Module: Numo::Linalg

Defined in:
ext/numo/linalg/lapack/lapack.c,
ext/numo/linalg/lapack/lapack_c.c,
ext/numo/linalg/lapack/lapack_d.c,
ext/numo/linalg/lapack/lapack_s.c,
ext/numo/linalg/lapack/lapack_z.c,
ext/numo/linalg/blas/blas.c,
ext/numo/linalg/blas/blas_c.c,
ext/numo/linalg/blas/blas_d.c,
ext/numo/linalg/blas/blas_s.c,
ext/numo/linalg/blas/blas_z.c,
lib/numo/linalg/function.rb,
lib/numo/linalg/loader.rb,
lib/numo/linalg/version.rb

Defined Under Namespace

Modules: Blas, Lapack, Loader Classes: LapackError

Constant Summary

BLAS_CHAR = 
{
 SFloat => "s",
 DFloat => "d",
 SComplex => "c",
 DComplex => "z",
}
VERSION = 
"0.1.0"

Class Method Summary collapse

Class Method Details

.blas_char(*args) ⇒ Object 



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# File 'lib/numo/linalg/function.rb', line 57

def blas_char(*args)
  t = Float
  args.each do |a|
    k =
      case a
      when NArray
        a.class
      when Array
        NArray.array_type(a)
      end
    if k && k < NArray
      t = k::UPCAST[t]
    end
  end
  BLAS_CHAR[t] || raise(TypeError,"invalid data type for BLAS/LAPACK")
end

.cho_fact(a, uplo: 'U') ⇒ Numo::NArray

Computes the Cholesky factorization of a symmetric/Hermitian positive definite matrix A. The factorization has the form

A = U**H * U,  if UPLO = 'U', or
A = L  * L**H,  if UPLO = 'L',

where U is an upper triangular matrix and L is lower triangular

Parameters:

  • a (Numo::NArray) —

    n-by-n symmetric matrix A (>= 2-dimensinal NArray)

  • uplo (String or Symbol) —

    optional, default=’U’. Access upper or (‘U’) lower (‘L’) triangle.

Returns:

  • (Numo::NArray) —

    the factor U or L.



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# File 'lib/numo/linalg/function.rb', line 418

def cho_fact(a, uplo:'U')
  Lapack.call(:potrf, a, uplo:uplo)[0]
end

.cho_inv(a, uplo: 'U') ⇒ Numo::NArray

Computes the inverse of a symmetric/Hermitian positive definite matrix A using the Cholesky factorization A = U**T*U or A = L*L**T computed by Linalg.cho_fact.

Parameters:

  • a (Numo::NArray) —

    the triangular factor U or L from the Cholesky factorization, as computed by Linalg.cho_fact.

  • uplo (String or Symbol) —

    optional, default=’U’. Access upper or (‘U’) lower (‘L’) triangle.

Returns:

  • (Numo::NArray) —

    the upper or lower triangle of the (symmetric) inverse of A.



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# File 'lib/numo/linalg/function.rb', line 434

def cho_inv(a, uplo:'U')
  Lapack.call(:potri, a, uplo:uplo)[0]
end

.cho_solve(a, b, uplo: 'U') ⇒ Numo::NArray

Solves a system of linear equations A*X = B with a symmetric/Hermitian positive definite matrix A using the Cholesky factorization A = U**T*U or A = L*L**T computed by Linalg.cho_fact.

Parameters:

  • a (Numo::NArray) —

    the triangular factor U or L from the Cholesky factorization, as computed by Linalg.cho_fact.

  • b (Numo::NArray) —

    the right hand side matrix B.

  • uplo (String or Symbol) —

    optional, default=’U’. Access upper or (‘U’) lower (‘L’) triangle.

Returns:

  • (Numo::NArray) —

    the solution matrix X.



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# File 'lib/numo/linalg/function.rb', line 450

def cho_solve(a, b, uplo:'U')
  Lapack.call(:potrs, a, b, uplo:uplo)[0]
end

.cond(a, ord = nil) ⇒ Numo::NArray

Compute the condition number of a matrix using the norm with one of the following order.

|  ord  |  matrix norm           |
| ----- | ---------------------- |
|  nil  | 2-norm using SVD       |
| 'fro' | Frobenius norm         |
| 'inf' | x.abs.sum(axis:-1).max |
|    1  | x.abs.sum(axis:-2).max |
|    2  | 2-norm (max sing_vals) |

Examples:

a = Numo::DFloat[[1, 0, -1], [0, 1, 0], [1, 0, 1]]
=> Numo::DFloat#shape=[3,3]
[[1, 0, -1],
 [0, 1, 0],
 [1, 0, 1]]
LA = Numo::Linalg
LA.cond(a)
=> 1.4142135623730951
LA.cond(a, 'fro')
=> 3.1622776601683795
LA.cond(a, 'inf')
=> 2.0
LA.cond(a, '-inf')
=> 1.0
LA.cond(a, 1)
=> 2.0
LA.cond(a, -1)
=> 1.0
LA.cond(a, 2)
=> 1.4142135623730951
LA.cond(a, -2)
=> 0.7071067811865475
(LA.svdvals(a)).min*(LA.svdvals(LA.inv(a))).min
=> 0.7071067811865475

Parameters:

  • a (Numo::NArray) —

    matrix or vector (>= 1-dimensinal NArray)

  • ord (String or Symbol) (defaults to: nil) —

    Order of the norm.

Returns:

  • (Numo::NArray) —

    cond result



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# File 'lib/numo/linalg/function.rb', line 711

def cond(a,ord=nil)
  if ord.nil?
    s = svdvals(a)
    s[false, 0]/s[false, -1]
  else
    norm(a, ord, axis:[-2,-1]) * norm(inv(a), ord, axis:[-2,-1])
  end
end

.det(a) ⇒ Float or Complex or Numo::NArray

Determinant of a matrix

Parameters:

  • a (Numo::NArray) —

    matrix (>= 2-dimensional NArray)

Returns:

  • (Float or Complex or Numo::NArray) 


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# File 'lib/numo/linalg/function.rb', line 725

def det(a)
  lu, piv, = Lapack.call(:getrf, a)
  idx = piv.new_narray.store(piv.class.new(piv.shape[-1]).seq(1))
  m = piv.eq(idx).count_false(axis:-1) % 2
  sign = m * -2 + 1
  lu.diagonal.prod(axis:-1) * sign
end

.dot(a, b) ⇒ Numo::NArray

Dot product.

Parameters:

  • a (Numo::NArray) —

    matrix or vector (>= 1-dimensinal NArray)

  • b (Numo::NArray) —

    matrix or vector (>= 1-dimensinal NArray)

Returns:

  • (Numo::NArray) —

    result of dot product



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# File 'lib/numo/linalg/function.rb', line 82

def dot(a, b)
  a = NArray.asarray(a)
  b = NArray.asarray(b)
  case a.ndim
  when 1
    case b.ndim
    when 1
      Blas.call(:dot, a, b)
    else
      Blas.call(:gemv, b, a, trans:'t')
    end
  else
    case b.ndim
    when 1
      Blas.call(:gemv, a, b)
    else
      Blas.call(:gemm, a, b)
    end
  end
end

.eig(a, left: false, right: true) ⇒ [w,vl,vr]

Computes the eigenvalues and, optionally, the left and/or right eigenvectors for a square nonsymmetric matrix A.

Parameters:

  • a (Numo::NArray) —

    square nonsymmetric matrix (>= 2-dimensinal NArray)

  • left (Bool) —

    (optional) If true, left eigenvectors are computed.

  • right (Bool) —

    (optional) If true, right eigenvectors are computed.

Returns:

  • ([w,vl,vr]) —
    • w [Numo::NArray] – The eigenvalues.
    • vl [Numo::NArray] – The left eigenvectors if left is true, otherwise nil.
    • vr [Numo::NArray] – The right eigenvectors if right is true, otherwise nil.


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# File 'lib/numo/linalg/function.rb', line 468

def eig(a, left:false, right:true)
  jobvl, jobvr = left, right
  case blas_char(a)
  when /c|z/
    w, vl, vr, info = Lapack.call(:geev, a, jobvl:jobvl, jobvr:jobvr)
  else
    wr, wi, vl, vr, info = Lapack.call(:geev, a, jobvl:jobvl, jobvr:jobvr)
    w  = wr + wi * Complex::I
    vl = _make_complex_eigvecs(w,vl) if left
    vr = _make_complex_eigvecs(w,vr) if right
  end
  [w,vl,vr] #.compact
end

.eigh(a, vals_only: false, uplo: false, turbo: false) ⇒ [w,v]

Computes the eigenvalues and, optionally, the left and/or right eigenvectors for a square symmetric/hermitian matrix A.

Parameters:

  • a (Numo::NArray) —

    square nonsymmetric matrix (>= 2-dimensinal NArray)

  • values_only (Bool) —

    (optional) If false, eigenvectors are computed.

  • uplo (String or Symbol) —

    (optional, default=’U’) Access upper (‘U’) or lower (‘L’) triangle.

Returns:

  • ([w,v]) —
    • w [Numo::NArray] – The eigenvalues.
    • v [Numo::NArray] – The eigenvectors if vals_only is false, otherwise nil.


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# File 'lib/numo/linalg/function.rb', line 493

def eigh(a, vals_only:false, uplo:false, turbo:false)
  jobz = vals_only ? 'N' : 'V' # jobz: Compute eigenvalues and eigenvectors.
  case blas_char(a)
  when /c|z/
    func = turbo ? :hegv : :heev
  else
    func = turbo ? :sygv : :syev
  end
  w, v, = Lapack.call(func, a, uplo:uplo, jobz:jobz)
  [w,v] #.compact
end

.eigvals(a) ⇒ Numo::NArray

Computes the eigenvalues only for a square nonsymmetric matrix A.

Parameters:

  • a (Numo::NArray) —

    square nonsymmetric matrix (>= 2-dimensinal NArray)

Returns:

  • (Numo::NArray) —

    eigenvalues



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# File 'lib/numo/linalg/function.rb', line 510

def eigvals(a)
  jobvl, jobvr = 'N','N'
  case blas_char(a)
  when /c|z/
    w, = Lapack.call(:geev, a, jobvl:jobvl, jobvr:jobvr)
  else
    wr, wi, = Lapack.call(:geev, a, jobvl:jobvl, jobvr:jobvr)
    w  = wr + wi * Complex::I
  end
  w
end

.eigvalsh(a, uplo: false, turbo: false) ⇒ Numo::NArray

Computes the eigenvalues for a square symmetric/hermitian matrix A.

Parameters:

  • a (Numo::NArray) —

    square symmetric/hermitian matrix (>= 2-dimensinal NArray)

  • uplo (String or Symbol) —

    (optional, default=’U’) Access upper (‘U’) or lower (‘L’) triangle.

Returns:

  • (Numo::NArray) —

    eigenvalues



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# File 'lib/numo/linalg/function.rb', line 530

def eigvalsh(a, uplo:false, turbo:false)
  jobz = 'N' # jobz: Compute eigenvalues and eigenvectors.
  case blas_char(a)
  when /c|z/
    func = turbo ? :hegv : :heev
  else
    func = turbo ? :sygv : :syev
  end
  Lapack.call(func, a, uplo:uplo, jobz:jobz)[0]
end

.inv(a, driver: "getrf", uplo: 'U') ⇒ Numo::NArray

Inverse matrix from square matrix a

Examples:

Numo::Linalg.inv(a,driver:'getrf')
=> Numo::DFloat#shape=[2,2]
[[-2, 1],
 [1.5, -0.5]]
a.dot(Numo::Linalg.inv(a,driver:'getrf'))
=> Numo::DFloat#shape=[2,2]
[[1, 0],
 [8.88178e-16, 1]]

Parameters:

  • a (Numo::NArray) —

    n-by-n square matrix (>= 2-dimensinal NArray)

  • driver (String or Symbol) —

    choose LAPACK diriver (‘ge’|’sy’|’he’|’po’) + (“sv”|”trf”) (optional, default=’getrf’)

  • uplo (String or Symbol) —

    optional, default=’U’. Access upper or (‘U’) lower (‘L’) triangle. (omitted when driver:”ge”)

Returns:

  • (Numo::NArray) —

    The inverse matrix.



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# File 'lib/numo/linalg/function.rb', line 829

def inv(a, driver:"getrf", uplo:'U')
  case driver
  when /(ge|sy|he|po)sv$/
    d = $1
    b = a.new_zeros.eye
    solve(a, b, driver:d, uplo:uplo)
  when /(ge|sy|he)tr[fi]$/
    d = $1
    lu, piv = lu_fact(a, driver:d, uplo:uplo)
    lu_inv(lu, piv, driver:d, uplo:uplo)
  when /potr[fi]$/
    lu = cho_fact(a, uplo:uplo)
    cho_inv(lu, uplo:uplo)
  else
    raise ArgumentError, "invalid driver: #{driver}"
  end
end

.lstsq(a, b, driver: 'lsd', rcond: -1)) ⇒ [x, resids, rank, s]

Computes the minimum-norm solution to a linear least squares problem:

    minimize 2-norm(| b - A*x |)

using the singular value decomposition (SVD) of A. A is an M-by-N matrix which may be rank-deficient.

Parameters:

  • a (Numo::NArray) —

    m-by-n matrix A (>= 2-dimensinal NArray)

  • b (Numo::NArray) —

    m-by-nrhs right-hand-side matrix b (>= 1-dimensinal NArray)

  • driver (String or Symbol) —

    choose LAPACK driver from ‘lsd’,’lss’,’lsy’ (optional, default=’lsd’)

  • rcond (Float) —

    (optional, default=-1) RCOND is used to determine the effective rank of A. Singular values S(i) <= RCOND*S(1) are treated as zero. If RCOND < 0, machine precision is used instead.

Returns:

  • ([x, resids, rank, s]) —
    • x – The solution matrix/vector X.
    • resids – Sums of residues, squared 2-norm for each column in b - a x. If matrix_rank(a) < N or > M, or ‘gelsy’ is used, this is an empty array.
    • rank – The effective rank of A, i.e., the number of singular values which are greater than RCOND*S(1).
    • s – The singular values of A in decreasing order. Returns nil if ‘gelsy’ is used.


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# File 'lib/numo/linalg/function.rb', line 873

def lstsq(a, b, driver:'lsd', rcond:-1)
  a = NArray.asarray(a)
  b = NArray.asarray(b)
  b_orig = nil
  if b.shape.size==1
    b_orig = b
    b = b_orig[true,:new]
  end
  m = a.shape[-2]
  n = a.shape[-1]
  #nrhs = b.shape[-1]
  if m != b.shape[-2]
    raise NArray::ShapeError, "size mismatch: A-row and B-row"
  end
  if m < n   # need to extend b matrix
    shp = b.shape
    shp[-2] = n
    b2 = b.class.zeros(*shp)
    b2[false,0...m,true] = b
    b = b2
  end
  case driver.to_s
  when /^(ge)?lsd$/i
    # x, s, rank, info
    x, s, rank, = Lapack.call(:gelsd, a, b, rcond:rcond)
  when /^(ge)?lss$/i
    # v, x, s, rank, info
    _, x, s, rank, = Lapack.call(:gelss, a, b, rcond:rcond)
  when /^(ge)?lsy$/i
    jpvt = Int32.zeros(*a[false,0,true].shape)
    # v, x, jpvt, rank, info
    _, x, _, rank, = Lapack.call(:gelsy, a, b, jpvt, rcond:rcond)
    s = nil
  else
    raise ArgumentError, "invalid driver: #{driver}"
  end
  resids = nil
  if m > n
    if /ls(d|s)$/i =~ driver
      case rank
      when n
        resids = (x[n..-1,true].abs**2).sum(axis:0)
      when NArray
        if true
          resids = (x[false,n..-1,true].abs**2).sum(axis:-2)
        else
          resids = x[false,0,true].new_zeros
          mask = rank.eq(n)
          # NArray does not suppurt this yet.
          resids[mask,true] = (x[mask,n..-1,true].abs**2).sum(axis:-2)
        end
      end
    end
    x = x[false,0...n,true]
  end
  if b_orig && b_orig.shape.size==1
    x = x[true,0]
    resids &&= resids[false,0]
  end
  [x, resids, rank, s]
end

.lu_fact(a, driver: "gen", uplo: "U") ⇒ [lu, ipiv]

Computes an LU factorization of a M-by-N matrix A using partial pivoting with row interchanges.

The factorization has the form

A = P * L * U

where P is a permutation matrix, L is lower triangular with unit diagonal elements (lower trapezoidal if m > n), and U is upper triangular (upper trapezoidal if m < n).

Parameters:

  • a (Numo::NArray) —

    m-by-n matrix A (>= 2-dimensinal NArray)

  • driver (String or Symbol) —

    choose LAPACK diriver from ‘gen’,’sym’,’her’. (optional, default=’gen’)

  • uplo (String or Symbol) —

    optional, default=’U’. Access upper or (‘U’) lower (‘L’) triangle. (omitted when driver:”gen”)

Returns:

  • ([lu, ipiv]) —
    • lu [Numo::NArray] – The factors L and U from the factorization A = P*L*U; the unit diagonal elements of L are not stored.
    • ipiv [Numo::NArray] – The pivot indices; for 1 <= i <= min(M,N), row i of the matrix was interchanged with row IPIV(i).


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# File 'lib/numo/linalg/function.rb', line 321

def lu_fact(a, driver:"gen", uplo:"U")
  case driver.to_s
  when /^gen?(trf)?$/i
    Lapack.call(:getrf, a)[0..1]
  when /^(sym?|her?)(trf)?$/i
    func = driver[0..2].downcase+"trf"
    Lapack.call(func, a, uplo:uplo)[0..1]
  else
    raise ArgumentError, "invalid driver: #{driver}"
  end
end

.lu_inv(lu, ipiv, driver: "gen", uplo: "U") ⇒ Numo::NArray

Computes the inverse of a matrix using the LU factorization computed by Numo::Linalg.lu_fact.

This method inverts U and then computes inv(A) by solving the system

inv(A)*L = inv(U)

for inv(A).

Parameters:

  • lu (Numo::NArray) —

    matrix containing the factors L and U from the factorization A = P*L*U as computed by Numo::Linalg.lu_fact.

  • ipiv (Numo::NArray) —

    The pivot indices from Numo::Linalg.lu_fact; for 1<=i<=N, row i of the matrix was interchanged with row IPIV(i).

  • driver (String or Symbol) —

    choose LAPACK diriver from ‘gen’,’sym’,’her’. (optional, default=’gen’)

  • uplo (String or Symbol) —

    optional, default=’U’. Access upper or (‘U’) lower (‘L’) triangle. (omitted when driver:”gen”)

Returns:

  • (Numo::NArray) —

    the inverse of the original matrix A.



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# File 'lib/numo/linalg/function.rb', line 354

def lu_inv(lu, ipiv, driver:"gen", uplo:"U")
  case driver.to_s
  when /^gen?(tri)?$/i
    Lapack.call(:getri, lu, ipiv)[0]
  when /^(sym?|her?)(tri)?$/i
    func = driver[0..2].downcase+"tri"
    Lapack.call(func, lu, ipiv, uplo:uplo)[0]
  else
    raise ArgumentError, "invalid driver: #{driver}"
  end
end

.lu_solve(lu, ipiv, b, driver: "gen", uplo: "U", trans: "N") ⇒ Numo::NArray

Solves a system of linear equations

A * X = B  or  A**T * X = B

with a N-by-N matrix A using the LU factorization computed by Numo::Linalg.lu_fact

Parameters:

  • lu (Numo::NArray) —

    matrix containing the factors L and U from the factorization A = P*L*U as computed by Numo::Linalg.lu_fact.

  • ipiv (Numo::NArray) —

    The pivot indices from Numo::Linalg.lu_fact; for 1<=i<=N, row i of the matrix was interchanged with row IPIV(i).

  • b (Numo::NArray) —

    the right hand side matrix B.

  • driver (String or Symbol) —

    choose LAPACK diriver from ‘gen’,’sym’,’her’. (optional, default=’gen’)

  • uplo (String or Symbol) —

    optional, default=’U’. Access upper or (‘U’) lower (‘L’) triangle. (omitted when driver:”gen”)

  • trans (String or Symbol) —

    Specifies the form of the system of equations (omitted if not driver:”gen”):

    • If ‘N’: A * X = B (No transpose).
    • If ‘T’: A*\*T* X = B (Transpose).
    • If ‘C’: A*\*T* X = B (Conjugate transpose = Transpose).

Returns:

  • (Numo::NArray) —

    the solution matrix X.



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# File 'lib/numo/linalg/function.rb', line 393

def lu_solve(lu, ipiv, b, driver:"gen", uplo:"U", trans:"N")
  case driver.to_s
  when /^gen?(trs)?$/i
    Lapack.call(:getrs, lu, ipiv, b, trans:trans)[0]
  when /^(sym?|her?)(trs)?$/i
    func = driver[0..2].downcase+"trs"
    Lapack.call(func, lu, ipiv, b, uplo:uplo)[0]
  else
    raise ArgumentError, "invalid driver: #{driver}"
  end
end

.matmul(a, b) ⇒ Numo::NArray

Matrix product.

Parameters:

  • a (Numo::NArray) —

    matrix (>= 2-dimensinal NArray)

  • b (Numo::NArray) —

    matrix (>= 2-dimensinal NArray)

Returns:

  • (Numo::NArray) —

    result of matrix product



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# File 'lib/numo/linalg/function.rb', line 107

def matmul(a, b)
  Blas.call(:gemm, a, b)
end

.matrix_power(a, n) ⇒ Object

Compute a square matrix a to the power n.

  • If n > 0: return a**n.
  • If n == 0: return identity matrix.
  • If n < 0: return (a*\*-1)*\*n.abs.

Examples:

i = Numo::DFloat[[0, 1], [-1, 0]]
=> Numo::DFloat#shape=[2,2]
[[0, 1],
 [-1, 0]]
Numo::Linalg.matrix_power(i,3)
=> Numo::DFloat#shape=[2,2]
[[0, -1],
 [1, 0]]
Numo::Linalg.matrix_power(i,0)
=> Numo::DFloat#shape=[2,2]
[[1, 0],
 [0, 1]]
Numo::Linalg.matrix_power(i,-3)
=> Numo::DFloat#shape=[2,2]
[[0, 1],
 [-1, 0]]

q = Numo::DFloat.zeros(4,4)
q[0..1,0..1] = -i
q[2..3,2..3] = i
q
=> Numo::DFloat#shape=[4,4]
[[-0, -1, 0, 0],
 [1, -0, 0, 0],
 [0, 0, 0, 1],
 [0, 0, -1, 0]]
Numo::Linalg.matrix_power(q,2)
=> Numo::DFloat#shape=[4,4]
[[-1, 0, 0, 0],
 [0, -1, 0, 0],
 [0, 0, -1, 0],
 [0, 0, 0, -1]]

Parameters:

  • a (Numo::NArray) —

    square matrix (>= 2-dimensinal NArray).

  • n (Integer) —

    the exponent.



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# File 'lib/numo/linalg/function.rb', line 153

def matrix_power(a, n)
  a = NArray.asarray(a)
  m,k = a.shape[-2..-1]
  unless m==k
    raise NArray::ShapeError, "input must be a square array"
  end
  unless Integer===n
    raise ArgumentError, "exponent must be an integer"
  end
  if n == 0
    return a.class.eye(m)
  elsif n < 0
    a = inv(a)
    n = n.abs
  end
  if n <= 3
    r = a
    (n-1).times do
      r = matmul(r,a)
    end
  else
    while (n & 1) == 0
      a = matmul(a,a)
      n >>= 1
    end
    r = a
    while n != 0
      a = matmul(a,a)
      n >>= 1
      if (n & 1) != 0
        r = matmul(r,a)
      end
    end
  end
  r
end

.matrix_rank(m, tol: nil, driver: 'svd') ⇒ Object

Compute matrix rank of array using SVD Rank is the number of singular values greater than tol.

Parameters:

  • m (Numo::NArray) —

    matrix (>= 2-dimensional NArray)

  • tol (Float) —

    threshold below which singular values are considered to be zero. If tol is nil, tol = sing_vals.max() * m.shape.max * EPSILON.

  • driver (String or Symbol) —

    choose LAPACK solver from ‘svd’, ‘sdd’. (optional, default=’svd’)



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# File 'lib/numo/linalg/function.rb', line 765

def matrix_rank(m, tol:nil, driver:'svd')
  m = Numo::NArray.asarray(m)
  if m.ndim < 2
    m.ne(0).any? ? 1 : 0
  else
    case driver.to_s
    when /^(ge)?sdd$/, "turbo"
      s = Lapack.call(:gesdd, m, jobz:'N')[0]
    when /^(ge)?svd$/
      s = Lapack.call(:gesvd, m, jobu:'N', jobvt:'N')[0]
    else
      raise ArgumentError, "invalid driver: #{driver}"
    end
    tol ||= s.max(axis:-1, keepdims:true) *
      (m.shape[-2..-1].max * s.class::EPSILON)
    (s > tol).count(axis:-1)
  end
end

.norm(a, ord = nil, axis: nil, keepdims: false) ⇒ Numo::NArray

Compute matrix or vector norm.

|  ord  |  matrix norm           | vector norm                 |
| ----- | ---------------------- | --------------------------- |
|  nil  | Frobenius norm         | 2-norm                      |
| 'fro' | Frobenius norm         |  -                          |
| 'inf' | x.abs.sum(axis:-1).max | x.abs.max                   |
|    0  |  -                     | (x.ne 0).sum                |
|    1  | x.abs.sum(axis:-2).max | same as below               |
|    2  | 2-norm (max sing_vals) | same as below               |
| other |  -                     | (x.abs**ord).sum**(1.0/ord) |

Parameters:

  • a (Numo::NArray) —

    matrix or vector (>= 1-dimensinal NArray)

  • ord (String or Symbol) (defaults to: nil) —

    Order of the norm .

  • axis (Integer or Array) —

    Applied axes (optional).

  • keepdims (Bool) —

    If true, the applied axes are left in result with size one (optional).

Returns:

  • (Numo::NArray) —

    norm result



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# File 'lib/numo/linalg/function.rb', line 563

def norm(a, ord=nil, axis:nil, keepdims:false)
  a = Numo::NArray.asarray(a)

  # check axis
  if axis
    case axis
    when Integer
      axis = [axis]
    when Array
      if axis.size < 1 || axis.size > 2
        raise ArgmentError, "axis option should be 1- or 2-element array"
      end
    else
      raise ArgumentError, "invalid option for axis: #{axis}"
    end
    # swap axes
    if a.ndim > 1
      idx = (0...a.ndim).to_a
      tmp = []
      axis.each do |i|
        x = idx[i]
        if x.nil?
          raise ArgmentError, "axis contains same dimension"
        end
        tmp << x
        idx[i] = nil
      end
      idx.compact!
      idx.concat(tmp)
      a = a.transpose(*idx)
    end
  else
    case a.ndim
    when 0
      raise ArgumentError, "zero-dimensional array"
    when 1
      axis = [-1]
    else
      axis = [-2,-1]
    end
  end

  # calculate norm
  case axis.size

  when 1  # vector
    k = keepdims
    ord ||= 2  # default
    case ord.to_s
    when "0"
      r = a.class.cast(a.ne(0)).sum(axis:-1, keepdims:k)
    when "1"
      r = a.abs.sum(axis:-1, keepdims:k)
    when "2"
      r = Blas.call(:nrm2, a, keepdims:k)
    when /^-?\d+$/
      o = ord.to_i
      r = (a.abs**o).sum(axis:-1, keepdims:k)**(1.0/o)
    when /^inf(inity)?$/i
      r = a.abs.max(axis:-1, keepdims:k)
    when /^-inf(inity)?$/i
      r = a.abs.min(axis:-1, keepdims:k)
    else
      raise ArgumentError, "ord (#{ord}) is invalid for vector norm"
    end

  when 2  # matrix
    if keepdims
      fixdims = [true] * a.ndim
      axis.each do |i|
        if i < -a.ndim || i >= a.ndim
          raise ArgmentError, "axis (%d) is out of range", i
        end
        fixdims[i] = :new
      end
    end
    ord ||= "fro"  # default
    case ord.to_s
    when "1"
      r, = Lapack.call(:lange, a, '1')
    when "-1"
      r = a.abs.sum(axis:-2).min(axis:-1)
    when "2"
      svd, = Lapack.call(:gesvd, a, jobu:'N', jobvt:'N')
      r = svd.max(axis:-1)
    when "-2"
      svd, = Lapack.call(:gesvd, a, jobu:'N', jobvt:'N')
      r = svd.min(axis:-1)
    when /^f(ro)?$/i
      r, = Lapack.call(:lange, a, 'F')
    when /^inf(inity)?$/i
      r, = Lapack.call(:lange, a, 'I')
    when /^-inf(inity)?$/i
      r = a.abs.sum(axis:-1).min(axis:-1)
    else
      raise ArgumentError, "ord (#{ord}) is invalid for matrix norm"
    end
    if keepdims
      if NArray===r
        r = r[*fixdims]
      else
        r = a.class.new(1,1).store(r)
      end
    end
  end
  return r
end

.pinv(a, driver: "svd", rcond: nil) ⇒ Numo::NArray

Compute the (Moore-Penrose) pseudo-inverse of a matrix using svd or lstsq.

Examples:

a = Numo::DFloat.new(5,3).rand_norm
=> Numo::DFloat#shape=[5,3]
[[-0.581255, -0.168354, 0.586895],
 [-0.595142, -0.802802, -0.326106],
 [0.282922, 1.68427, 0.918499],
 [-0.0485384, -0.464453, -0.992194],
 [0.413794, -0.60717, -0.699695]]
b = Numo::Linalg.pinv(a,driver:"svd")
=> Numo::DFloat(view)#shape=[3,5]
[[-0.360863, -0.813125, -0.353367, -0.891963, 0.877253],
 [-0.227645, 0.162939, 0.696655, 0.787685, -0.469346],
 [0.408671, -0.308323, -0.337807, -1.13833, 0.228051]]
(a-a.dot(b.dot(a))).abs.max
=> 5.551115123125783e-16

Parameters:

  • a (Numo::NArray) —

    m-by-n matrix A (>= 2-dimensinal NArray)

  • driver (String or Symbol) —

    choose LAPACK driver from SVD (‘svd’, ‘sdd’) or Least square (‘lsd’,’lss’,’lsy’) (optional, default=’svd’)

  • rcond (Float) —

    (optional, default=-1) RCOND is used to determine the effective rank of A. Singular values S(i) <= RCOND*S(1) are treated as zero. If RCOND < 0, machine precision is used instead.

Returns:

  • (Numo::NArray) 


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# File 'lib/numo/linalg/function.rb', line 963

def pinv(a, driver:"svd", rcond:nil)
  a = NArray.asarray(a)
  if a.ndim < 2
    raise NArray::ShapeError, "2-d array is required"
  end
  case driver
  when /^(ge)?s[dv]d$/
    s, u, vh = svd(a, driver:driver, job:'S')
    if rcond.nil? || rcond < 0
      rcond = ((SFloat===s) ? 1e3 : 1e6) * s.class::EPSILON
    elsif ! Numeric === rcond
      raise ArgumentError, "rcond must be Numeric"
    end
    cond = (s > rcond * s.max(axis:-1, keepdims:true))
    if cond.all?
      r = s.reciprocal
    else
      r = s.new_zeros
      r[cond] = s[cond].reciprocal
    end
    u *= r[false,:new,true]
    dot(u,vh).conj.swapaxes(-2,-1)
  when /^(ge)?ls[dsy]$/
    b = a.class.eye(a.shape[-2])
    x, = lstsq(a, b, driver:driver, rcond:rcond)
    x
  else
    raise ArgumentError, "#{driver.inspect} is not one of drivers: "+
      "svd, sdd, lsd, lss, lsy"
  end
end

.qr(a, mode: "reduce") ⇒ r, ...

Computes a QR factorization of a complex M-by-N matrix A: A = Q * R.

Parameters:

  • a (Numo::NArray) —

    m-by-n matrix A (>= 2-dimensinal NArray)

  • mode (String) —
    • “reduce” – returns both Q and R,
    • “r” – returns only R,
    • “economy” – returns both Q and R but computed in economy-size,
    • “raw” – returns QR and TAU used in LAPACK.

Returns:

  • (r) —

    if mode:”r”

  • ([q,r]) —

    if mode:”reduce” or “economic”

  • ([qr,tau]) —

    if mode:”raw” (LAPACK geqrf result)



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# File 'lib/numo/linalg/function.rb', line 205

def qr(a, mode:"reduce")
  qr,tau, = Lapack.call(:geqrf, a)
  *shp,m,n = qr.shape
  r = (m >= n && %w[economic raw].include?(mode)) ?
    qr[false, 0...n, true].triu : qr.triu
  mode = mode.to_s.downcase
  case mode
  when "r"
    return r
  when "raw"
    return [qr,tau]
  when "reduce","economic"
    # skip
  else
    raise ArgumentError, "invalid mode:#{mode}"
  end
  if m < n
    q, = Lapack.call(:orgqr, qr[false, 0...m], tau)
  elsif mode == "economic"
    q, = Lapack.call(:orgqr, qr, tau)
  else
    qqr = qr.class.zeros(*(shp+[m,m]))
    qqr[false,0...n] = qr
    q, = Lapack.call(:orgqr, qqr, tau)
  end
  return [q,r]
end

.slogdet(a) ⇒ [sign,logdet]

Natural logarithm of the determinant of a matrix

Parameters:

  • a (Numo::NArray) —

    matrix (>= 2-dimensional NArray)

Returns:

  • ([sign,logdet]) —
    • sign – A number representing the sign of the determinant.
    • logdet – The natural log of the absolute value of the determinant.


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# File 'lib/numo/linalg/function.rb', line 740

def slogdet(a)
  lu, piv, = Lapack.call(:getrf, a)
  idx = piv.new_narray.store(piv.class.new(piv.shape[-1]).seq(1))
  m = piv.eq(idx).count_false(axis:-1) % 2
  sign = m * -2 + 1

  lud = lu.diagonal
  if (lud.eq 0).any?
    return 0, (-Float::INFINITY)
  end
  lud_abs = lud.abs
  sign *= (lud/lud_abs).prod
  [sign, NMath.log(lud_abs).sum(axis:-1)]
end

.solve(a, b, driver: "gen", uplo: 'U') ⇒ Numo::NArray

Solves linear equation a * x = b for x from square matrix a

Parameters:

  • a (Numo::NArray) —

    n-by-n square matrix (>= 2-dimensinal NArray)

  • b (Numo::NArray) —

    n-by-nrhs right-hand-side matrix (>= 1-dimensinal NArray)

  • driver (String or Symbol) —

    choose LAPACK diriver from ‘gen’,’sym’,’her’ or ‘pos’. (optional, default=’gen’)

  • uplo (String or Symbol) —

    optional, default=’U’. Access upper or (‘U’) lower (‘L’) triangle. (omitted when driver:”gen”)

Returns:

  • (Numo::NArray) —

    The solusion matrix/vector X.



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# File 'lib/numo/linalg/function.rb', line 798

def solve(a, b, driver:"gen", uplo:'U')
  case driver.to_s
  when /^gen?(sv)?$/i
    # returns lu, x, ipiv, info
    Lapack.call(:gesv, a, b)[1]
  when /^(sym?|her?|pos?)(sv)?$/i
    func = driver[0..2].downcase+"sv"
    Lapack.call(func, a, b, uplo:uplo)[1]
  else
    raise ArgumentError, "invalid driver: #{driver}"
  end
end

.svd(a, driver: 'svd', job: 'A') ⇒ [sigma,u,vt]

Computes the Singular Value Decomposition (SVD) of a M-by-N matrix A, and the left and/or right singular vectors. The SVD is written

A = U * SIGMA * transpose(V)

where SIGMA is an M-by-N matrix which is zero except for its min(m,n) diagonal elements, U is an M-by-M orthogonal matrix, and V is an N-by-N orthogonal matrix. The diagonal elements of SIGMA are the singular values of A; they are real and non-negative, and are returned in descending order. The first min(m,n) columns of U and V are the left and right singular vectors of A. Note that the routine returns V**T, not V.

Parameters:

  • a (Numo::NArray) —

    m-by-n matrix A (>= 2-dimensinal NArray)

  • driver (String or Symbol) —

    choose LAPACK solver from ‘svd’, ‘sdd’. (optional, default=’svd’)

  • job (String or Symbol) —
    • ‘A’: all M columns of U and all N rows of V**T are returned in the arrays U and VT.
    • ‘S’: the first min(M,N) columns of U and the first min(M,N) rows of V**T are returned in the arrays U and VT.
    • ‘N’: no columns of U or rows of V**T are computed.

Returns:

  • ([sigma,u,vt]) —

    SVD result. Array



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# File 'lib/numo/linalg/function.rb', line 258

def svd(a, driver:'svd', job:'A')
  unless /^[ASN]/i =~ job
    raise ArgumentError, "invalid job: #{job.inspect}"
  end
  case driver.to_s
  when /^(ge)?sdd$/i, "turbo"
    Lapack.call(:gesdd, a, jobz:job)[0..2]
  when /^(ge)?svd$/i
    Lapack.call(:gesvd, a, jobu:job, jobvt:job)[0..2]
  else
    raise ArgumentError, "invalid driver: #{driver}"
  end
end

.svdvals(a, driver: 'svd') ⇒ Numo::NArray

Computes the Singular Values of a M-by-N matrix A. The SVD is written

A = U * SIGMA * transpose(V)

where SIGMA is an M-by-N matrix which is zero except for its min(m,n) diagonal elements. The diagonal elements of SIGMA are the singular values of A; they are real and non-negative, and are returned in descending order.

Parameters:

  • a (Numo::NArray) —

    m-by-n matrix A (>= 2-dimensinal NArray)

  • driver (String or Symbol) —

    choose LAPACK solver from ‘svd’, ‘sdd’. (optional, default=’svd’)

Returns:

  • (Numo::NArray) —

    returns SIGMA (singular values).



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# File 'lib/numo/linalg/function.rb', line 287

def svdvals(a, driver:'svd')
  case driver.to_s
  when /^(ge)?sdd$/i, "turbo"
    Lapack.call(:gesdd, a, jobz:'N')[0]
  when /^(ge)?svd$/i
    Lapack.call(:gesvd, a, jobu:'N', jobvt:'N')[0]
  else
    raise ArgumentError, "invalid driver: #{driver}"
  end
end