provides arbitrary precision for arithmetic operations

When using Fast Fourier Transform to implement biginteger, It is a static method that calls the bigarithmetic class. The SQRT method calculates the square root using the Newton Iteration Method:

Ui + 1 = UI (3-vui2)/2

U∞ secondary convergence at 1/√V, Multiply by V to get √.V. The square root is faster than the binary search method in this article.