Linear Algebra Opcodes
Linear Algebra Opcodes — Scalar, vector, and matrix arithmetic on real and complex values.
Description
These opcodes implement many linear algebra operations, from scalar, vector, and matrix arithmetic up to and including QR based eigenvalue decompositions. The opcodes are designed for digital signal processing, and of course other mathematical operations, in the Csound orchestra language.
The numerical implementation uses the gmm++ library from home.gna.org/getfem/gmm_intro.
![[Warning]](images/warning.png)
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Warning |
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For applications with f-sig variables, array arithmetic must be performed only when the f-sig is "current," because f-rate is some fraction of k-rate; currency can be determined with the la_k_current_f opcode. For applications using assignments between real vectors and a-rate variables, array arithmetic must be performed only when the vectors are "current", because the size of the vector may be some integral multiple of ksmps; currency can be determined by means of the la_k_current_vr opcode. |
Table 4. Linear Algebra Data Types
| Mathematical Type | Code | Corresponding Csound Type or Types |
|---|---|---|
| real scalar | r | i-rate or k-rate variable |
| complex scalar | c | pair of i-rate or k-rate variables, e.g. "kr, ki" |
| real vector | vr | i-rate variable holding address of array |
| real vector | a | a-rate variable |
| real vector | t | function table number |
| complex vector | vc | i-rate variable holding address of array |
| complex vector | f | fsig variable |
| real matrix | mr | i-rate variable holding address of array |
| complex matrix | mc | i-rate variable holding address of array |
All arrays are 0-based; the first index iterates rows to give columns, the second index iterates columns to give elements.
All arrays are general and dense; banded, Hermitian, symmetric and sparse routines are not implemented.
An array can be of type code vr, vc, mr, or mc and is stored in an i-rate object. In orchestra code, an array is passed as a MYFLT i-rate variable that contains the address of the array object, which is actually stored in the allocator opcode instance. Although array variables are i-rate, of course their values and even shapes may change at i-rate or k-rate.
All operands must be pre-allocated; except for the creation opcodes, no opcode ever allocates any arrays. This is true even if the array appears on the left-hand side of an opcode! However, some operations may reshape arrays to hold results.
Arrays are automatically deallocated when their instrument is deallocated.
Not only for more efficient performance, but also to make it easier to remember opcode names, the performance rate, output value types, operation names, and input value types are deterministically encoded into the opcode name:
- "la" for "linear algebra opcode family".
- "i" or "k" for performance rate.
- Type code(s) (see above table) for output value(s), but only if the type is not implicit from the input values.
- Operation name: common mathematical name (preferred) or abbreviation.
- Type code(s) for input values, if not implicit.
For additional details, see the gmm++ documentation at http://download.gna.org/getfem/doc/gmmuser.pdf.
Syntax
Array Creation
ivr la_i_vr_create irows
Create a real vector with irows rows.
ivc la_i_vc_create irows
Create a complex vector with irows rows.
imr la_i_mr_create irows, icolumns [, odiagonal]
Create a real matrix with irows rows and icolumns columns, with an optional value on the diagonal.
imc la_i_mc_create irows, icolumns [, odiagonal_r, odiagonal_i]
Create a complex matrix with irows rows and icolumns columns, with an optional complex value on the diagonal.
Array Introspection
irows la_i_size_vr ivr
Return the number of rows in real vector ivr.
irows la_i_size_vc ivc
Return the number of rows in complex vector ivc.
irows, icolumns la_i_size_mr imr
Return the number of rows and columns in real matrix imr.
irows, icolumns la_i_size_mc imc
Return the number of rows and columns in complex matrix imc.
kfiscurrent la_k_current_f fsig
Return 1 if fsig is current, that is, if the value of fsig will change on the next kperiod.
kvriscurrent la_k_current_vr ivr
Return 1 if the real vector ivr is current, that is, if Csound's current audio sample frame stands at index 0 of the vector.
la_i_print_vr ivr
Print the value of real vector ivr.
la_i_print_vc ivc
Print the value of complex vector ivc.
la_i_print_mr imr
Print the value of real matrix imr.
la_i_print_mc imc
Print the value of complex matrix imc.
Array Assignment and Conversion
ivr la_i_assign_vr ivr
Assign the value of the real vector on the right-hand side to the real vector on the left-hand side, at i-rate.
ivr la_k_assign_vr ivr
Assign the value of the real vector on the right-hand side to the real vector on the left-hand side, at k-rate.
ivc la_i_assign_vc ivc
ivc la_k_assign_vc ivr
imr la_i_assign_mr imr
imr la_k_assign_mr imr
imc la_i_assign_mc imc
imc la_k_assign_mc imr
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Warning |
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Assignments to vectors from tables or fsigs may resize the vectors. Assignments to vectors from a-rate variables, or to a-rate variables from vectors, will be performed incrementally, one chunk of ksmps elements per kperiod. Therefore, array arithmetic on such vectors should only be performed when the vectors are current, as determined by the la_k_currrent_vr opcode. |
ivr la_k_assign_a asig
ivr la_i_assign_t itablenumber
ivr la_k_assign_t itablenumber
ivc la_k_assign_f fsig
asig la_k_a_assign ivr
itablenum la_i_t_assign ivr
itablenum la_k_t_assign ivr
fsig la_k_f_assign ivc
Fill Arrays with Random Elements
ivr la_i_random_vr [ifill_fraction]
ivr la_k_random_vr [kfill_fraction]
ivc la_i_random_vc [ifill_fraction]
ivc la_k_random_vc [kfill_fraction]
imr la_i_random_mr [ifill_fraction]
imr la_k_random_mr [kfill_fraction]
imc la_i_random_mc [ifill_fraction]
imc la_k_random_mc [kfill_fraction]
Array Element Access
ivr la_i_vr_set irow, ivalue
kvr la_k_vr_set krow, kvalue
ivc la_i_vc_set irow, ivalue_r, ivalue_i
kvc la_k_vc_set krow, kvalue_r, kvalue_i
imr la_i mr_set irow, icolumn, ivalue
kmr la_k mr_set krow, kcolumn, ivalue
imc la_i_mc_set irow, icolumn, ivalue_r, ivalue_i
kmc la_k_mc_set krow, kcolumn, kvalue_r, kvalue_i
ivalue la_i_get_vr ivr, irow
kvalue la_k_get_vr ivr, krow
ivalue_r, ivalue_i la_i_get_vc ivc, irow
kvalue_r, kvalue_i la_k_get_vc ivc, krow
ivalue la_i_get_mr imr, irow, icolumn
kvalue la_k_get_mr imr, krow, kcolumn
ivalue_r, ivalue_i la_i_get_mc imc, irow, icolumn
kvalue_r, kvalue_i la_k_get_mc imc, krow, kcolumn
Single Array Operations
imr la_i_transpose_mr imr
imr la_k_transpose_mr imr
imc la_i_transpose_mc imc
imc la_k_transpose_mc imc
ivr la_i_conjugate_vr ivr
ivr la_k_conjugate_vr ivr
ivc la_i_conjugate_vc ivc
ivc la_k_conjugate_vc ivc
imr la_i_conjugate_mr imr
imr la_k_conjugate_mr imr
imc la_i_conjugate_mc imc
imc la_k_conjugate_mc imc
Scalar Operations
ir la_i_norm1_vr ivr
kr la_k_norm1_vr ivc
ir la_i_norm1_vc ivc
kr la_k_norm1_vc ivc
ir la_i_norm1_mr imr
kr la_k_norm1_mr imr
ir la_i_norm1_mc imc
kr la_k_norm1_mc imc
ir la_i_norm_euclid_vr ivr
kr la_k_norm_euclid_vr ivr
ir la_i_norm_euclid_vc ivc
kr la_k_norm_euclid_vc ivc
ir la_i_norm_euclid_mr mvr
kr la_k_norm_euclid_mr mvr
ir la_i_norm_euclid_mc mvc
kr la_k_norm_euclid_mc mvc
ir la_i_distance_vr ivr
kr la_k_distance_vr ivr
ir la_i_distance_vc ivc
kr la_k_distance_vc ivc
ir la_i_norm_max imr
kr la_k_norm_max imc
ir la_i_norm_max imr
kr la_k_norm_max imc
ir la_i_norm_inf_vr ivr
kr la_k_norm_inf_vr ivr
ir la_i_norm_inf_vc ivc
kr la_k_norm_inf_vc ivc
ir la_i_norm_inf_mr imr
kr la_k_norm_inf_mr imr
ir la_i_norm_inf_mc imc
kr la_k_norm_inf_mc imc
ir la_i_trace_mr imr
kr la_k_trace_mr imr
ir, ii la_i_trace_mc imc
kr, ki la_k_trace_mc imc
ir la_i_lu_det imr
kr la_k_lu_det imr
ir la_i_lu_det imc
kr la_k_lu_det imc
Elementwise Array-Array Operations
ivr la_i_add_vr ivr_a, ivr_b
ivc la_k_add_vc ivc_a, ivc_b
imr la_i_add_mr imr_a, imr_b
imc la_k_add_mc imc_a, imc_b
ivr la_i_subtract_vr ivr_a, ivr_b
ivc la_k_subtract_vc ivc_a, ivc_b
imr la_i_subtract_mr imr_a, imr_b
imc la_k_subtract_mc imc_a, imc_b
ivr la_i_multiply_vr ivr_a, ivr_b
ivc la_k_multiply_vc ivc_a, ivc_b
imr la_i_multiply_mr imr_a, imr_b
imc la_k_multiply_mc imc_a, imc_b
ivr la_i_divide_vr ivr_a, ivr_b
ivc la_k_divide_vc ivc_a, ivc_b
imr la_i_divide_mr imr_a, imr_b
imc la_k_divide_mc imc_a, imc_b
Inner Products
ir la_i_dot_vr ivr_a, ivr_b
kr la_k_dot_vr ivr_a, ivr_b
ir, ii la_i_dot_vc ivc_a, ivc_b
kr, ki la_k_dot_vc ivc_a, ivc_b
imr la_i_dot_mr imr_a, imr_b
imr la_k_dot_mr imr_a, imr_b
imc la_i_dot_mc imc_a, imc_b
imc la_k_dot_mc imc_a, imc_b
ivr la_i_dot_mr_vr imr_a, ivr_b
ivr la_k_dot_mr_vr imr_a, ivr_b
ivc la_i_dot_mc_vc imc_a, ivc_b
ivc la_k_dot_mc_vc imc_a, ivc_b
Matrix Inversion
imr, icondition la_i_invert_mr imr
imr, kcondition la_k_invert_mr imr
imc, icondition la_i_invert_mc imc
imc, kcondition la_k_invert_mc imc
Matrix Decompositions and Solvers
ivr la_i_upper_solve_mr imr [, j_1_diagonal]
ivr la_k_upper_solve_mr imr [, j_1_diagonal]
ivc la_i_upper_solve_mc imc [, j_1_diagonal]
ivc la_k_upper_solve_mc imc [, j_1_diagonal]
ivr la_i_lower_solve_mr imr [, j_1_diagonal]
ivr la_k_lower_solve_mr imr [, j_1_diagonal]
ivc la_i_lower_solve_mc imc [, j_1_diagonal]
ivc la_k_lower_solve_mc imc [, j_1_diagonal]
imr, ivr_pivot, isize la_i_lu_factor_mr imr
imr, ivr_pivot, ksize la_k_lu_factor_mr imr
imc, ivr_pivot, isize la_i_lu_factor_mc imc
imc, ivr_pivot, ksize la_k_lu_factor_mc imc
ivr_x la_i_lu_solve_mr imr, ivr_b
ivr_x la_k_lu_solve_mr imr, ivr_b
ivc_x la_i_lu_solve_mc imc, ivc_b
ivc_x la_k_lu_solve_mc imc, ivc_b
imr_q, imr_r la_i_qr_factor_mr imr
imr_q, imr_r la_k_qr_factor_mr imr
imc_q, imc_r la_i_qr_factor_mc imc
imc_q, imc_r la_k_qr_factor_mc imc
ivr_eig_vals la_i_qr_eigen_mr imr, i_tolerance
ivr_eig_vals la_k_qr_eigen_mr imr, k_tolerance
ivr_eig_vals la_i_qr_eigen_mc imc, i_tolerance
ivr_eig_vals la_k_qr_eigen_mc imc, k_tolerance
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Warning |
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| Matrix must be Hermitian in order to compute eigenvectors. |
ivr_eig_vals, imr_eig_vecs la_i_qr_sym_eigen_mr imr, i_tolerance
ivr_eig_vals, imr_eig_vecs la_k_qr_sym_eigen_mr imr, k_tolerance
ivc_eig_vals, imc_eig_vecs la_i_qr_sym_eigen_mc imc, i_tolerance
ivc_eig_vals, imc_eig_vecs la_k_qr_sym_eigen_mc imc, k_tolerance