The goal of this project is to progressively write a comprehensive mathematical library for Jai, similar to the Gnu Scientific Library or Java Apache Math.
The focus lies in providing the functionality first, performance second.
Feel free to add functionality and performance upgrades.
- Add
Quaternion64(maybe evenOctonion64) - I'm going through [2] now to improve the algorithms since that book is actually considering special properties of matrices early on and also writes out EVERY algorithm used.
There are more thoughts written down in the document outlining some of my thoughts.
Complex and reals scalars, vectors, matrices interoperate automatically (in most cases). Many functions specialize during compile-time on the real or complex variant (e.g. all the elementary functions).
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Utils
calculation(Code); will create and release a Pool of memory for all the allocations in the Code block.
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Complex numbers
+,-,*,/,str; for pretty printing
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Elementary (
z∈ ℂ)sign(r)factorial(z)binomial(from, choose)conjugate(z)abs_sq(z)abs(z)arg(z)exp(z)log(z)pow(z)sqrt(z)phase(magnitude ∈ ℂ, angle ∈ ℂ)sin,cos,tan,cot,sec,cscon ℂasin,acos,atan,acot,asec,acscon ℂsinh,cosh,tanh,coth,sech,cschon ℂasinh,acosh,atanh,acoth,asech,acschon ℂ
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polynomials
solve_quadratic-> ℂsolve_quadratic_real-> ℝpolynom(x, ..a)= a[0] x^n + a[1] x^{n-1} + ... + a[n] x^0, with a,x ∈ ℝ,ℂsynthetic_divisionrepeated_synthetic_division
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vector:
VectorType; real or complex vector withVectorTypeis a generic interface/trate that is implemented by any concrete vector struct, e.g.DenseVector($Type, $Dimensions).str; for pretty printing- operators
[],==,+,-,*,/ - in-place functions add, sub, neg, mul, div
- multiple initialization functions (ones, basis, varargs, etc.)
conjugateouter_productreflectnorm(vec, n)with n ∈ ℝ, specialisationsnorm_2,norm_1,norm_infcross, for 3-dim vectorsanglepermuteswap
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matrix:
MatrixType; real or complexMatrixTypeis a generic interface/trate that is implemented by any concrete matrix struct, e.g.DenseMatrix($Type, $Columns, $Rows).pstr,strfor pretty printing- operators
[],[][],==,+,-,*,/ row,column- in-place functions
add,sub,neg,mul,div - multiple initialization functions (1, ones, hadamard, varargs, etc.)
reflectorsubmatrixtransposeconjugateconjugate_transpose=daggertensornorm_1,norm_inf,norm_frobeniuspermute_rows,permute_columns,permuteswap_columns,swap_rows
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checks
is_diagonal_unit(M)is_(left, right)\_triangular(M)is_right_quasi_triangular(M)is_unitary(M)is_(left, right)_trapezoidal(M)
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linear algebra
solve_linear_2x2=sl2solve_linear_(left, right)_triangular=sl(l, r)tasolve_linear_right_quasi_triangular=slrqtasolve_linear_orthogonal_projection=solve_linear_unitary=slopsolve_linear_successive_orthogonal_projection=slsopsolve_linear_(left, right)_trapezoidal=sl(l, r)tzfor vectors and matricesdecompose_LRgaussian_factorization_(no, full, row)_pivotsolve_linear_gaussian_factorization_(no, full, row)_pivot=slgf_(n, f, r)p=solve_LR(n, f, r)pinverse(M); M quadratic matrix ∈ ℂdeterminant=det; ∈ ℂ
- linear algebra
- LAPACK Scaling for Gaussian factorization (Algorithm 3.9.1 [1])
- Iterative Improvement for Gaussian factorization (Algorithm 3.9.2 [1])
Give sources for algorithms written so that we can take a look and help debugging.
Write (at least some) tests contained in each file.
[1] Scientific Computing, Vol I: Linear and nonlinear equations, Texts in computational science and engineering 18, Springer
[2] Matrix Computations, 4th Edition; Gene H. Golub & Charles F. Van Loan; Johns Hopkins University Press, Baltimore; 2013