This file explains the functions of the library routines.
Note:
Type FiniteField is defined as unsigned long
Type Double is defined as double
extern long nullspaceLong(const long n,
const long m,
const long *A,
mpz_t * *mp_N_pass);
/*
* Calling Sequence:
* nullspaceLong(n, m, A, mp_N_pass)
*
* Summary:
* Compute the right nullspace of A.
*
* Input: n: long, row dimension of A
* m: long, column dimension of A
* A: 1-dim signed long array length n*m, representing n x m matrix
* in row major order
*
* Output:
* - *mp_N_pass: points to a 1-dim mpz_t array of length m*s, where s is the
* dimension of the right nullspace of A
* - the dimension s of the nullspace is returned
*
* Notes:
* - The matrix A is represented by one-dimension array in row major order.
* - Space for what mp_N_points to is allocated by this procedure: if the
* nullspace is empty, mp_N_pass is set to NULL.
*/
extern long nullspaceMP(const long n,
const long m,
const mpz_t *A,
mpz_t * *mp_N_pass);
/*
* Calling Sequence:
* nullspaceMP(n, m, A, mp_N_pass)
*
* Summary: Compute the right nullspace of A. In this function A is a
* 1-dimensional mpz_t array.
*
* Input: n: long, row dimension of A
* m: long, column dimension of A
* A: 1-dim mpz_t array length n*m, representing n x m matrix
* in row major order
*
* Output:
* - *mp_N_pass: points to a 1-dim mpz_t array of length m*s, where s is the
* dimension of the right nullspace of A
* - the dimension s of the nullspace is returned
*
* Notes:
* - The matrix A is represented by one-dimension array in row major order.
* - Space for what mp_N_points to is allocated by this procedure: if the
* nullspace is empty, mp_N_pass is set to NULL.
*/
extern void nonsingSolvMM (const enum SOLU_POS solupos, const long n,
const long m, const long *A, mpz_t *mp_B,
mpz_t *mp_N, mpz_t mp_D);
/*
* Calling Sequence:
* nonsingSolvMM(solupos, n, m, A, mp_B, mp_N, mp_D)
*
* Summary:
* Solve nonsingular system of linear equations, where the left hand side
* input matrix is a signed long matrix.
*
* Description:
* Given a n x n nonsingular signed long matrix A, a n x m or m x n mpz_t
* matrix mp_B, this function will compute the solution of the system
* 1. AX = mp_B
* 2. XA = mp_B.
* The parameter solupos controls whether the system is in the type of 1
* or 2.
*
* Since the unique solution X is a rational matrix, the function will
* output the numerator matrix mp_N and denominator mp_D respectively,
* such that A(mp_N) = mp_D*mp_B or (mp_N)A = mp_D*mp_B.
*
* Input:
* solupos: enumerate, flag to indicate the system to be solved
* - solupos = LeftSolu: solve XA = mp_B
* - solupos = RightSolu: solve AX = mp_B
* n: long, dimension of A
* m: long, column or row dimension of mp_B depending on solupos
* A: 1-dim signed long array length n*n, representing the n x n
* left hand side input matrix
* mp_B: 1-dim mpz_t array length n*m, representing the right hand side
* matrix of the system
* - solupos = LeftSolu: mp_B a m x n matrix
* - solupos = RightSolu: mp_B a n x m matrix
*
* Output:
* mp_N: 1-dim mpz_t array length n*m, representing the numerator matrix
* of the solution
* - solupos = LeftSolu: mp_N a m x n matrix
* - solupos = RightSolu: mp_N a n x m matrix
* mp_D: mpz_t, denominator of the solution
*
* Precondition:
* A must be a nonsingular matrix.
*
* Note:
* - It is necessary to make sure the input parameters are correct,
* expecially the dimension, since there is no parameter checks in the
* function.
* - Input and output matrices are row majored and represented by
* one-dimension array.
* - It is needed to preallocate the memory space of mp_N and mp_D.
*
*/
extern void nonsingSolvLlhsMM (const enum SOLU_POS solupos, const long n,
const long m, mpz_t *mp_A, mpz_t *mp_B,
mpz_t *mp_N, mpz_t mp_D);
/*
* Calling Sequence:
* nonsingSolvLlhsMM(solupos, n, m, mp_A, mp_B, mp_N, mp_D)
*
* Summary:
* Solve nonsingular system of linear equations, where the left hand side
* input matrix is a mpz_t matrix.
*
* Description:
* Given a n x n nonsingular mpz_t matrix A, a n x m or m x n mpz_t
* matrix mp_B, this function will compute the solution of the system
* 1. (mp_A)X = mp_B
* 2. X(mp_A) = mp_B.
* The parameter solupos controls whether the system is in the type of 1
* or 2.
*
* Since the unique solution X is a rational matrix, the function will
* output the numerator matrix mp_N and denominator mp_D respectively,
* such that mp_Amp_N = mp_D*mp_B or mp_Nmp_A = mp_D*mp_B.
*
* Input:
* solupos: enumerate, flag to indicate the system to be solved
* - solupos = LeftSolu: solve XA = mp_B
* - solupos = RightSolu: solve AX = mp_B
* n: long, dimension of A
* m: long, column or row dimension of mp_B depending on solupos
* mp_A: 1-dim mpz_t array length n*n, representing the n x n left hand
* side input matrix
* mp_B: 1-dim mpz_t array length n*m, representing the right hand side
* matrix of the system
* - solupos = LeftSolu: mp_B a m x n matrix
* - solupos = RightSolu: mp_B a n x m matrix
*
* Output:
* mp_N: 1-dim mpz_t array length n*m, representing the numerator matrix
* of the solution
* - solupos = LeftSolu: mp_N a m x n matrix
* - solupos = RightSolu: mp_N a n x m matrix
* mp_D: mpz_t, denominator of the solution
*
* Precondition:
* mp_A must be a nonsingular matrix.
*
* Note:
* - It is necessary to make sure the input parameters are correct,
* expecially the dimension, since there is no parameter checks in the
* function.
* - Input and output matrices are row majored and represented by
* one-dimension array.
* - It is needed to preallocate the memory space of mp_N and mp_D.
*
*/
extern void nonsingSolvRNSMM (const enum SOLU_POS solupos, const long n,
const long m, const long basislen,
const FiniteField *basis, Double **ARNS,
mpz_t *mp_B, mpz_t *mp_N, mpz_t mp_D);
/*
* Calling Sequence:
* nonsingSolvRNSMM(solupos, basislen, n, m, basis, ARNS, mp_B, mp_N, mp_D)
*
* Summary:
* Solve nonsingular system of linear equations, where the left hand side
* input matrix is represented in a RNS.
*
* Description:
* Given a n x n nonsingular matrix A represented in a RNS, a n x m or m x n
* mpz_t matrix mp_B, this function will compute the solution of the system
* 1. AX = mp_B
* 2. XA = mp_B.
* The parameter solupos controls whether the system is in the type of 1
* or 2.
*
* Since the unique solution X is a rational matrix, the function will
* output the numerator matrix mp_N and denominator mp_D respectively,
* such that A(mp_N) = mp_D*mp_B or (mp_N)A = mp_D*mp_B.
*
* Input:
* solupos: enumerate, flag to indicate the system to be solved
* - solupos = LeftSolu: solve XA = mp_B
* - solupos = RightSolu: solve AX = mp_B
* basislen: long, dimension of RNS basis
* n: long, dimension of A
* m: long, column or row dimension of mp_B depending on solupos
* basis: 1-dim FiniteField array length basislen, RNS basis
* ARNS: 2-dim Double array, dimension basislen x n*n, representation of
* n x n input matrix A in RNS, where ARNS[i] = A mod basis[i]
* mp_B: 1-dim mpz_t array length n*m, representing the right hand side
* matrix of the system
* - solupos = LeftSolu: mp_B a m x n matrix
* - solupos = RightSolu: mp_B a n x m matrix
*
* Output:
* mp_N: 1-dim mpz_t array length n*m, representing the numerator matrix
* of the solution
* - solupos = LeftSolu: mp_N a m x n matrix
* - solupos = RightSolu: mp_N a n x m matrix
* mp_D: mpz_t, denominator of the solution
*
* Precondition:
* - A must be a nonsingular matrix.
* - Any element p in RNS basis must satisfy 2*(p-1)^2 <= 2^53-1.
*
* Note:
* - It is necessary to make sure the input parameters are correct,
* expecially the dimension, since there is no parameter checks in the
* function.
* - Input and output matrices are row majored and represented by
* one-dimension array.
* - It is needed to preallocate the memory space of mp_N and mp_D.
*
*/
extern long certSolveLong (const long certflag, const long n, const long m,
const long *A, mpz_t *mp_b, mpz_t *mp_N,
mpz_t mp_D, mpz_t *mp_NZ, mpz_t mp_DZ);
/*
*
* Calling Sequence:
* 1/2/3 <-- certSolveLong(certflag, n, m, A, mp_b, mp_N, mp_D,
* mp_NZ, mp_DZ)
*
* Summary:
* Certified solve a system of linear equations without reducing the
* solution size, where the left hand side input matrix is represented
* by signed long integers
*
* Description:
* Let the system of linear equations be Av = b, where A is a n x m matrix,
* and b is a n x 1 vector. There are three possibilities:
*
* 1. The system has more than one rational solution
* 2. The system has a unique rational solution
* 3. The system has no solution
*
* In the first case, there exist a solution vector v with minimal
* denominator and a rational certificate vector z to certify that the
* denominator of solution v is really the minimal denominator.
*
* The 1 x n certificate vector z satisfies that z.A is an integer vector
* and z.b has the same denominator as the solution vector v.
* In this case, the function will output the solution with minimal
* denominator and optional certificate vector z (if certflag = 1).
*
* Note: if choose not to compute the certificate vector z, the solution
* will not garantee, but with high probability, to be the minimal
* denominator solution, and the function will run faster.
*
* In the second case, the function will only compute the unique solution
* and the contents in the space for certificate vector make no sense.
*
* In the third case, there exists a certificate vector q to certify that
* the system has no solution. The 1 x n vector q satisfies q.A = 0 but
* q.b <> 0. In this case, the function will output this certificate vector
* q and store it into the same space for certificate z. The value of
* certflag also controls whether to output q or not.
*
* Note: if the function returns 3, then the system determinately does not
* exist solution, no matter whether to output certificate q or not.
*
* Input:
* certflag: 1/0, flag to indicate whether or not to compute the certificate
* vector z or q.
* - If certflag = 1, compute the certificate.
* - If certflag = 0, not compute the certificate.
* n: long, row dimension of the system
* m: long, column dimension of the system
* A: 1-dim signed long array length n*m, representation of n x m
* matrix A
* mp_b: 1-dim mpz_t array length n, representation of n x 1 vector b
*
* Return:
* 1: the first case, system has more than one solution
* 2: the second case, system has a unique solution
* 3: the third case, system has no solution
*
* Output:
* mp_N: 1-dim mpz_t array length m,
* - numerator vector of the solution with minimal denominator
* in the first case
* - numerator vector of the unique solution in the second case
* - make no sense in the third case
* mp_D: mpz_t,
* - minimal denominator of the solutions in the first case
* - denominator of the unique solution in the second case
* - make no sense in the third case
*
* The following will only be computed when certflag = 1
* mp_NZ: 1-dim mpz_t array length n,
* - numerator vector of the certificate z in the first case
* - make no sense in the second case
* - numerator vector of the certificate q in the third case
* mp_DZ: mpz_t,
* - denominator of the certificate z if in the first case
* - make no sense in the second case
* - denominator of the certificate q in the third case
*
* Note:
* - The space of (mp_N, mp_D) is needed to be preallocated, and entries in
* mp_N and integer mp_D are needed to be initiated as any integer values.
* - If certflag is specified to be 1, then also needs to preallocate space
* for (mp_NZ, mp_DZ), and initiate integer mp_DZ and entries in mp_NZ to
* be any integer values.
* Otherwise, set mp_NZ = NULL, and mp_DZ = any integer
*
*/
extern long certSolveRedLong (const long certflag, const long nullcol,
const long n, const long m, const long *A,
mpz_t *mp_b, mpz_t *mp_N, mpz_t mp_D,
mpz_t *mp_NZ, mpz_t mp_DZ);
/*
*
* Calling Sequence:
* 1/2/3 <-- certSolveRedLong(certflag, nullcol, n, m, A, mp_b, mp_N, mp_D,
* mp_NZ, mp_DZ)
*
* Summary:
* Certified solve a system of linear equations and reduce the solution
* size, where the left hand side input matrix is represented by signed
* long integers
*
* Description:
* Let the system of linear equations be Av = b, where A is a n x m matrix,
* and b is a n x 1 vector. There are three possibilities:
*
* 1. The system has more than one rational solution
* 2. The system has a unique rational solution
* 3. The system has no solution
*
* In the first case, there exist a solution vector v with minimal
* denominator and a rational certificate vector z to certify that the
* denominator of solution v is really the minimal denominator.
*
* The 1 x n certificate vector z satisfies that z.A is an integer vector
* and z.b has the same denominator as the solution vector v.
* In this case, the function will output the solution with minimal
* denominator and optional certificate vector z (if certflag = 1).
*
* Note: if choose not to compute the certificate vector z, the solution
* will not garantee, but with high probability, to be the minimal
* denominator solution, and the function will run faster.
*
* Lattice reduction will be used to reduce the solution size. Parameter
* nullcol designates the dimension of kernal basis we use to reduce the
* solution size as well as the dimension of nullspace we use to compute
* the minimal denominator. The heuristic results show that the solution
* size will be reduced by factor 1/nullcol.
*
* To find the minimum denominator as fast as possible, nullcol cannot be
* too small. We use NULLSPACE_COLUMN as the minimal value of nullcol. That
* is, if the input nullcol is less than NULLSPACE_COLUMN, NULLSPACE_COLUMN
* will be used instead. However, if the input nullcol becomes larger, the
* function will be slower. Meanwhile, it does not make sense to make
* nullcol greater than the dimension of nullspace of the input system.
*
* As a result, the parameter nullcol will not take effect unless
* NULLSPACE_COLUMN < nullcol < dimnullspace is satisfied, where
* dimnullspace is the dimension of nullspace of the input system. If the
* above condition is not satisfied, the boundary value NULLSPACE_COLUMN or
* dimnullspace will be used.
*
* In the second case, the function will only compute the unique solution
* and the contents in the space for certificate vector make no sense.
*
* In the third case, there exists a certificate vector q to certify that
* the system has no solution. The 1 x n vector q satisfies q.A = 0 but
* q.b <> 0. In this case, the function will output this certificate vector
* q and store it into the same space for certificate z. The value of
* certflag also controls whether to output q or not.
*
* Note: if the function returns 3, then the system determinately does not
* exist solution, no matter whether to output certificate q or not.
*
* Input:
* certflag: 1/0, flag to indicate whether or not to compute the certificate
* vector z or q.
* - If certflag = 1, compute the certificate.
* - If certflag = 0, not compute the certificate.
* nullcol: long, dimension of nullspace and kernel basis of conditioned
* system,
* if nullcol < NULLSPACE_COLUMN, use NULLSPACE_COLUMN instead
* n: long, row dimension of the system
* m: long, column dimension of the system
* A: 1-dim signed long array length n*m, representation of n x m
* matrix A
* mp_b: 1-dim mpz_t array length n, representation of n x 1 vector b
*
* Return:
* 1: the first case, system has more than one solution
* 2: the second case, system has a unique solution
* 3: the third case, system has no solution
*
* Output:
* mp_N: 1-dim mpz_t array length m,
* - numerator vector of the solution with minimal denominator
* in the first case
* - numerator vector of the unique solution in the second case
* - make no sense in the third case
* mp_D: mpz_t,
* - minimal denominator of the solutions in the first case
* - denominator of the unique solution in the second case
* - make no sense in the third case
*
* The following will only be computed when certflag = 1
* mp_NZ: 1-dim mpz_t array length n,
* - numerator vector of the certificate z in the first case
* - make no sense in the second case
* - numerator vector of the certificate q in the third case
* mp_DZ: mpz_t,
* - denominator of the certificate z if in the first case
* - make no sense in the second case
* - denominator of the certificate q in the third case
*
* Note:
* - The space of (mp_N, mp_D) is needed to be preallocated, and entries in
* mp_N and integer mp_D are needed to be initiated as any integer values.
* - If certflag is specified to be 1, then also needs to preallocate space
* for (mp_NZ, mp_DZ), and initiate integer mp_DZ and entries in mp_NZ to
* be any integer values.
* Otherwise, set mp_NZ = NULL, and mp_DZ = any integer
*
*/
extern long certSolveMP (const long certflag, const long n, const long m,
mpz_t *mp_A, mpz_t *mp_b, mpz_t *mp_N,
mpz_t mp_D, mpz_t *mp_NZ, mpz_t mp_DZ);
/*
*
* Calling Sequence:
* 1/2/3 <-- certSolveMP(certflag, n, m, mp_A, mp_b, mp_N, mp_D,
* mp_NZ, mp_DZ)
*
* Summary:
* Certified solve a system of linear equations without reducing the
* solution size, where the left hand side input matrix is represented
* by mpz_t integers
*
* Description:
* Let the system of linear equations be Av = b, where A is a n x m matrix,
* and b is a n x 1 vector. There are three possibilities:
*
* 1. The system has more than one rational solution
* 2. The system has a unique rational solution
* 3. The system has no solution
*
* In the first case, there exist a solution vector v with minimal
* denominator and a rational certificate vector z to certify that the
* denominator of solution v is really the minimal denominator.
*
* The 1 x n certificate vector z satisfies that z.A is an integer vector
* and z.b has the same denominator as the solution vector v.
* In this case, the function will output the solution with minimal
* denominator and optional certificate vector z (if certflag = 1).
*
* Note: if choose not to compute the certificate vector z, the solution
* will not garantee, but with high probability, to be the minimal
* denominator solution, and the function will run faster.
*
* In the second case, the function will only compute the unique solution
* and the contents in the space for certificate vector make no sense.
*
* In the third case, there exists a certificate vector q to certify that
* the system has no solution. The 1 x n vector q satisfies q.A = 0 but
* q.b <> 0. In this case, the function will output this certificate vector
* q and store it into the same space for certificate z. The value of
* certflag also controls whether to output q or not.
*
* Note: if the function returns 3, then the system determinately does not
* exist solution, no matter whether to output certificate q or not.
* In the first case, there exist a solution vector v with minimal
* denominator and a rational certificate vector z to certify that the
* denominator of solution v is the minimal denominator.
*
* Input:
* certflag: 1/0, flag to indicate whether or not to compute the certificate
* vector z or q.
* - If certflag = 1, compute the certificate.
* - If certflag = 0, not compute the certificate.
* n: long, row dimension of the system
* m: long, column dimension of the system
* mp_A: 1-dim mpz_t array length n*m, representation of n x m matrix A
* mp_b: 1-dim mpz_t array length n, representation of n x 1 vector b
*
* Return:
* 1: the first case, system has more than one solution
* 2: the second case, system has a unique solution
* 3: the third case, system has no solution
*
* Output:
* mp_N: 1-dim mpz_t array length m,
* - numerator vector of the solution with minimal denominator
* in the first case
* - numerator vector of the unique solution in the second case
* - make no sense in the third case
* mp_D: mpz_t,
* - minimal denominator of the solutions in the first case
* - denominator of the unique solution in the second case
* - make no sense in the third case
*
* The following will only be computed when certflag = 1
* mp_NZ: 1-dim mpz_t array length n,
* - numerator vector of the certificate z in the first case
* - make no sense in the second case
* - numerator vector of the certificate q in the third case
* mp_DZ: mpz_t,
* - denominator of the certificate z if in the first case
* - make no sense in the second case
* - denominator of the certificate q in the third case
*
* Note:
* - The space of (mp_N, mp_D) is needed to be preallocated, and entries in
* mp_N and integer mp_D are needed to be initiated as any integer values.
* - If certflag is specified to be 1, then also needs to preallocate space
* for (mp_NZ, mp_DZ), and initiate integer mp_DZ and entries in mp_NZ to
* be any integer values.
* Otherwise, set mp_NZ = NULL, and mp_DZ = any integer
*
*/
extern long certSolveRedMP (const long certflag, const long nullcol,
const long n, const long m, mpz_t *mp_A,
mpz_t *mp_b, mpz_t *mp_N, mpz_t mp_D,
mpz_t *mp_NZ, mpz_t mp_DZ);
/*
*
* Calling Sequence:
* 1/2/3 <-- certSolveRedMP(certflag, nullcol, n, m, mp_A, mp_b, mp_N, mp_D,
* mp_NZ, mp_DZ)
*
* Summary:
* Certified solve a system of linear equations and reduce the solution
* size, where the left hand side input matrix is represented by signed
* mpz_t integers
*
* Description:
* Let the system of linear equations be Av = b, where A is a n x m matrix,
* and b is a n x 1 vector. There are three possibilities:
*
* 1. The system has more than one rational solution
* 2. The system has a unique rational solution
* 3. The system has no solution
*
* In the first case, there exist a solution vector v with minimal
* denominator and a rational certificate vector z to certify that the
* denominator of solution v is really the minimal denominator.
*
* The 1 x n certificate vector z satisfies that z.A is an integer vector
* and z.b has the same denominator as the solution vector v.
* In this case, the function will output the solution with minimal
* denominator and optional certificate vector z (if certflag = 1).
*
* Note: if choose not to compute the certificate vector z, the solution
* will not garantee, but with high probability, to be the minimal
* denominator solution, and the function will run faster.
*
* Lattice reduction will be used to reduce the solution size. Parameter
* nullcol designates the dimension of kernal basis we use to reduce the
* solution size as well as the dimension of nullspace we use to compute
* the minimal denominator. The heuristic results show that the solution
* size will be reduced by factor 1/nullcol.
*
* To find the minimum denominator as fast as possible, nullcol cannot be
* too small. We use NULLSPACE_COLUMN as the minimal value of nullcol. That
* is, if the input nullcol is less than NULLSPACE_COLUMN, NULLSPACE_COLUMN
* will be used instead. However, if the input nullcol becomes larger, the
* function will be slower. Meanwhile, it does not make sense to make
* nullcol greater than the dimension of nullspace of the input system.
*
* As a result, the parameter nullcol will not take effect unless
* NULLSPACE_COLUMN < nullcol < dimnullspace is satisfied, where
* dimnullspace is the dimension of nullspace of the input system. If the
* above condition is not satisfied, the boundary value NULLSPACE_COLUMN or
* dimnullspace will be used.
*
* In the second case, the function will only compute the unique solution
* and the contents in the space for certificate vector make no sense.
*
* In the third case, there exists a certificate vector q to certify that
* the system has no solution. The 1 x n vector q satisfies q.A = 0 but
* q.b <> 0. In this case, the function will output this certificate vector
* q and store it into the same space for certificate z. The value of
* certflag also controls whether to output q or not.
*
* Note: if the function returns 3, then the system determinately does not
* exist solution, no matter whether to output certificate q or not.
* In the first case, there exist a solution vector v with minimal
* denominator and a rational certificate vector z to certify that the
* denominator of solution v is the minimal denominator.
*
* Input:
* certflag: 1/0, flag to indicate whether or not to compute the certificate
* vector z or q.
* - If certflag = 1, compute the certificate.
* - If certflag = 0, not compute the certificate.
* nullcol: long, dimension of nullspace and kernel basis of conditioned
* system,
* if nullcol < NULLSPACE_COLUMN, use NULLSPACE_COLUMN instead
* n: long, row dimension of the system
* m: long, column dimension of the system
* mp_A: 1-dim mpz_t array length n*m, representation of n x m matrix A
* mp_b: 1-dim mpz_t array length n, representation of n x 1 vector b
*
* Return:
* 1: the first case, system has more than one solution
* 2: the second case, system has a unique solution
* 3: the third case, system has no solution
*
* Output:
* mp_N: 1-dim mpz_t array length m,
* - numerator vector of the solution with minimal denominator
* in the first case
* - numerator vector of the unique solution in the second case
* - make no sense in the third case
* mp_D: mpz_t,
* - minimal denominator of the solutions in the first case
* - denominator of the unique solution in the second case
* - make no sense in the third case
*
* The following will only be computed when certflag = 1
* mp_NZ: 1-dim mpz_t array length n,
* - numerator vector of the certificate z in the first case
* - make no sense in the second case
* - numerator vector of the certificate q in the third case
* mp_DZ: mpz_t,
* - denominator of the certificate z if in the first case
* - make no sense in the second case
* - denominator of the certificate q in the third case
*
* Note:
* - The space of (mp_N, mp_D) is needed to be preallocated, and entries in
* mp_N and integer mp_D are needed to be initiated as any integer values.
* - If certflag is specified to be 1, then also needs to preallocate space
* for (mp_NZ, mp_DZ), and initiate integer mp_DZ and entries in mp_NZ to
* be any integer values.
* Otherwise, set mp_NZ = NULL, and mp_DZ = any integer
*
*/
extern void RowEchelonTransform (const FiniteField p, Double *A, const long n,
const long m, const long frows,
const long lrows, const long redflag,
const long eterm, long *Q, long *rp,
FiniteField *d);
/*
* Calling Sequence:
* RowEchelonTransform(p, A, n, m, frows, lrows, redflag, eterm, Q, rp, d)
*
* Summary:
* Compute a mod p row-echelon transform of a mod p input matrix
*
* Description:
* Given a n x m mod p matrix A, a row-echelon transform of A is a 4-tuple
* (U,P,rp,d) with rp the rank profile of A (the unique and strictly
* increasing list [j1,j2,...jr] of column indices of the row-echelon form
* which contain the pivots), P a permutation matrix such that all r leading
* submatrices of (PA)[0..r-1,rp] are nonsingular, U a nonsingular matrix
* such that UPA is in row-echelon form, and d the determinant of
* (PA)[0..r-1,rp].
*
* Generally, it is required that p be a prime, as inverses are needed, but
* in some cases it is possible to obtain an echelon transform when p is
* composite. For the cases where the echelon transform cannot be obtained
* for p composite, the function returns an error indicating that p is
* composite.
*
* The matrix U is structured, and has last n-r columns equal to the last n-r
* columns of the identity matrix, n the row dimension of A.
*
* The first r rows of UPA comprise a basis in echelon form for the row
* space of A, while the last n-r rows of U comprise a basis for the left
* nullspace of PA.
*
* For efficiency, this function does not output an echelon transform
* (U,P,rp,d) directly, but rather the expression sequence (Q,rp,d).
* Q, rp, d are the form of arrays and pointers in order to operate inplace,
* which require to preallocate spaces and initialize them. Initially,
* Q[i] = i (i=0..n), rp[i] = 0 (i=0..n), and *d = 1. Upon completion, rp[0]
* stores the rank r, rp[1..r] stores the rank profile. i<=Q[i]<=n for
* i=1..r. The input Matrix A is modified inplace and used to store U.
* Let A' denote the state of A on completion. Then U is obtained from the
* identity matrix by replacing the first r columns with those of A', and P
* is obtained from the identity matrix by swapping row i with row Q[i], for
* i=1..r in succession.
*
* Parameters flrows, lrows, redflag, eterm control the specific operations
* this function will perform. Let (U,P,rp,d) be as constructed above. If
* frows=0, the first r rows of U will not be correct. If lrows=0, the last
* n-r rows of U will not be correct. The computation can be up to four
* times faster if these flags are set to 0.
*
* If redflag=1, the row-echelon form is reduced, that is (UPA)[0..r-1,rp]
* will be the identity matrix. If redflag=0, the row-echelon form will not
* be reduced, that is (UPA)[1..r,rp] will be upper triangular and U is unit
* lower triangular. If frows=0 then redflag has no effect.
*
* If eterm=1, then early termination is triggered if a column of the
* input matrix is discovered that is linearly dependant on the previous
* columns. In case of early termination, the third return value d will be 0
* and the remaining components of the echelon transform will not be correct.
*
* Input:
* p: FiniteField, modulus
* A: 1-dim Double array length n*m, representation of a n x m input
* matrix
* n: long, row dimension of A
* m: long, column dimension of A
* frows: 1/0,
* - if frows = 1, the first r rows of U will be correct
* - if frows = 0, the first r rows of U will not be correct
* lrows: 1/0,
* - if lrows = 1, the last n-r rows of U will be correct
* - if lrows = 0, the last n-r rows of U will not be correct
* redflag: 1/0,
* - if redflag = 1, compute row-echelon form
* - if redflag = 0, not compute reow-echelon form
* eterm: 1/0,
* - if eterm = 1, terminate early if not in full rank
* - if eterm = 0, not terminate early
* Q: 1-dim long array length n+1, compact representation of
* permutation vector, initially Q[i] = i, 0 <= i <= n
* rp: 1-dim long array length n+1, representation of rank profile,
* initially rp[i] = 0, 0 <= i <= n
* d: pointer to FiniteField, storing determinant of the matrix,
* initially *d = 1
*
* Precondition:
* ceil(n/2)*(p-1)^2+(p-1) <= 2^53-1 = 9007199254740991 (n >= 2)
*
*/
extern Double * mAdjoint (const FiniteField p, Double *A, const long n);
/*
* Calling Sequence:
* Adj <-- mAdjoint(p, A, n)
*
* Summary:
* Compute the adjoint of a mod p square matrix
*
* Description:
* Given a n x n mod p matrix A, the function computes adjoint of A. Input
* A is not modified upon completion.
*
* Input:
* p: FiniteField, prime modulus
* if p is a composite number, the routine will still work if no error
* message is returned
* A: 1-dim Double array length n*n, representation of a n x n mod p matrix.
* The entries of A are casted from integers
* n: long, dimension of A
*
* Return:
* 1-dim Double matrix length n*n, repesentation of a n x n mod p matrix,
* adjoint of A
*
* Precondition:
* n*(p-1)^2 <= 2^53-1 = 9007199254740991
*
*/
extern long mBasis (const FiniteField p, Double *A, const long n,
const long m, const long basis, const long nullsp,
Double **B, Double **N);
/*
* Calling Sequence:
* r/-1 <-- mBasis(p, A, n, m, basis, nullsp, B, N)
*
* Summary:
* Compute a basis for the rowspace and/or a basis for the left nullspace
* of a mod p matrix
*
* Description:
* Given a n x m mod p matrix A, the function computes a basis for the
* rowspace B and/or a basis for the left nullspace N of A. Row vectors in
* the r x m matrix B consist of basis of A, where r is the rank of A in
* Z/pZ. If r is zero, then B will be NULL. Row vectors in the n-r x n
* matrix N consist of the left nullspace of A. N will be NULL if A is full
* rank.
*
* The pointers are passed into argument lists to store the computed basis
* and nullspace. Upon completion, the rank r will be returned. The
* parameters basis and nullsp control whether to compute basis and/or
* nullspace. If set basis and nullsp in the way that both basis and
* nullspace will not be computed, an error message will be printed and
* instead of rank r, -1 will be returned.
*
* Input:
* p: FiniteField, prime modulus
* if p is a composite number, the routine will still work if no
* error message is returned
* A: 1-dim Double array length n*m, representation of a n x m mod p
* matrix. The entries of A are casted from integers
* n: long, row dimension of A
* m: long, column dimension of A
* basis: 1/0, flag to indicate whether to compute basis for rowspace or
* not
* - basis = 1, compute the basis
* - basis = 0, not compute the basis
* nullsp: 1/0, flag to indicate whether to compute basis for left nullspace
* or not
* - nullsp = 1, compute the nullspace
* - nullsp = 0, not compute the nullspace
*
* Output:
* B: pointer to (Double *), if basis = 1, *B will be a 1-dim r*m Double
* array, representing the r x m basis matrix. If basis = 1 and r = 0,
* *B = NULL
*
* N: pointer to (Double *), if nullsp = 1, *N will be a 1-dim (n-r)*n Double
* array, representing the n-r x n nullspace matrix. If nullsp = 1 and
* r = n, *N = NULL.
*
* Return:
* - if basis and/or nullsp are set to be 1, then return the rank r of A
* - if both basis and nullsp are set to be 0, then return -1
*
* Precondition:
* n*(p-1)^2 <= 2^53-1 = 9007199254740991
*
* Note:
* - In case basis = 0, nullsp = 1, A will be destroyed inplace. Otherwise,
* A will not be changed.
* - Space of B and/or N will be allocated in the function
*
*/
extern long mDeterminant (const FiniteField p, Double *A, const long n);
/*
* Calling Sequence:
* det <-- mDeterminant(p, A, n)
*
* Summary:
* Compute the determinant of a square mod p matrix
*
* Input:
* p: FiniteField, prime modulus
* if p is a composite number, the routine will still work if no error
* message is returned
* A: 1-dim Double array length n*n, representation of a n x n mod p matrix.
* The entries of A are casted from integers
* n: long, dimension of A
*
* Output:
* det(A) mod p, the determinant of square matrix A
*
* Precondition:
* ceil(n/2)*(p-1)^2+(p-1) <= 2^53-1 = 9007199254740991 (n >= 2)
*
* Note:
* A is destroyed inplace
*
*/
extern long mInverse (const FiniteField p, Double *A, const long n);
/*
* Calling Sequence:
* 1/0 <-- mInverse(p, A, n)
*
* Summary:
* Certified compute the inverse of a mod p matrix inplace
*
* Description:
* Given a n x n mod p matrix A, the function computes A^(-1) mod p
* inplace in case A is a nonsingular matrix in Z/Zp. If the inverse does
* not exist, the function returns 0.
*
* A will be destroyed at the end in both cases. If the inverse exists, A is
* inplaced by its inverse. Otherwise, the inplaced A is not the inverse.
*
* Input:
* p: FiniteField, prime modulus
* if p is a composite number, the routine will still work if no error
* message is returned
* A: 1-dim Double array length n*n, representation of a n x n mod p matrix.
* The entries of A are casted from integers
* n: long, dimension of A
*
* Return:
* - 1, if A^(-1) mod p exists
* - 0, if A^(-1) mod p does not exist
*
* Precondition:
* ceil(n/2)*(p-1)^2+(p-1) <= 2^53-1 = 9007199254740991 (n >= 2)
*
* Note:
* A is destroyed inplace
*
*/
extern long mRank (const FiniteField p, Double *A, const long n, const long m);
/*
* Calling Sequence:
* r <-- mRank(p, A, n, m)
*
* Summary:
* Compute the rank of a mod p matrix
*
* Input:
* p: FiniteField, prime modulus
* if p is a composite number, the routine will still work if no
* error message is returned
* A: 1-dim Double array length n*m, representation of a n x m mod p
* matrix. The entries of A are casted from integers
* n: long, row dimension of A
* m: long, column dimension of A
*
* Return:
* r: long, rank of matrix A
*
* Precondition:
* ceil(n/2)*(p-1)^2+(p-1) <= 2^53-1 = 9007199254740991 (n >= 2)
*
* Note:
* A is destroyed inplace
*
*/
extern long * mRankProfile (const FiniteField p, Double *A,
const long n, const long m);
/*
* Calling Sequence:
* rp <-- mRankProfile(p, A, n, m)
*
* Summary:
* Compute the rank profile of a mod p matrix
*
* Input:
* p: FiniteField, prime modulus
* if p is a composite number, the routine will still work if no
* error message is returned
* A: 1-dim Double array length n*m, representation of a n x m mod p
* matrix. The entries of A are casted from integers
* n: long, row dimension of A
* m: long, column dimension of A
*
* Return:
* rp: 1-dim long array length n+1, where
* - rp[0] is the rank of matrix A
* - rp[1..r] is the rank profile of matrix A
*
* Precondition:
* ceil(n/2)*(p-1)^2+(p-1) <= 2^53-1 = 9007199254740991 (n >= 2)
*
* Note:
* A is destroyed inplace
*
*/
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