Module: Numo::TinyLinalg

Defined in:
lib/numo/tiny_linalg.rb,
lib/numo/tiny_linalg/version.rb,
ext/numo/tiny_linalg/tiny_linalg.cpp,
ext/numo/tiny_linalg/tiny_linalg.cpp

Overview

Numo::TinyLinalg is a subset library from Numo::Linalg consisting only of methods used in Machine Learning algorithms.

Constant Summary collapse

VERSION =

The version of Numo::TinyLinalg you install.

'0.3.6'
OPENBLAS_VERSION =

The version of OpenBLAS used in background library.

rb_str_new_cstr(OPENBLAS_VERSION)

Class Method Summary collapse

Class Method Details

.blas_char(a, ...) ⇒ String

Returns BLAS char ([sdcz]) defined by data-type of arguments.

Parameters:

  • a (Numo::NArray)

Returns:

  • (String) 


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# File 'ext/numo/tiny_linalg/tiny_linalg.cpp', line 107

static VALUE tiny_linalg_blas_char(int argc, VALUE* argv, VALUE self) {
  VALUE nary_arr = Qnil;
  rb_scan_args(argc, argv, "*", &nary_arr);

  const char type = blas_char(nary_arr);
  if (type == 'n') {
    rb_raise(rb_eTypeError, "invalid data type for BLAS/LAPACK");
    return Qnil;
  }

  return rb_str_new(&type, 1);
}

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

Solves linear equation ‘A * x = b` or `A * X = B` for `x` with the Cholesky factorization of `A`.

Examples:

require 'numo/tiny_linalg'

Numo::Linalg = Numo::TinyLinalg unless defined?(Numo::Linalg)

s = Numo::DFloat.new(3, 3).rand - 0.5
a = s.transpose.dot(s)
u = Numo::Linalg.cholesky(a)

b = Numo::DFloat.new(3).rand
x = Numo::Linalg.cho_solve(u, b)

puts (b - a.dot(x)).abs.max
=> 0.0

Parameters:

  • a (Numo::NArray) —

    The n-by-n cholesky factor.

  • b (Numo::NArray) —

    The n right-hand side vector, or n-by-nrhs right-hand side matrix.

  • uplo (String) (defaults to: 'U') —

    Whether to compute the upper- or lower-triangular Cholesky factor (‘U’ or ‘L’).

Returns:

  • (Numo::NArray) —

    The solution vector or matrix ‘X`.

Raises:

  • (ArgumentError) 


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# File 'lib/numo/tiny_linalg.rb', line 341

def cho_solve(a, b, uplo: 'U')
  raise ArgumentError, 'input array a must be 2-dimensional' if a.ndim != 2
  raise ArgumentError, 'input array a must be square' if a.shape[0] != a.shape[1]
  raise ArgumentError, "incompatible dimensions: a.shape[0] = #{a.shape[0]} != b.shape[0] = #{b.shape[0]}" if a.shape[0] != b.shape[0]

  bchr = blas_char(a, b)
  raise ArgumentError, "invalid array type: #{a.class}" if bchr == 'n'

  fnc = :"#{bchr}potrs"
  x, _info = Numo::TinyLinalg::Lapack.send(fnc, a, b.dup, uplo: uplo)
  x
end

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

Computes the Cholesky decomposition of a symmetric / Hermitian positive-definite matrix.

Examples:

require 'numo/tiny_linalg'

Numo::Linalg = Numo::TinyLinalg unless defined?(Numo::Linalg)

s = Numo::DFloat.new(3, 3).rand - 0.5
a = s.transpose.dot(s)
u = Numo::Linalg.cholesky(a)

pp u
# =>
# Numo::DFloat#shape=[3,3]
# [[0.532006, 0.338183, -0.18036],
#  [0, 0.325153, 0.011721],
#  [0, 0, 0.436738]]

pp (a - u.transpose.dot(u)).abs.max
# => 1.3877787807814457e-17

l = Numo::Linalg.cholesky(a, uplo: 'L')

pp l
# =>
# Numo::DFloat#shape=[3,3]
# [[0.532006, 0, 0],
#  [0.338183, 0.325153, 0],
#  [-0.18036, 0.011721, 0.436738]]

pp (a - l.dot(l.transpose)).abs.max
# => 1.3877787807814457e-17

Parameters:

  • a (Numo::NArray) —

    The n-by-n symmetric matrix.

  • uplo (String) (defaults to: 'U') —

    Whether to compute the upper- or lower-triangular Cholesky factor (‘U’ or ‘L’).

Returns:

  • (Numo::NArray) —

    The upper- or lower-triangular Cholesky factor of a.

Raises:

  • (ArgumentError) 


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# File 'lib/numo/tiny_linalg.rb', line 300

def cholesky(a, uplo: 'U')
  raise ArgumentError, 'input array a must be 2-dimensional' if a.ndim != 2
  raise ArgumentError, 'input array a must be square' if a.shape[0] != a.shape[1]

  bchr = blas_char(a)
  raise ArgumentError, "invalid array type: #{a.class}" if bchr == 'n'

  fnc = :"#{bchr}potrf"
  c, _info = Numo::TinyLinalg::Lapack.send(fnc, a.dup, uplo: uplo)

  case uplo
  when 'U'
    c.triu
  when 'L'
    c.tril
  else
    raise ArgumentError, "invalid uplo: #{uplo}"
  end
end

.det(a) ⇒ Float/Complex

Computes the determinant of matrix.

Examples:

require 'numo/tiny_linalg'

Numo::Linalg = Numo::TinyLinalg unless defined?(Numo::Linalg)

a = Numo::DFloat[[0, 2, 3], [4, 5, 6], [7, 8, 9]]
pp (3.0 - Numo::Linalg.det(a)).abs
# => 1.3322676295501878e-15

Parameters:

  • a (Numo::NArray) —

    The n-by-n square matrix.

Returns:

  • (Float/Complex) —

    The determinant of ‘a`.

Raises:

  • (ArgumentError) 


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# File 'lib/numo/tiny_linalg.rb', line 367

def det(a)
  raise ArgumentError, 'input array a must be 2-dimensional' if a.ndim != 2
  raise ArgumentError, 'input array a must be square' if a.shape[0] != a.shape[1]

  bchr = blas_char(a)
  raise ArgumentError, "invalid array type: #{a.class}" if bchr == 'n'

  getrf = :"#{bchr}getrf"
  lu, piv, info = Numo::TinyLinalg::Lapack.send(getrf, a.dup)

  if info.zero?
    det_l = 1
    det_u = lu.diagonal.prod
    det_p = piv.map_with_index { |v, i| v == i + 1 ? 1 : -1 }.prod
    det_l * det_u * det_p
  elsif info.positive?
    raise 'the factor U is singular, and the inverse matrix could not be computed.'
  else
    raise "the #{-info}-th argument of getrf had illegal value"
  end
end

.dot(a, b) ⇒ Float|Complex|Numo::NArray

Calculates dot product of two vectors / matrices.

Parameters:

  • a (Numo::NArray)
  • b (Numo::NArray)

Returns:

  • (Float|Complex|Numo::NArray) 


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# File 'ext/numo/tiny_linalg/tiny_linalg.cpp', line 155

static VALUE tiny_linalg_dot(VALUE self, VALUE a_, VALUE b_) {
  VALUE a = IsNArray(a_) ? a_ : rb_funcall(numo_cNArray, rb_intern("asarray"), 1, a_);
  VALUE b = IsNArray(b_) ? b_ : rb_funcall(numo_cNArray, rb_intern("asarray"), 1, b_);

  VALUE arg_arr = rb_ary_new3(2, a, b);
  const char type = blas_char(arg_arr);
  if (type == 'n') {
    rb_raise(rb_eTypeError, "invalid data type for BLAS/LAPACK");
    return Qnil;
  }

  VALUE ret = Qnil;
  narray_t* a_nary = NULL;
  narray_t* b_nary = NULL;
  GetNArray(a, a_nary);
  GetNArray(b, b_nary);
  const int a_ndim = NA_NDIM(a_nary);
  const int b_ndim = NA_NDIM(b_nary);

  if (a_ndim == 1) {
    if (b_ndim == 1) {
      ID fn_id = type == 'c' || type == 'z' ? rb_intern("dotu") : rb_intern("dot");
      ret = rb_funcall(rb_mTinyLinalgBlas, rb_intern("call"), 3, ID2SYM(fn_id), a, b);
    } else {
      VALUE kw_args = rb_hash_new();
      if (!RTEST(nary_check_contiguous(b)) && RTEST(rb_funcall(b, rb_intern("fortran_contiguous?"), 0))) {
        b = rb_funcall(b, rb_intern("transpose"), 0);
        rb_hash_aset(kw_args, ID2SYM(rb_intern("trans")), rb_str_new_cstr("N"));
      } else {
        rb_hash_aset(kw_args, ID2SYM(rb_intern("trans")), rb_str_new_cstr("T"));
      }
      char fn_name[] = "xgemv";
      fn_name[0] = type;
      VALUE argv[3] = { b, a, kw_args };
      ret = rb_funcallv_kw(rb_mTinyLinalgBlas, rb_intern(fn_name), 3, argv, RB_PASS_KEYWORDS);
    }
  } else {
    if (b_ndim == 1) {
      VALUE kw_args = rb_hash_new();
      if (!RTEST(nary_check_contiguous(a)) && RTEST(rb_funcall(b, rb_intern("fortran_contiguous?"), 0))) {
        a = rb_funcall(a, rb_intern("transpose"), 0);
        rb_hash_aset(kw_args, ID2SYM(rb_intern("trans")), rb_str_new_cstr("T"));
      } else {
        rb_hash_aset(kw_args, ID2SYM(rb_intern("trans")), rb_str_new_cstr("N"));
      }
      char fn_name[] = "xgemv";
      fn_name[0] = type;
      VALUE argv[3] = { a, b, kw_args };
      ret = rb_funcallv_kw(rb_mTinyLinalgBlas, rb_intern(fn_name), 3, argv, RB_PASS_KEYWORDS);
    } else {
      VALUE kw_args = rb_hash_new();
      if (!RTEST(nary_check_contiguous(a)) && RTEST(rb_funcall(b, rb_intern("fortran_contiguous?"), 0))) {
        a = rb_funcall(a, rb_intern("transpose"), 0);
        rb_hash_aset(kw_args, ID2SYM(rb_intern("transa")), rb_str_new_cstr("T"));
      } else {
        rb_hash_aset(kw_args, ID2SYM(rb_intern("transa")), rb_str_new_cstr("N"));
      }
      if (!RTEST(nary_check_contiguous(b)) && RTEST(rb_funcall(b, rb_intern("fortran_contiguous?"), 0))) {
        b = rb_funcall(b, rb_intern("transpose"), 0);
        rb_hash_aset(kw_args, ID2SYM(rb_intern("transb")), rb_str_new_cstr("T"));
      } else {
        rb_hash_aset(kw_args, ID2SYM(rb_intern("transb")), rb_str_new_cstr("N"));
      }
      char fn_name[] = "xgemm";
      fn_name[0] = type;
      VALUE argv[3] = { a, b, kw_args };
      ret = rb_funcallv_kw(rb_mTinyLinalgBlas, rb_intern(fn_name), 3, argv, RB_PASS_KEYWORDS);
    }
  }

  RB_GC_GUARD(a);
  RB_GC_GUARD(b);

  return ret;
}

.eigh(a, b = nil, vals_only: false, vals_range: nil, uplo: 'U', turbo: false) ⇒ Array<Numo::NArray>

Computes the eigenvalues and eigenvectors of a symmetric / Hermitian matrix by solving an ordinary or generalized eigenvalue problem.

Examples:

require 'numo/tiny_linalg'

Numo::Linalg = Numo::TinyLinalg unless defined?(Numo::Linalg)

x = Numo::DFloat.new(5, 3).rand - 0.5
c = x.dot(x.transpose)
vals, vecs = Numo::Linalg.eigh(c, vals_range: [2, 4])

pp vals
# =>
# Numo::DFloat#shape=[3]
# [0.118795, 0.434252, 0.903245]

pp vecs
# =>
# Numo::DFloat#shape=[5,3]
# [[0.154178, 0.60661, -0.382961],
#  [-0.349761, -0.141726, -0.513178],
#  [0.739633, -0.468202, 0.105933],
#  [0.0519655, -0.471436, -0.701507],
#  [-0.551488, -0.412883, 0.294371]]

pp (x - vecs.dot(vals.diag).dot(vecs.transpose)).abs.max
# => 3.3306690738754696e-16

Parameters:

  • a (Numo::NArray) —

    The n-by-n symmetric / Hermitian matrix.

  • b (Numo::NArray) (defaults to: nil) —

    The n-by-n symmetric / Hermitian matrix. If nil, identity matrix is assumed.

  • vals_only (Boolean) (defaults to: false) —

    The flag indicating whether to return only eigenvalues.

  • vals_range (Range/Array) (defaults to: nil) —

    The range of indices of the eigenvalues (in ascending order) and corresponding eigenvectors to be returned. If nil, all eigenvalues and eigenvectors are computed.

  • uplo (String) (defaults to: 'U') —

    This argument is for compatibility with Numo::Linalg.solver, and is not used.

  • turbo (Bool) (defaults to: false) —

    The flag indicating whether to use a divide and conquer algorithm. If vals_range is given, this flag is ignored.

Returns:

  • (Array<Numo::NArray>) —

    The eigenvalues and eigenvectors.

Raises:

  • (ArgumentError) 


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# File 'lib/numo/tiny_linalg.rb', line 51

def eigh(a, b = nil, vals_only: false, vals_range: nil, uplo: 'U', turbo: false) # rubocop:disable Metrics/AbcSize, Metrics/CyclomaticComplexity, Metrics/ParameterLists, Metrics/PerceivedComplexity, Lint/UnusedMethodArgument
  raise ArgumentError, 'input array a must be 2-dimensional' if a.ndim != 2
  raise ArgumentError, 'input array a must be square' if a.shape[0] != a.shape[1]

  b_given = !b.nil?
  raise ArgumentError, 'input array b must be 2-dimensional' if b_given && b.ndim != 2
  raise ArgumentError, 'input array b must be square' if b_given && b.shape[0] != b.shape[1]
  raise ArgumentError, "invalid array type: #{b.class}" if b_given && blas_char(b) == 'n'

  bchr = blas_char(a)
  raise ArgumentError, "invalid array type: #{a.class}" if bchr == 'n'

  jobz = vals_only ? 'N' : 'V'

  if b_given
    fnc = %w[d s].include?(bchr) ? "#{bchr}sygv" : "#{bchr}hegv"
    if vals_range.nil?
      fnc << 'd' if turbo
      vecs, _b, vals, _info = Numo::TinyLinalg::Lapack.send(fnc.to_sym, a.dup, b.dup, jobz: jobz)
    else
      fnc << 'x'
      il = vals_range.first(1)[0] + 1
      iu = vals_range.last(1)[0] + 1
      _a, _b, _m, vals, vecs, _ifail, _info = Numo::TinyLinalg::Lapack.send(
        fnc.to_sym, a.dup, b.dup, jobz: jobz, range: 'I', il: il, iu: iu
      )
    end
  else
    fnc = %w[d s].include?(bchr) ? "#{bchr}syev" : "#{bchr}heev"
    if vals_range.nil?
      fnc << 'd' if turbo
      vecs, vals, _info = Numo::TinyLinalg::Lapack.send(fnc.to_sym, a.dup, jobz: jobz)
    else
      fnc << 'r'
      il = vals_range.first(1)[0] + 1
      iu = vals_range.last(1)[0] + 1
      _a, _m, vals, vecs, _isuppz, _info = Numo::TinyLinalg::Lapack.send(
        fnc.to_sym, a.dup, jobz: jobz, range: 'I', il: il, iu: iu
      )
    end
  end

  vecs = nil if vals_only

  [vals, vecs]
end

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

Computes the inverse matrix of a square matrix.

Examples:

require 'numo/tiny_linalg'

Numo::Linalg = Numo::TinyLinalg unless defined?(Numo::Linalg)

a = Numo::DFloat.new(5, 5).rand

inv_a = Numo::Linalg.inv(a)

pp (inv_a.dot(a) - Numo::DFloat.eye(5)).abs.max
# => 7.019165976816745e-16

pp inv_a.dot(a).sum
# => 5.0

Parameters:

  • a (Numo::NArray) —

    The n-by-n square matrix.

  • driver (String) (defaults to: 'getrf') —

    This argument is for compatibility with Numo::Linalg.solver, and is not used.

  • uplo (String) (defaults to: 'U') —

    This argument is for compatibility with Numo::Linalg.solver, and is not used.

Returns:

  • (Numo::NArray) —

    The inverse matrix of ‘a`.

Raises:

  • (ArgumentError) 


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# File 'lib/numo/tiny_linalg.rb', line 410

def inv(a, driver: 'getrf', uplo: 'U') # rubocop:disable Lint/UnusedMethodArgument
  raise ArgumentError, 'input array a must be 2-dimensional' if a.ndim != 2
  raise ArgumentError, 'input array a must be square' if a.shape[0] != a.shape[1]

  bchr = blas_char(a)
  raise ArgumentError, "invalid array type: #{a.class}" if bchr == 'n'

  getrf = :"#{bchr}getrf"
  getri = :"#{bchr}getri"

  lu, piv, info = Numo::TinyLinalg::Lapack.send(getrf, a.dup)
  if info.zero?
    Numo::TinyLinalg::Lapack.send(getri, lu, piv)[0]
  elsif info.positive?
    raise 'the factor U is singular, and the inverse matrix could not be computed.'
  else
    raise "the #{-info}-th argument of getrf had illegal value"
  end
end

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

Computes the matrix or vector norm.

|  ord  |  matrix norm           | vector norm                 |
| ----- | ---------------------- | --------------------------- |
|  nil  | Frobenius norm         | 2-norm                      |
| 'fro' | Frobenius norm         |  -                          |
| 'nuc' | nuclear 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) |

Examples:

require 'numo/tiny_linalg'
Numo::Linalg = Numo::TinyLinalg unless defined?(Numo::Linalg)

# matrix norm
x = Numo::DFloat[[1, 2, -3, 1], [-4, 1, 8, 2]]
pp Numo::Linalg.norm(x)
# => 10

# vector norm
x = Numo::DFloat[3, -4]
pp Numo::Linalg.norm(x)
# => 5

Parameters:

  • a (Numo::NArray) —

    The matrix or vector (>= 1-dimensinal NArray)

  • ord (String/Numeric) (defaults to: nil) —

    The order of the norm.

  • axis (Integer/Array) (defaults to: nil) —

    The applied axes.

  • keepdims (Bool) (defaults to: false) —

    The flag indicating whether to leave the normed axes in the result as dimensions with size one.

Returns:

  • (Numo::NArray/Numeric) —

    The norm of the matrix or vectors.

Raises:

  • (ArgumentError) 


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# File 'lib/numo/tiny_linalg.rb', line 130

def norm(a, ord = nil, axis: nil, keepdims: false) # rubocop:disable Metrics/AbcSize, Metrics/CyclomaticComplexity, Metrics/MethodLength, Metrics/PerceivedComplexity
  a = Numo::NArray.asarray(a) unless a.is_a?(Numo::NArray)

  return 0.0 if a.empty?

  # for compatibility with Numo::Linalg.norm
  if ord.is_a?(String)
    if ord == 'inf'
      ord = Float::INFINITY
    elsif ord == '-inf'
      ord = -Float::INFINITY
    end
  end

  if axis.nil?
    norm = case a.ndim
           when 1
             Numo::TinyLinalg::Blas.send(:"#{blas_char(a)}nrm2", a) if ord.nil? || ord == 2
           when 2
             if ord.nil? || ord == 'fro'
               Numo::TinyLinalg::Lapack.send(:"#{blas_char(a)}lange", a, norm: 'F')
             elsif ord.is_a?(Numeric)
               if ord == 1
                 Numo::TinyLinalg::Lapack.send(:"#{blas_char(a)}lange", a, norm: '1')
               elsif !ord.infinite?.nil? && ord.infinite?.positive?
                 Numo::TinyLinalg::Lapack.send(:"#{blas_char(a)}lange", a, norm: 'I')
               end
             end
           else
             if ord.nil?
               b = a.flatten.dup
               Numo::TinyLinalg::Blas.send(:"#{blas_char(b)}nrm2", b)
             end
           end
    unless norm.nil?
      norm = Numo::NArray.asarray(norm).reshape(*([1] * a.ndim)) if keepdims
      return norm
    end
  end

  if axis.nil?
    axis = Array.new(a.ndim) { |d| d }
  else
    case axis
    when Integer
      axis = [axis]
    when Array, Numo::NArray
      axis = axis.flatten.to_a
    else
      raise ArgumentError, "invalid axis: #{axis}"
    end
  end

  raise ArgumentError, "the number of dimensions of axis is inappropriate for the norm: #{axis.size}" unless axis.size == 1 || axis.size == 2
  raise ArgumentError, "axis is out of range: #{axis}" unless axis.all? { |ax| (-a.ndim...a.ndim).cover?(ax) }

  if axis.size == 1
    ord ||= 2
    raise ArgumentError, "invalid ord: #{ord}" unless ord.is_a?(Numeric)

    ord_inf = ord.infinite?
    if ord_inf.nil?
      case ord
      when 0
        a.class.cast(a.ne(0)).sum(axis: axis, keepdims: keepdims)
      when 1
        a.abs.sum(axis: axis, keepdims: keepdims)
      else
        (a.abs**ord).sum(axis: axis, keepdims: keepdims)**1.fdiv(ord)
      end
    elsif ord_inf.positive?
      a.abs.max(axis: axis, keepdims: keepdims)
    else
      a.abs.min(axis: axis, keepdims: keepdims)
    end
  else
    ord ||= 'fro'
    raise ArgumentError, "invalid ord: #{ord}" unless ord.is_a?(String) || ord.is_a?(Numeric)
    raise ArgumentError, "invalid axis: #{axis}" if axis.uniq.size == 1

    r_axis, c_axis = axis.map { |ax| ax.negative? ? ax + a.ndim : ax }

    norm = if ord.is_a?(String)
             raise ArgumentError, "invalid ord: #{ord}" unless %w[fro nuc].include?(ord)

             if ord == 'fro'
               Numo::NMath.sqrt((a.abs**2).sum(axis: axis))
             else
               b = a.transpose(c_axis, r_axis).dup
               gesvd = :"#{blas_char(b)}gesvd"
               s, = Numo::TinyLinalg::Lapack.send(gesvd, b, jobu: 'N', jobvt: 'N')
               s.sum(axis: -1)
             end
           else
             ord_inf = ord.infinite?
             if ord_inf.nil?
               case ord
               when -2
                 b = a.transpose(c_axis, r_axis).dup
                 gesvd = :"#{blas_char(b)}gesvd"
                 s, = Numo::TinyLinalg::Lapack.send(gesvd, b, jobu: 'N', jobvt: 'N')
                 s.min(axis: -1)
               when -1
                 c_axis -= 1 if c_axis > r_axis
                 a.abs.sum(axis: r_axis).min(axis: c_axis)
               when 1
                 c_axis -= 1 if c_axis > r_axis
                 a.abs.sum(axis: r_axis).max(axis: c_axis)
               when 2
                 b = a.transpose(c_axis, r_axis).dup
                 gesvd = :"#{blas_char(b)}gesvd"
                 s, = Numo::TinyLinalg::Lapack.send(gesvd, b, jobu: 'N', jobvt: 'N')
                 s.max(axis: -1)
               else
                 raise ArgumentError, "invalid ord: #{ord}"
               end
             else
               r_axis -= 1 if r_axis > c_axis
               if ord_inf.positive?
                 a.abs.sum(axis: c_axis).max(axis: r_axis)
               else
                 a.abs.sum(axis: c_axis).min(axis: r_axis)
               end
             end
           end
    if keepdims
      norm = Numo::NArray.asarray(norm) unless norm.is_a?(Numo::NArray)
      norm = norm.reshape(*([1] * a.ndim))
    end

    norm
  end
end

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

Computes the (Moore-Penrose) pseudo-inverse of a matrix using singular value decomposition.

Examples:

require 'numo/tiny_linalg'

Numo::Linalg = Numo::TinyLinalg unless defined?(Numo::Linalg)

a = Numo::DFloat.new(5, 3).rand

inv_a = Numo::Linalg.pinv(a)

pp (inv_a.dot(a) - Numo::DFloat.eye(3)).abs.max
# => 1.1102230246251565e-15

pp inv_a.dot(a).sum
# => 3.0

Parameters:

  • a (Numo::NArray) —

    The m-by-n matrix to be pseudo-inverted.

  • driver (String) (defaults to: 'svd') —

    The LAPACK driver to be used (‘svd’ or ‘sdd’).

  • rcond (Float) (defaults to: nil) —

    The threshold value for small singular values of ‘a`, default value is `a.shape.max * EPS`.

Returns:

  • (Numo::NArray) —

    The pseudo-inverse of ‘a`.



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# File 'lib/numo/tiny_linalg.rb', line 451

def pinv(a, driver: 'svd', rcond: nil)
  s, u, vh = svd(a, driver: driver, job: 'S')
  rcond = a.shape.max * s.class::EPSILON if rcond.nil?
  rank = s.gt(rcond * s[0]).count

  u = u[true, 0...rank] / s[0...rank]
  u.dot(vh[0...rank, true]).conj.transpose
end

.qr(a, mode: 'reduce') ⇒ Numo::NArray+

Computes the QR decomposition of a matrix.

Examples:

require 'numo/tiny_linalg'

Numo::Linalg = Numo::TinyLinalg unless defined?(Numo::Linalg)

x = Numo::DFloat.new(5, 3).rand

q, r = Numo::Linalg.qr(x, mode: 'economic')

pp q
# =>
# Numo::DFloat#shape=[5,3]
# [[-0.0574417, 0.635216, 0.707116],
#  [-0.187002, -0.073192, 0.422088],
#  [-0.502239, 0.634088, -0.537489],
#  [-0.0473292, 0.134867, -0.0223491],
#  [-0.840979, -0.413385, 0.180096]]

pp r
# =>
# Numo::DFloat#shape=[3,3]
# [[-1.07508, -0.821334, -0.484586],
#  [0, 0.513035, 0.451868],
#  [0, 0, 0.678737]]

pp (q.dot(r) - x).abs.max
# => 3.885780586188048e-16

Parameters:

  • a (Numo::NArray) —

    The m-by-n matrix to be decomposed.

  • mode (String) (defaults to: 'reduce') —

    The mode of decomposition.

    • “reduce” – returns both Q [m, m] and R [m, n],

    • “r” – returns only R,

    • “economic” – returns both Q [m, n] and R [n, n],

    • “raw” – returns QR and TAU (LAPACK geqrf results).

Returns:

  • (Numo::NArray) —

    if mode=‘r’.

  • (Array<Numo::NArray>) —

    if mode=‘reduce’ or ‘economic’ or ‘raw’.

Raises:

  • (ArgumentError) 


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# File 'lib/numo/tiny_linalg.rb', line 498

def qr(a, mode: 'reduce')
  raise ArgumentError, 'input array a must be 2-dimensional' if a.ndim != 2
  raise ArgumentError, "invalid mode: #{mode}" unless %w[reduce r economic raw].include?(mode)

  bchr = blas_char(a)
  raise ArgumentError, "invalid array type: #{a.class}" if bchr == 'n'

  geqrf = :"#{bchr}geqrf"
  qr, tau, = Numo::TinyLinalg::Lapack.send(geqrf, a.dup)

  return [qr, tau] if mode == 'raw'

  m, n = qr.shape
  r = m > n && %w[economic raw].include?(mode) ? qr[0...n, true].triu : qr.triu

  return r if mode == 'r'

  org_ung_qr = %w[d s].include?(bchr) ? :"#{bchr}orgqr" : :"#{bchr}ungqr"

  q = if m < n
        Numo::TinyLinalg::Lapack.send(org_ung_qr, qr[true, 0...m], tau)[0]
      elsif mode == 'economic'
        Numo::TinyLinalg::Lapack.send(org_ung_qr, qr, tau)[0]
      else
        qqr = a.class.zeros(m, m)
        qqr[0...m, 0...n] = qr
        Numo::TinyLinalg::Lapack.send(org_ung_qr, qqr, tau)[0]
      end

  [q, r]
end

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

Solves linear equation ‘A * x = b` or `A * X = B` for `x` from square matrix `A`.

Examples:

require 'numo/tiny_linalg'

Numo::Linalg = Numo::TinyLinalg unless defined?(Numo::Linalg)

a = Numo::DFloat.new(3, 3).rand
b = Numo::DFloat.eye(3)

x = Numo::Linalg.solve(a, b)

pp x
# =>
# Numo::DFloat#shape=[3,3]
# [[-2.12332, 4.74868, 0.326773],
#  [1.38043, -3.79074, 1.25355],
#  [0.775187, 1.41032, -0.613774]]

pp (b - a.dot(x)).abs.max
# => 2.1081041547796492e-16

Parameters:

  • a (Numo::NArray) —

    The n-by-n square matrix.

  • b (Numo::NArray) —

    The n right-hand side vector, or n-by-nrhs right-hand side matrix.

  • driver (String) (defaults to: 'gen') —

    This argument is for compatibility with Numo::Linalg.solver, and is not used.

  • uplo (String) (defaults to: 'U') —

    This argument is for compatibility with Numo::Linalg.solver, and is not used.

Returns:

  • (Numo::NArray) —

    The solusion vector / matrix ‘X`.

Raises:

  • (ArgumentError) 


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# File 'lib/numo/tiny_linalg.rb', line 557

def solve(a, b, driver: 'gen', uplo: 'U') # rubocop:disable Lint/UnusedMethodArgument
  raise ArgumentError, 'input array a must be 2-dimensional' if a.ndim != 2
  raise ArgumentError, 'input array a must be square' if a.shape[0] != a.shape[1]

  bchr = blas_char(a, b)
  raise ArgumentError, "invalid array type: #{a.class}, #{b.class}" if bchr == 'n'

  gesv = :"#{bchr}gesv"
  Numo::TinyLinalg::Lapack.send(gesv, a.dup, b.dup)[1]
end

.solve_triangular(a, b, lower: false) ⇒ Numo::NArray

Solves linear equation ‘A * x = b` or `A * X = B` for `x` assuming `A` is a triangular matrix.

Examples:

require 'numo/tiny_linalg'

Numo::Linalg = Numo::TinyLinalg unless defined?(Numo::Linalg)

a = Numo::DFloat.new(3, 3).rand.triu
b = Numo::DFloat.eye(3)

x = Numo::Linalg.solve(a, b)

pp x
# =>
# Numo::DFloat#shape=[3,3]
# [[16.1932, -52.0604, 30.5283],
#  [0, 8.61765, -17.9585],
#  [0, 0, 6.05735]]

pp (b - a.dot(x)).abs.max
# => 4.071100642430302e-16

Parameters:

  • a (Numo::NArray) —

    The n-by-n triangular matrix.

  • b (Numo::NArray) —

    The n right-hand side vector, or n-by-nrhs right-hand side matrix.

  • lower (Boolean) (defaults to: false) —

    The flag indicating whether to use the lower-triangular part of ‘a`.

Returns:

  • (Numo::NArray) —

    The solusion vector / matrix ‘X`.

Raises:

  • (ArgumentError) 


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# File 'lib/numo/tiny_linalg.rb', line 594

def solve_triangular(a, b, lower: false)
  raise ArgumentError, 'input array a must be 2-dimensional' if a.ndim != 2
  raise ArgumentError, 'input array a must be square' if a.shape[0] != a.shape[1]

  bchr = blas_char(a, b)
  raise ArgumentError, "invalid array type: #{a.class}, #{b.class}" if bchr == 'n'

  trtrs = :"#{bchr}trtrs"
  uplo = lower ? 'L' : 'U'
  x, info = Numo::TinyLinalg::Lapack.send(trtrs, a, b.dup, uplo: uplo)
  raise "wrong value is given to the #{info}-th argument of #{trtrs} used internally" if info.negative?

  x
end

.svd(a, driver: 'svd', job: 'A') ⇒ Array<Numo::NArray>

Computes the Singular Value Decomposition (SVD) of a matrix: ‘A = U * S * V^T`

Examples:

require 'numo/tiny_linalg'

Numo::Linalg = Numo::TinyLinalg unless defined?(Numo::Linalg)

x = Numo::DFloat.new(5, 2).rand.dot(Numo::DFloat.new(2, 3).rand)
pp x
# =>
# Numo::DFloat#shape=[5,3]
# [[0.104945, 0.0284236, 0.117406],
#  [0.862634, 0.210945, 0.922135],
#  [0.324507, 0.0752655, 0.339158],
#  [0.67085, 0.102594, 0.600882],
#  [0.404631, 0.116868, 0.46644]]

s, u, vt = Numo::Linalg.svd(x, job: 'S')

z = u.dot(s.diag).dot(vt)
pp z
# =>
# Numo::DFloat#shape=[5,3]
# [[0.104945, 0.0284236, 0.117406],
#  [0.862634, 0.210945, 0.922135],
#  [0.324507, 0.0752655, 0.339158],
#  [0.67085, 0.102594, 0.600882],
#  [0.404631, 0.116868, 0.46644]]

pp (x - z).abs.max
# => 4.440892098500626e-16

Parameters:

  • a (Numo::NArray) —

    Matrix to be decomposed.

  • driver (String) (defaults to: 'svd') —

    The LAPACK driver to be used (‘svd’ or ‘sdd’).

  • job (String) (defaults to: 'A') —

    The job option (‘A’, ‘S’, or ‘N’).

Returns:

  • (Array<Numo::NArray>) —

    The singular values and singular vectors ([s, u, vt]).

Raises:

  • (ArgumentError) 


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# File 'lib/numo/tiny_linalg.rb', line 645

def svd(a, driver: 'svd', job: 'A')
  raise ArgumentError, "invalid job: #{job}" unless /^[ASN]/i.match?(job.to_s)

  bchr = blas_char(a)
  raise ArgumentError, "invalid array type: #{a.class}" if bchr == 'n'

  case driver.to_s
  when 'sdd'
    gesdd = :"#{bchr}gesdd"
    s, u, vt, info = Numo::TinyLinalg::Lapack.send(gesdd, a.dup, jobz: job)
  when 'svd'
    gesvd = :"#{bchr}gesvd"
    s, u, vt, info = Numo::TinyLinalg::Lapack.send(gesvd, a.dup, jobu: job, jobvt: job)
  else
    raise ArgumentError, "invalid driver: #{driver}"
  end

  raise "the #{info.abs}-th argument had illegal value" if info.negative?
  raise 'input array has a NAN entry' if info == -4
  raise 'svd did not converge' if info.positive?

  [s, u, vt]
end