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System of linear congruences

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System of linear congruences


Mike Sheppard (view profile)


11 Sep 2011 (Updated )

Solution to Simultaneous Linear Congruences

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X = LINCON(A,B,N) solves the system of linear congruences
A(1) * X == B(1) (mod N(1))
A(2) * X == B(2) (mod N(2))
A(m) * X == B(m) (mod N(m))
The solution, X, will be given as a vector [x1 x2] representing the general solution
   X == x1 (mod x2)
Any specific answer can be found by letting X = x1 + x2 * k for any integer value of k.
If no solution exists [NaN NaN] will be returned.
A scalar input functions as a constant vector of the same size as the other inputs.
Program is compatible with Variable Precision Integer Arithmetic Toolbox available on File Exchange (#22725) Use of VPI is recommended for large magnitude inputs or outputs. If VPI is not used and internal calculations exceed largest consecutive flint a warning will be given that results may be inaccurate, along with [NaN NaN].
Example #1:
Solve the following system of linear congruences
   2x == 2 (mod 6)
   3x == 2 (mod 7)
   2x == 4 (mod 8)
   a=[2 3 2]; b=[2 2 4]; n=[6 7 8];
   isequal( mod(a*x(1),n) , b)
Example #2:
   Use of VPI for large magnitude numbers. Solve the following system of linear congruences
   (1234567)x == 89 (mod 2^32)
   (9876543)x == 21 (mod 3^50)
   (5555)x == 62830211 (mod 7^10)
   a=[1234567 9876543 5555]; b=[89 21 62830211];
   n=[vpi(2)^32 vpi(3)^50 vpi(7)^10];
   isequal( mod(a*x(1),n) , b)


Variable Precision Integer Arithmetic inspired this file.

MATLAB release MATLAB 7.12 (R2011a)
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01 Dec 2011 Su Dongcai

Su Dongcai (view profile)

04 Apr 2012 1.1

Updated help section

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