Linalg.xtl

1⍝# Linalg.xtl -- the facade of the linalg extension: numerical linear
2⍝# algebra through nalgebra. Matrices are Float arrays (row by column);
3⍝# a right-hand side may be a vector or a matrix of columns.
4⍝# Import: "la:" u_se< "Linalg"
5⍝# Run with xetal-x (xetal-x --ext DIR puts this directory on XETAL_PATH).
6⍝# Each signature line is written out by the binding macro (lib/Ffi.xtlm).
7
8ᶠᶠⁱ⁼u̲se< "Ffi"
9
10⍝# a la:s_olve b: x with a x = b (a square and not singular).
11"s_olve : float float -> floats"    ᶠᶠⁱb̲ind< "linalg/solve"
ᶠᶠⁱb̲ind< expands toˡs̲olve ← { a b →
  pa ← (f̲ormat r̲avel f̲loat a) ⎕N̲PUT "ext:linalg/solve?float=" c̲at f̲ormat s̲hape a
  pb ← (f̲ormat r̲avel f̲loat b) ⎕N̲PUT "ext:linalg/solve?float=" c̲at f̲ormat s̲hape b
  v ← n̲umbers ⎕N̲GET "ext:linalg/solve"
  (f̲loor n̲umbers ⎕N̲GET "ext:linalg/solve?shape") r̲eshape v
}
12⍝# la:i_nverse a: the inverse of a square matrix.
13"i_nverse : float -> floats"        ᶠᶠⁱb̲ind< "linalg/inverse"
ᶠᶠⁱb̲ind< expands toˡi̲nverse ← { a →
  pa ← (f̲ormat r̲avel f̲loat a) ⎕N̲PUT "ext:linalg/inverse?float=" c̲at f̲ormat s̲hape a
  v ← n̲umbers ⎕N̲GET "ext:linalg/inverse"
  (f̲loor n̲umbers ⎕N̲GET "ext:linalg/inverse?shape") r̲eshape v
}
14⍝# la:d_et a: the determinant.
15"d_et : float -> float"             ᶠᶠⁱb̲ind< "linalg/det"
ᶠᶠⁱb̲ind< expands toˡd̲et ← { a →
  pa ← (f̲ormat r̲avel f̲loat a) ⎕N̲PUT "ext:linalg/det?float=" c̲at f̲ormat s̲hape a
  f̲irst n̲umbers ⎕N̲GET "ext:linalg/det"
}
16⍝# a la:l_stsq b: the x minimizing |a x - b| (least squares, by the SVD).
17"l_stsq : float float -> floats"    ᶠᶠⁱb̲ind< "linalg/lstsq"
ᶠᶠⁱb̲ind< expands toˡl̲stsq ← { a b →
  pa ← (f̲ormat r̲avel f̲loat a) ⎕N̲PUT "ext:linalg/lstsq?float=" c̲at f̲ormat s̲hape a
  pb ← (f̲ormat r̲avel f̲loat b) ⎕N̲PUT "ext:linalg/lstsq?float=" c̲at f̲ormat s̲hape b
  v ← n̲umbers ⎕N̲GET "ext:linalg/lstsq"
  (f̲loor n̲umbers ⎕N̲GET "ext:linalg/lstsq?shape") r̲eshape v
}
18⍝## Eigenvalues
19
20⍝# la:e_ig a: a symmetric matrix's eigenvalues, ascending.
21"e_ig : float -> floats"            ᶠᶠⁱb̲ind< "linalg/eigsym"
ᶠᶠⁱb̲ind< expands toˡe̲ig ← { a →
  pa ← (f̲ormat r̲avel f̲loat a) ⎕N̲PUT "ext:linalg/eigsym?float=" c̲at f̲ormat s̲hape a
  v ← n̲umbers ⎕N̲GET "ext:linalg/eigsym"
  (f̲loor n̲umbers ⎕N̲GET "ext:linalg/eigsym?shape") r̲eshape v
}
22⍝# la:e_igVecs a: a symmetric matrix's eigenvectors, one column each, in
23⍝# the order of ˡᵃe̲ig.
24"e_igVecs : float -> floats"        ᶠᶠⁱb̲ind< "linalg/eigvecs"
ᶠᶠⁱb̲ind< expands toˡe̲igVecs ← { a →
  pa ← (f̲ormat r̲avel f̲loat a) ⎕N̲PUT "ext:linalg/eigvecs?float=" c̲at f̲ormat s̲hape a
  v ← n̲umbers ⎕N̲GET "ext:linalg/eigvecs"
  (f̲loor n̲umbers ⎕N̲GET "ext:linalg/eigvecs?shape") r̲eshape v
}
25
26⍝## The singular value decomposition
27
28⍝# a = U diag(s) V', thin (r, the smaller side, columns) with s
29⍝# descending; each function computes it again, so call each once.
30
31⍝# la:s_vdS a: the singular values s, descending.
32"s_vdS : float -> floats"           ᶠᶠⁱb̲ind< "linalg/svd_s"
ᶠᶠⁱb̲ind< expands toˡs̲vdS ← { a →
  pa ← (f̲ormat r̲avel f̲loat a) ⎕N̲PUT "ext:linalg/svd_s?float=" c̲at f̲ormat s̲hape a
  v ← n̲umbers ⎕N̲GET "ext:linalg/svd_s"
  (f̲loor n̲umbers ⎕N̲GET "ext:linalg/svd_s?shape") r̲eshape v
}
33⍝# la:s_vdU a: the left singular vectors U, m by r, one column each.
34"s_vdU : float -> floats"           ᶠᶠⁱb̲ind< "linalg/svd_u"
ᶠᶠⁱb̲ind< expands toˡs̲vdU ← { a →
  pa ← (f̲ormat r̲avel f̲loat a) ⎕N̲PUT "ext:linalg/svd_u?float=" c̲at f̲ormat s̲hape a
  v ← n̲umbers ⎕N̲GET "ext:linalg/svd_u"
  (f̲loor n̲umbers ⎕N̲GET "ext:linalg/svd_u?shape") r̲eshape v
}
35⍝# la:s_vdV a: the right singular vectors V, n by r, one column each.
36"s_vdV : float -> floats"           ᶠᶠⁱb̲ind< "linalg/svd_v"
ᶠᶠⁱb̲ind< expands toˡs̲vdV ← { a →
  pa ← (f̲ormat r̲avel f̲loat a) ⎕N̲PUT "ext:linalg/svd_v?float=" c̲at f̲ormat s̲hape a
  v ← n̲umbers ⎕N̲GET "ext:linalg/svd_v"
  (f̲loor n̲umbers ⎕N̲GET "ext:linalg/svd_v?shape") r̲eshape v
}
37
38⍝# The native package behind these functions.
39ˡpackage ← "linalg"