![\[ (A_r + i*A_i)*(x_r + i*x_i) = (b_r + i*b_i) \]](form_0.png)
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Classes | |
| struct | AZ_KOMPLEX_STRUCT |
Defines | |
| #define | AZK_True_Complex 1 |
| #define | AZK_Komplex_Same_Structure 2 |
| #define | AZK_Komplex_General 3 |
| #define | AZK_Komplex_No_Copy 4 |
Typedefs | |
| typedef AZ_KOMPLEX_STRUCT | AZ_KOMPLEX |
Functions | |
| void | AZK_create_linsys_c2k (double *xc, double *bc, int *options, double *params, int *proc_config, AZ_MATRIX *Amat_complex, double **x, double **b, AZ_MATRIX **Amat_komplex) |
| Create Komplex System from Complex System. | |
| void | AZK_create_linsys_g2k (double *xr, double *xi, double *br, double *bi, int *options, double *params, int *proc_config, double c0r, double c0i, AZ_MATRIX *Amat_mat0, double c1r, double c1i, AZ_MATRIX *Amat_mat1, double **x, double **b, AZ_MATRIX **Amat_komplex) |
| Create Komplex System from General System. | |
| void | AZK_create_linsys_ri2k (double *xr, double *xi, double *br, double *bi, int *options, double *params, int *proc_config, AZ_MATRIX *Amat_real, double *val_imag, double **x, double **b, AZ_MATRIX **Amat_komplex) |
| Create Komplex System from Real and Imaginary Parts. | |
| void | AZK_create_linsys_no_copy (double *xr, double *xi, double *br, double *bi, int *options, double *params, int *proc_config, AZ_MATRIX *Amat_real, AZ_MATRIX *Amat_imag, double **x, double **b, AZ_MATRIX **Amat) |
| void | AZK_create_matrix_g2k (int options[], double params[], int proc_config[], double c0r, double c0i, AZ_MATRIX *Amat_mat0, double c1r, double c1i, AZ_MATRIX *Amat_mat1, AZ_MATRIX **Amat_komplex) |
| Create Komplex Matrix from General Matrix. | |
| void | AZK_create_matrix_g2k_fill_entry (int nrow, int ncol, double c11, double c12, double *mat0v, double c21, double c22, double *mat1v, double *komplex) |
| void | AZK_create_matrix_c2k (int options[], double params[], int proc_config[], AZ_MATRIX *Amat_complex, AZ_MATRIX **Amat_komplex) |
| Create Komplex matrix from Complex matrix. | |
| void | AZK_create_matrix_c2k_fill_entry (int nrow, int ncol, double *cur_complex, double *cur_komplex) |
| void | AZK_create_matrix_ri2k (int options[], double params[], int proc_config[], AZ_MATRIX *Amat_real, double *val_imag, AZ_MATRIX **Amat_komplex) |
| Create Komplex Matrix from Real and Imaginary Parts. | |
| void | AZK_create_matrix_ri2k_fill_entry (int nrow, int ncol, double *realv, double *imagv, double *komplex) |
| void | AZK_create_precon (int *options, double *params, int *proc_config, double *x, double *b, AZ_MATRIX *Amat, AZ_PRECOND **Prec) |
| Create a Preconditioner for a Komplex matrix. | |
| void | AZK_create_vector_c2k (int *options, double *params, int *proc_config, AZ_MATRIX *Amat_komplex, double *vc, double **vk) |
| Create Komplex vector from Complex vector. | |
| void | AZK_create_vector_g2k (int *options, double *params, int *proc_config, AZ_MATRIX *Amat_komplex, double *vr, double *vi, double **vk) |
| Create Komplex vector from Real and Imaginary Parts. | |
| void | AZK_create_vector_ri2k (int *options, double *params, int *proc_config, AZ_MATRIX *Amat_komplex, double *vr, double *vi, double **vk) |
| Create Komplex vector from Real and Imaginary Parts. | |
| void | AZK_destroy_linsys (int *options, double *params, int *proc_config, double **x, double **b, AZ_MATRIX **Amat_komplex) |
| Destroy a Komplex System. | |
| void | AZK_destroy_matrix (int options[], double params[], int proc_config[], AZ_MATRIX **Amat_komplex) |
| Destroy a Komplex Matrix. | |
| void | AZK_destroy_precon (int *options, double *params, int *proc_config, AZ_MATRIX *Amat, AZ_PRECOND **Prec) |
| Destroy a Komplex preconditioner. | |
| void | AZK_destroy_vector (int *options, double *params, int *proc_config, AZ_MATRIX *Amat_komplex, double **vk) |
| Destroy a Komplex vector. | |
| void | AZK_extract_solution_k2c (int *options, double *params, int *proc_config, AZ_MATRIX *Amat_komplex, AZ_PRECOND *Prec, double *vk, double *vc) |
| Extract a Complex vector from a Komplex vector. | |
| void | AZK_extract_solution_k2g (int *options, double *params, int *proc_config, AZ_MATRIX *Amat_komplex, AZ_PRECOND *Prec, double *vk, double *vr, double *vi) |
| Extract real/imaginary parts of a complex vector from a Komplex vector. | |
| void | AZK_extract_solution_k2ri (int *options, double *params, int *proc_config, AZ_MATRIX *Amat_komplex, AZ_PRECOND *Prec, double *vk, double *vr, double *vi) |
| Extract real/imaginary parts of a complex vector from a Komplex vector. | |
| void | AZK_iterate (double *xr, double *xi, double *br, double *bi, int *options, double *params, double *status, int *proc_config, AZ_MATRIX *Amat_real, AZ_MATRIX *Amat_imag) |
| void | AZK_matvec_no_copy (double *x, double *y, AZ_MATRIX *Amat, int proc_config[]) |
| void | AZK_permute_ri (int *options, double *params, int *proc_config, double *b, AZ_MATRIX *Amat_komplex) |
| Permute a Komplex system for better numerical stability. | |
| void | AZK_precon (double x[], int options[], int proc_config[], double params[], AZ_MATRIX *Amat, AZ_PRECOND *Prec) |
| double | AZK_residual_norm_no_copy (double *xr, double *xi, double *br, double *bi, int *options, double *params, int *proc_config, AZ_MATRIX *Amat_real, AZ_MATRIX *Amat_imag) |
| double | AZK_residual_norm (double *xk, double *bk, int *options, double *params, int *proc_config, AZ_MATRIX *Amat_komplex) |
The file, along with az_aztec.h must be included in every file that uses the Komplex library.
KOMPLEX is an add-on module to AZTEC that allows users to solve complex-valued linear systems.
KOMPLEX solves a complex-valued linear system Ax = b by solving an equivalent real-valued system of twice the dimension. Specifically, writing in terms of real and imaginary parts, we have
or by separating into real and imaginary equations we have
![\[ \left( \begin{array}{rr} A_r & -A_i\\ A_i & A_r \end{array} \right) \left( \begin{array}{r} x_r\\ x_i \end{array} \right) = \left( \begin{array}{r} b_r\\ b_i \end{array} \right) \]](form_1.png)
which is a real-valued system of twice the size. If we find xr and xi, we can form the solution to the original system as x = xr +i*xi.
KOMPLEX accept user linear systems in three forms with either global or local index values.
1) The first form is true complex. The user passes in an MSR or VBR format matrix where the values are stored like Fortran complex numbers. Thus, the values array is of type double that is twice as long as the number of complex values. Each complex entry is stored with real part followed by imaginary part (as in Fortran).
2) The second form stores real and imaginary parts separately, but the pattern for each is identical. Thus only the values of the imaginary part are passed to the creation routines.
3) The third form accepts two real-valued matrices with no assumption about the structure of the matrices. Each matrix is multiplied by a user-supplied complex constant. This is the most general form.
Each of the above forms supports a global or local index set. By this we mean that the index values (stored in bindx) refer to the global problem indices, or the local indices (for example after calling AZ_transform).
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Create Komplex System from Complex System. Transforms a complex-valued system Amat_complex * xc = bc where double precision arrays hold the complex values of Amat_complex, xc and bc in Fortran complex format, i.e., if dimension of complex system is N then xc is of length 2*N and the first complex value is stored with the real part in xc[0] and the imaginary part in xc[1] and so on.
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Create Komplex System from General System. Transforms a complex-valued system (c0r+i*c0i)*A0 +(c1r+i*c1i)*A1) * (xr+i*xi) = (br+i*bi) to a Komplex system.
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Create Komplex System from Real and Imaginary Parts. Transforms a complex-valued system (Ar +i*Ai) * (xr + i*xi) = (br + i*bi) where double precision arrays hold the real and imaginary parts separately. The pattern of the imaginary part matches the real part. Thus no structure for the imaginary part is passed in.
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Create Komplex matrix from Complex matrix. Transforms a complex-valued matrix where double precision array hold the complex values of Amat_complex in Fortran complex format.
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Create Komplex Matrix from General Matrix. Transforms a complex-valued Matrix (c0r+i*c0i)*A0 +(c1r+i*c1i)*A1) to a Komplex matrix.
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Create Komplex Matrix from Real and Imaginary Parts. Transforms a complex-valued matrix (Ar +i*Ai) where double precision arrays hold the real and imaginary parts separately. The pattern of the imaginary part matches the real part. Thus no structure for the imaginary part is passed in.
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Create a Preconditioner for a Komplex matrix. Constructs a preconditioner for a Komplex matrix Amat. All preconditioning options available in Aztec are supported.
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Create Komplex vector from Complex vector. Transforms a complex-valued vector vc to a real vector where vc in Fortran complex format, i.e., if dimension of complex system is N then vc is of length 2*N and the first complex value is stored with the real part in vc[0] and the imaginary part in vc[1] and so on.
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Create Komplex vector from Real and Imaginary Parts. Transforms a complex vector where double precision arrays hold the real and imaginary parts separately.
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Create Komplex vector from Real and Imaginary Parts. Transforms a complex vector where double precision arrays hold the real and imaginary parts separately.
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Destroy a Komplex System. Destroys a komplex system created by any of the AZK_create_linsys functions. Deletes any memory allocated by creation routine.
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Destroy a Komplex Matrix. Destroys a komplex matrix created by any of the AZK_create_matrix functions. Deletes any memory allocated by creation routine.
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Destroy a Komplex preconditioner. Destroys a komplex preconditioner created by the AZK_create_preconditioner function. Deletes any memory allocated by creation routine.
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Destroy a Komplex vector. Destroys a komplex vector created by any of the AZK_create_vector functions. Deletes any memory allocated by creation routine.
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Extract a Complex vector from a Komplex vector. Transforms a komplex vector to a complex vector.
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Extract real/imaginary parts of a complex vector from a Komplex vector. Transforms a komplex vector to real and imaginary parts.
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Extract real/imaginary parts of a complex vector from a Komplex vector. Transforms a komplex vector to real and imaginary parts.
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Permute a Komplex system for better numerical stability. An alternative to the standard Komplex formulation is to permute the block rows so that the imaginary part is on the main diagonal. For example:
This action may be desirable, or necessary in situations where the real part has small or zero diagonal entries. This routine looks at each real/imaginary pair and, based on a heuristic may swap the real and imaginary parts. This action does not affect the sparsity pattern, but only the mapping from the complex (or real/imaginary) mapping to the komplex mapping, and back.
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1.3.9.1
![\[ \left( \begin{array}{rr} A_i & A_r\\ A_r & -A_i \end{array} \right) \left( \begin{array}{r} x_r\\ x_i \end{array} \right) = \left( \begin{array}{r} b_i\\ b_r \end{array} \right) \]](form_2.png)