Module: Numo::Linalg

Defined in:
lib/numo/linalg.rb,
lib/numo/linalg/version.rb,
ext/numo/linalg/linalg.c,
ext/numo/linalg/linalg.c

Overview

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

Defined Under Namespace

Classes: LapackError

Constant Summary collapse

VERSION =

The version of numo-linalg-alt you install.

'0.9.1'
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/linalg/linalg.c', line 76

static VALUE 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_fact(a, uplo: 'U') ⇒ Numo::NArray

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

Parameters:

  • a (Numo::NArray) —

    The n-by-n symmetric / Hermitian positive-definite matrix.

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

    The part of the matrix to be used (‘U’ or ‘L’).

Returns:

  • (Numo::NArray) —

    The upper- / lower-triangular matrix ‘U` / `L`.

Raises:

  • (Numo::NArray::ShapeError) 


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# File 'lib/numo/linalg.rb', line 1448

def cho_fact(a, uplo: 'U')
  raise Numo::NArray::ShapeError, 'input array a must be 2-dimensional' if a.ndim != 2
  raise Numo::NArray::ShapeError, 'input array a must be square' if a.shape[0] != a.shape[1]
  raise ArgumentError, 'uplo must be "U" or "L"' unless %w[U L].include?(uplo)

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

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

  raise LapackError, "the #{-info}-th argument of #{fnc} had illegal value" if info.negative?

  if info.positive?
    raise LapackError, "the leading principal minor of order #{info} is not positive, " \
                       'and the factorization could not be completed.'
  end

  c
end

.cho_inv(a, uplo: 'U') ⇒ Numo::NArray

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

Examples:

require 'numo/linalg'

a = Numo::DFloat.new(3, 5).rand - 0.5
a = a.dot(a.transpose)
c = Numo::Linalg.cho_fact(a)
tri_inv_a = Numo::Linalg.cho_inv(c)
tri_inv_a = tri_inv_a.triu
inv_a = tri_inv_a + tri_inv_a.transpose - tri_inv_a.diagonal.diag
error = (inv_a.dot(a) - Numo::DFloat.eye(3)).abs.max
pp error
# => 1.923726113137665e-15

Parameters:

  • a (Numo::NArray) —

    The Cholesky decomposition of the n-by-n symmetric / Hermitian positive-definite matrix.

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

    The part of the matrix to be used (‘U’ or ‘L’).

Returns:

  • (Numo::NArray) —

    The upper- / lower-triangular matrix of the inverse of ‘a`.

Raises:

  • (ArgumentError) 


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# File 'lib/numo/linalg.rb', line 2210

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

  fnc = :"#{bchr}potri"
  inv, info = Numo::Linalg::Lapack.send(fnc, a.dup, uplo: uplo)

  raise LapackError, "the #{info.abs}-th argument of #{fnc} had illegal value" if info.negative?

  if info.positive?
    raise LapackError, "the (#{info - 1}, #{info - 1})-th element of the factor U or L is zero, " \
                       'and the inverse could not be computed.'
  end

  inv
end

.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/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:

  • (Numo::NArray::ShapeError) 


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# File 'lib/numo/linalg.rb', line 363

def cho_solve(a, b, uplo: 'U')
  raise Numo::NArray::ShapeError, 'input array a must be 2-dimensional' if a.ndim != 2
  raise Numo::NArray::ShapeError, 'input array a must be square' if a.shape[0] != a.shape[1]
  raise Numo::NArray::ShapeError, "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::Linalg::Lapack.send(fnc, a, b.dup, uplo: uplo)
  raise LapackError, "the #{-info}-th argument of potrs had illegal value" if info.negative?

  x
end

.cho_solve_banded(ab, b, uplo: 'U') ⇒ Numo::NArray

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

Examples:

require 'numo/linalg'

a = Numo::DFloat.new(4, 4).rand - 0.5
a = a.dot(a.transpose)
a = a.tril(1) * a.triu(-1)
ab = Numo::DFloat.zeros(2, 4)
ab[0, 1...] = a[a.diag_indices(1)]
ab[1, true] = a[a.diag_indices]
c = Numo::Linalg.cholesky_banded(ab)
b = Numo::DFloat.new(4, 2).rand
x = Numo::Linalg.cho_solve_banded(c, b)
pp (b - a.dot(x)).abs.max
# => 1.1102230246251565e-16

Parameters:

  • ab (Numo::NArray) —

    The m-by-n banded 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:

  • (Numo::NArray::ShapeError) 


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# File 'lib/numo/linalg.rb', line 399

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

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

  fnc = :"#{bchr}pbtrs"
  x, info = Numo::Linalg::Lapack.send(fnc, ab, b.dup, uplo: uplo)
  raise LapackError, "the #{-info}-th argument of potrs had illegal value" if info.negative?

  x
end

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

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

Examples:

require '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:

  • (Numo::NArray::ShapeError) 


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# File 'lib/numo/linalg.rb', line 301

def cholesky(a, uplo: 'U')
  raise Numo::NArray::ShapeError, 'input array a must be 2-dimensional' if a.ndim != 2
  raise Numo::NArray::ShapeError, '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::Linalg::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

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

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

Parameters:

  • a (Numo::NArray) —

    The banded matrix.

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

    Is the matrix form upper or lower (‘U’ or ‘L’).

Returns:

  • (Numo::NArray) —

    The Cholesky factor of the banded matrix.

Raises:

  • (Numo::NArray::ShapeError) 


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# File 'lib/numo/linalg.rb', line 326

def cholesky_banded(a, uplo: 'U')
  raise Numo::NArray::ShapeError, 'input array a must be 2-dimensional' if a.ndim != 2

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

  fnc = :"#{bchr}pbtrf"
  c, info = Numo::Linalg::Lapack.send(fnc, a.dup, uplo: uplo)
  raise LapackError, "the #{-info}-th argument of potrs had illegal value" if info.negative?

  if info.positive?
    raise LapackError, "the leading principal minor of order #{info} is not positive, " \
                       'and the factorization could not be completed.'
  end

  c
end

.cond(a, ord = nil) ⇒ Numo::NArray

Compute the condition number of a matrix.

nil or 2: 2-norm using singular values, ‘fro’: Frobenius norm, ‘info’: infinity norm, and 1: 1-norm.

Parameters:

  • a (Numo::NArray) —

    The input matrix.

  • ord (String/Symbol/Integer) (defaults to: nil) —

    The order of the norm.

Returns:

  • (Numo::NArray) —

    The condition number of the matrix.



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# File 'lib/numo/linalg.rb', line 1906

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

.coshm(a) ⇒ Numo::NArray

Computes the matrix hyperbolic cosine using the matrix exponential.

Parameters:

  • a (Numo::NArray) —

    The n-by-n square matrix.

Returns:

  • (Numo::NArray) —

    The matrix hyperbolic cosine of ‘a`.

Raises:

  • (Numo::NArray::ShapeError) 


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# File 'lib/numo/linalg.rb', line 2125

def coshm(a)
  raise Numo::NArray::ShapeError, 'input array a must be 2-dimensional' if a.ndim != 2
  raise Numo::NArray::ShapeError, 'input array a must be square' if a.shape[0] != a.shape[1]

  0.5 * (expm(a) + expm(-a))
end

.cosm(a) ⇒ Numo::NArray

Computes the matrix cosine using the matrix exponential.

Parameters:

  • a (Numo::NArray) —

    The n-by-n square matrix.

Returns:

  • (Numo::NArray) —

    The matrix cosine of ‘a`.

Raises:

  • (Numo::NArray::ShapeError) 


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# File 'lib/numo/linalg.rb', line 2079

def cosm(a)
  raise Numo::NArray::ShapeError, 'input array a must be 2-dimensional' if a.ndim != 2
  raise Numo::NArray::ShapeError, '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'

  b = a * 1.0i
  if %w[z c].include?(bchr)
    0.5 * (expm(b) + expm(-b))
  else
    expm(b).real
  end
end

.det(a) ⇒ Float/Complex

Computes the determinant of matrix.

Examples:

require '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:

  • (Numo::NArray::ShapeError) 


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# File 'lib/numo/linalg.rb', line 424

def det(a)
  raise Numo::NArray::ShapeError, 'input array a must be 2-dimensional' if a.ndim != 2
  raise Numo::NArray::ShapeError, '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::Linalg::Lapack.send(getrf, a.dup)
  raise LapackError, "the #{-info}-th argument of getrf had illegal value" if info.negative?

  # info > 0 means the factor U has a zero diagonal element and is singular.
  # In this case, the determinant is zero. The method should simply return 0.0.
  # Therefore, the error is not raised here.
  # raise 'the factor U is singular, ...' if info.positive?

  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
end

.diagsvd(s, m, n) ⇒ Numo::NArray

Creates a diagonal matrix from the given singular values.

Examples:

require 'numo/linalg'

s = Numo::DFloat[4, 2, 1]
d = Numo::Linalg.diagsvd(s, 3, 4)
pp d
# =>
# Numo::DFloat#shape=[3,4]
# [[4, 0, 0, 0],
#  [0, 2, 0, 0],
#  [0, 0, 1, 0]]
d = Numo::Linalg.diagsvd(s, 4, 3)
pp d
# =>
# Numo::DFloat#shape=[4,3]
# [[4, 0, 0],
#  [0, 2, 0],
#  [0, 0, 1],
#  [0, 0, 0]]

Parameters:

  • s (Numo::NArray) —

    The singular values.

  • m (Integer) —

    The number of rows of the constructed matrix.

  • n (Integer) —

    The number of columns of the constructed matrix.

Returns:

  • (Numo::NArray) —

    The m-by-n diagonal matrix with the singular values on the diagonal.

Raises:

  • (ArgumentError) 


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# File 'lib/numo/linalg.rb', line 1261

def diagsvd(s, m, n)
  sz_s = s.size
  raise ArgumentError, "size of s must be equal to m or n: s.size=#{sz_s}, m=#{m}, n=#{n}" if sz_s != m && sz_s != n

  mat = s.class.zeros(m, n)
  if sz_s == m
    mat[true, 0...m] = s.diag
  else
    mat[0...n, true] = s.diag
  end
  mat
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/linalg/linalg.c', line 125

static VALUE 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_mLinalgBlas, 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_mLinalgBlas, 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_mLinalgBlas, 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_mLinalgBlas, rb_intern(fn_name), 3, argv, RB_PASS_KEYWORDS);
    }
  }

  RB_GC_GUARD(a);
  RB_GC_GUARD(b);

  return ret;
}

.eig(a, left: false, right: true) ⇒ Array<Numo::NArray>

Computes the eigenvalues and right and/or left eigenvectors of a general square matrix.

Examples:

require 'numo/linalg'

a = Numo::DFloat.new(5, 5).rand - 0.5
w, _vl, vr = Numo::Linalg.eig(a)
error = (a.dot(vr) - vr.dot(w.diag)).abs.max
pp error
# => 4.718447854656915e-16

Parameters:

  • a (Numo::NArray) —

    The n-by-n square matrix.

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

    The flag indicating whether to compute the left eigenvectors.

  • right (Boolean) (defaults to: true) —

    The flag indicating whether to compute the right eigenvectors.

Returns:

  • (Array<Numo::NArray>) —

    The eigenvalues, left eigenvectors, and right eigenvectors.

Raises:

  • (Numo::NArray::ShapeError) 


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# File 'lib/numo/linalg.rb', line 1620

def eig(a, left: false, right: true) # rubocop:disable  Metrics/AbcSize, Metrics/PerceivedComplexity
  raise Numo::NArray::ShapeError, '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]

  jobvl = left ? 'V' : 'N'
  jobvr = right ? 'V' : 'N'

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

  fnc = :"#{bchr}geev"
  if %w[z c].include?(bchr)
    w, vl, vr, info = Numo::Linalg::Lapack.send(fnc, a.dup, jobvl: jobvl, jobvr: jobvr)
  else
    wr, wi, vl, vr, info = Numo::Linalg::Lapack.send(fnc, a.dup, jobvl: jobvl, jobvr: jobvr)
  end

  raise LapackError, "the #{info.abs}-th argument of #{fnc} had illegal value" if info.negative?
  raise LapackError, 'the QR algorithm failed to compute all the eigenvalues.' if info.positive?

  if %w[d s].include?(bchr)
    w = wr + (wi * 1.0i)
    ids = wi.gt(0).where
    unless ids.empty?
      cast_class = bchr == 'd' ? Numo::DComplex : Numo::SComplex
      if left
        tmp = cast_class.cast(vl)
        tmp[true, ids].imag = vl[true, ids + 1]
        tmp[true, ids + 1] = tmp[true, ids].conj
        vl = tmp
      end
      if right
        tmp = cast_class.cast(vr)
        tmp[true, ids].imag = vr[true, ids + 1]
        tmp[true, ids + 1] = tmp[true, ids].conj
        vr = tmp
      end
    end
  end

  [w, left ? vl : nil, right ? vr : nil]
end

.eig_banded(ab, vals_only: false, vals_range: nil, lower: false) ⇒ Array<Numo::NArray>

Computes the eigenvalues and eigenvectors of a symmetric / Hermitian banded matrix.

Examples:

require 'numo/linalg'

# Banded matrix A:
# [4 2 1 0 0]
# [2 5 2 1 0]
# [1 2 6 2 1]
# [0 1 2 7 2]
# [0 0 1 2 8]
#
ab = Numo::DFloat[[0, 0, 1, 1, 1],
                  [0, 2, 2, 2, 2],
                  [4, 5, 6, 7, 8]]

w, v = Numo::Linalg.eig_banded(ab)

b = v.dot(w.diag).dot(v.transpose)
b[b<1e-10] = 0.0
pp b
# =>
# Numo::DFloat#shape=[5,5]
# [[4, 2, 1, 0, 0],
#  [2, 5, 2, 1, 0],
#  [1, 2, 6, 2, 1],
#  [0, 1, 2, 7, 2],
#  [0, 0, 1, 2, 8]]

Parameters:

  • ab (Numo::NArray) —

    The (kd+1)-by-n array representing the banded matrix.

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

    The flag indicating whether to compute 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.

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

    The flag indicating whether to be in the lower-banded form.

Returns:

  • (Array<Numo::NArray>) —

    The eigenvalues and eigenvectors.

Raises:

  • (Numo::NArray::ShapeError) 


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# File 'lib/numo/linalg.rb', line 1738

def eig_banded(ab, vals_only: false, vals_range: nil, lower: false)
  raise Numo::NArray::ShapeError, 'input array ab must be 2-dimensional' if ab.ndim != 2

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

  jobz = vals_only ? 'N' : 'V'
  uplo = lower ? 'L' : 'U'
  fnc = %w[d s].include?(bchr) ? "#{bchr}sbevx" : "#{bchr}hbevx"

  if vals_range.nil?
    _, _, vals, vecs, _, info = Numo::Linalg::Lapack.send(fnc.to_sym, ab.dup, range: 'A', jobz: jobz, uplo: uplo)
  else
    il = vals_range.first(1)[0] + 1
    iu = vals_range.last(1)[0] + 1
    _, _, vals, vecs, _, info = Numo::Linalg::Lapack.send(fnc.to_sym, ab.dup,
                                                          range: 'I', jobz: jobz, uplo: uplo, il: il, iu: iu)
  end

  raise LapackError, "the #{info.abs}-th argument of #{fnc} had illegal value" if info.negative?

  vecs = nil if vals_only

  [vals, vecs]
end

.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/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:

  • (Numo::NArray::ShapeError) 


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# File 'lib/numo/linalg.rb', line 55

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 Numo::NArray::ShapeError, 'input array a must be 2-dimensional' if a.ndim != 2
  raise Numo::NArray::ShapeError, 'input array a must be square' if a.shape[0] != a.shape[1]

  b_given = !b.nil?
  raise Numo::NArray::ShapeError, 'input array b must be 2-dimensional' if b_given && b.ndim != 2
  raise Numo::NArray::ShapeError, '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::Linalg::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::Linalg::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::Linalg::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::Linalg::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

.eigh_tridiagonal(d, e, vals_only: false, vals_range: nil) ⇒ Array<Numo::NArray>

Computes the eigenvalues and eigenvectors of a real symmetric tridiagonal matrix.

Examples:

require 'numo/linalg'

# symmetric tridiagonal matrix A:
# [4 2 0 0 0]
# [2 5 2 0 0]
# [0 2 6 2 0]
# [0 0 2 7 2]
# [0 0 0 2 8]
d = Numo::DFloat[4, 5, 6, 7, 8]
e = Numo::DFloat[2, 2, 2, 2]

w, v = Numo::Linalg.eigh_tridiagonal(d, e)

b = v.dot(w.diag).dot(v.transpose)
b[b<1e-10] = 0.0
pp b
# =>
# Numo::DFloat#shape=[5,5]
# [[4, 2, 0, 0, 0],
#  [2, 5, 2, 0, 0],
#  [0, 2, 6, 2, 0],
#  [0, 0, 2, 7, 2],
#  [0, 0, 0, 2, 8]]

Parameters:

  • d (Numo::NArray) —

    The 1-dimensional array of length n representing the diagonal elements of the matrix.

  • e (Numo::NArray) —

    The 1-dimensional array of length n-1 representing the off-diagonal elements of the matrix.

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

    The flag indicating whether to compute 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.

Returns:

  • (Array<Numo::NArray>) —

    The eigenvalues and eigenvectors.

Raises:

  • (Numo::NArray::ShapeError) 


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# File 'lib/numo/linalg.rb', line 1810

def eigh_tridiagonal(d, e, vals_only: false, vals_range: nil) # rubocop:disable Metrics/AbcSize
  raise Numo::NArray::ShapeError, 'input array d must be 1-dimensional' if d.ndim != 1
  raise Numo::NArray::ShapeError, 'input array e must be 1-dimensional' if e.ndim != 1

  if d.shape[0] != e.shape[0] + 1
    raise Numo::NArray::ShapeError,
          "incompatible dimensions: d.shape[0] (#{d.shape[0]}) != e.shape[0] + 1 (#{e.shape[0] + 1})"
  end

  if d.is_a?(Numo::DComplex) || d.is_a?(Numo::SComplex) || e.is_a?(Numo::DComplex) || e.is_a?(Numo::SComplex)
    raise ArgumentError, 'eigh_tridiagonal does not support complex arrays'
  end

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

  jobz = vals_only ? 'N' : 'V'
  fnc = "#{bchr}stevx"

  e_w_zero = e.concatenate(0)

  if vals_range.nil?
    _, vals, vecs, _, info = Numo::Linalg::Lapack.send(fnc.to_sym, d.dup, e_w_zero, range: 'A', jobz: jobz)
  else
    il = vals_range.first(1)[0] + 1
    iu = vals_range.last(1)[0] + 1
    _, vals, vecs, _, info = Numo::Linalg::Lapack.send(fnc.to_sym, d.dup, e_w_zero,
                                                       range: 'I', jobz: jobz, il: il, iu: iu)
  end

  raise LapackError, "the #{info.abs}-th argument of #{fnc} had illegal value" if info.negative?

  vecs = nil if vals_only

  [vals, vecs]
end

.eigvals(a) ⇒ Numo::NArray

Computes the eigenvalues of a general square matrix.

Parameters:

  • a (Numo::NArray) —

    The n-by-n square matrix.

Returns:

  • (Numo::NArray) —

    The eigenvalues.

Raises:

  • (Numo::NArray::ShapeError) 


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# File 'lib/numo/linalg.rb', line 1667

def eigvals(a)
  raise Numo::NArray::ShapeError, 'input array a must be 2-dimensional' if a.ndim != 2
  raise Numo::NArray::ShapeError, '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}geev"
  if %w[z c].include?(bchr)
    w, _vl, _vr, info = Numo::Linalg::Lapack.send(fnc, a.dup, jobvl: 'N', jobvr: 'N')
  else
    wr, wi, _vl, _vr, info = Numo::Linalg::Lapack.send(fnc, a.dup, jobvl: 'N', jobvr: 'N')
    w = wr + (wi * 1.0i)
  end

  raise LapackError, "the #{info.abs}-th argument of #{fnc} had illegal value" if info.negative?
  raise LapackError, 'the QR algorithm failed to compute all the eigenvalues.' if info.positive?

  w
end

.eigvals_banded(ab, vals_range: nil, lower: false) ⇒ Numo::NArray

Computes the eigenvalues a symmetric / Hermitian banded matrix.

Parameters:

  • ab (Numo::NArray) —

    The (kd+1)-by-n array representing the banded matrix.

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

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

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

    The flag indicating whether to be in the lower-banded form.

Returns:

  • (Numo::NArray) —

    The eigenvalues.



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# File 'lib/numo/linalg.rb', line 1772

def eigvals_banded(ab, vals_range: nil, lower: false)
  eig_banded(ab, vals_only: true, vals_range: vals_range, lower: lower)[0]
end

.eigvalsh(a, b = nil, vals_range: nil, uplo: 'U', turbo: false) ⇒ Numo::NArray

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

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_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:

  • (Numo::NArray) —

    The eigenvalues.



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# File 'lib/numo/linalg.rb', line 1698

def eigvalsh(a, b = nil, vals_range: nil, uplo: 'U', turbo: false)
  eigh(a, b, vals_only: true, vals_range: vals_range, uplo: uplo, turbo: turbo)[0]
end

.eigvalsh_tridiagonal(d, e, vals_range: nil) ⇒ Numo::NArray

Computes the eigenvalues of a real symmetric tridiagonal matrix.

Parameters:

  • d (Numo::NArray) —

    The 1-dimensional array of length n representing the diagonal elements of the matrix.

  • e (Numo::NArray) —

    The 1-dimensional array of length n-1 representing the off-diagonal elements of the matrix.

  • 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.

Returns:

  • (Numo::NArray) —

    The eigenvalues.



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# File 'lib/numo/linalg.rb', line 1855

def eigvalsh_tridiagonal(d, e, vals_range: nil)
  eigh_tridiagonal(d, e, vals_only: true, vals_range: vals_range)[0]
end

.expm(a, ord = 8) ⇒ Numo::NArray

Computes the matrix exponential using a scaling and squaring algorithm with a Pade approximation.

Reference:

    1. Moler and C. Van Loan, “Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later,” SIAM Review, vol. 45, no. 1, pp. 3-49, 2003.

Parameters:

  • a (Numo::NArray) —

    The n-by-n square matrix.

  • ord (Integer) (defaults to: 8) —

    The order of the Padé approximation.

Returns:

  • (Numo::NArray) —

    The matrix exponential of ‘a`.

Raises:

  • (Numo::NArray::ShapeError) 


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# File 'lib/numo/linalg.rb', line 2013

def expm(a, ord = 8) # rubocop:disable Metrics/AbcSize
  raise Numo::NArray::ShapeError, 'input array a must be 2-dimensional' if a.ndim != 2
  raise Numo::NArray::ShapeError, 'input array a must be square' if a.shape[0] != a.shape[1]

  norm = a.abs.max
  n_sqr = norm.positive? ? [0, Math.log2(norm).to_i + 1].max : 0
  a /= 2**n_sqr

  x = a.dup
  c = 0.5
  sgn = 1
  nume = a.class.eye(a.shape[0]) + (c * a)
  deno = a.class.eye(a.shape[0]) - (c * a)
  (2..ord).each do |k|
    c *= (ord - k + 1).fdiv(k * ((2 * ord) - k + 1))
    x = a.dot(x)
    c_x = c * x
    nume += c_x
    deno += sgn * c_x
    sgn = -sgn
  end

  a_expm = Numo::Linalg.solve(deno, nume)
  n_sqr.times { a_expm = a_expm.dot(a_expm) }
  a_expm
end

.hessenberg(a, calc_q: false) ⇒ Numo::NArray, Array<Numo::NArray, Numo::NArray>

Computes the Hessenberg decomposition of a square matrix. The Hessenberg decomposition is given by ‘A = Q * H * Q^H`, where `A` is the input matrix, `Q` is a unitary matrix, and `H` is an upper Hessenberg matrix.

Examples:

require 'numo/linalg'

a = Numo::DFloat[[1, 2, 3], [4, 5, 6], [7, 8, 9]] * 0.5
h, q = Numo::Linalg.hessenberg(a, calc_q: true)

pp h
# => Numo::DFloat#shape=[3,3]
# [[0.5, -1.7985, -0.124035],
#  [-4.03113, 7.02308, 1.41538],
#  [0, 0.415385, -0.0230769]]
pp q
# => Numo::DFloat#shape=[3,3]
# [[1, 0, 0],
#  [0, -0.496139, -0.868243],
#  [0, -0.868243, 0.496139]]
pp (a - q.dot(h).dot(q.transpose)).abs.max
# => 1.7763568394002505e-15

Parameters:

  • a (Numo::NArray) —

    The n-by-n square matrix.

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

    The flag indicating whether to calculate the unitary matrix ‘Q`.

Returns:

  • (Numo::NArray) —

    if calc_q=false, the Hessenberg form ‘H`.

  • (Array<Numo::NArray, Numo::NArray>) —

    if calc_q=true, the Hessenberg form ‘H` and the unitary matrix `Q`.

Raises:

  • (Numo::NArray::ShapeError) 


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# File 'lib/numo/linalg.rb', line 856

def hessenberg(a, calc_q: false)
  raise Numo::NArray::ShapeError, 'input array a must be 2-dimensional' if a.ndim != 2
  raise Numo::NArray::ShapeError, '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'

  func = :"#{bchr}gebal"
  b, ilo, ihi, _, info = Numo::Linalg::Lapack.send(func, a.dup)

  raise LapackError, "the #{-info}-th argument of #{func} had illegal value" if info.negative?

  func = :"#{bchr}gehrd"
  hq, tau, info = Numo::Linalg::Lapack.send(func, b, ilo: ilo, ihi: ihi)

  raise LapackError, "the #{-info}-th argument of #{func} had illegal value" if info.negative?

  h = hq.triu(-1)
  return h unless calc_q

  func = %w[d s].include?(bchr) ? :"#{bchr}orghr" : :"#{bchr}unghr"
  q, info = Numo::Linalg::Lapack.send(func, hq, tau, ilo: ilo, ihi: ihi)

  raise LapackError, "the #{-info}-th argument of #{func} had illegal value" if info.negative?

  [h, q]
end

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

Computes the inverse matrix of a square matrix.

Examples:

require '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:

  • (Numo::NArray::ShapeError) 


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# File 'lib/numo/linalg.rb', line 465

def inv(a, driver: 'getrf', uplo: 'U') # rubocop:disable Lint/UnusedMethodArgument
  raise Numo::NArray::ShapeError, 'input array a must be 2-dimensional' if a.ndim != 2
  raise Numo::NArray::ShapeError, '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::Linalg::Lapack.send(getrf, a.dup)
  raise LapackError, "the #{-info}-th argument of getrf had illegal value" if info.negative?

  a_inv, info = Numo::Linalg::Lapack.send(getri, lu, piv)
  raise LapackError, "the #{-info}-th argument of getrf had illegal value" if info.negative?
  raise LapackError, 'The matrix is singular, and the inverse matrix could not be computed.' if info.positive?

  a_inv
end

.ldl(a, uplo: 'U', hermitian: true) ⇒ Array<Numo::NArray>

Computes the Bunch-Kaufman decomposition of a symmetric / Hermitian matrix. The factorization has the form ‘A = U * D * U^T` or `A = L * D * L^T`, where `U` (or `L`) is a product of permutation and unit upper (lower) triangular matrices, and `D` is a block diagonal matrix.

Examples:

require 'numo/linalg'

a = Numo::DFloat.new(5, 5).rand
a = 0.5 * (a + a.transpose)
u, d, _perm = Numo::Linalg.ldl(a)
error = (a - u.dot(d).dot(u.transpose)).abs.max
pp error
# => 5.551115123125783e-17

Parameters:

  • a (Numo::NArray) —

    The n-by-n symmetric / Hermitian matrix.

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

    The part of the matrix to be used (‘U’ or ‘L’).

  • hermitian (Boolean) (defaults to: true) —

    The flag indicating whether ‘a` is Hermitian.

Returns:

  • (Array<Numo::NArray>) —

    The permutated upper (lower) triangular matrix, the block diagonal matrix, and the permutation indices.

Raises:

  • (Numo::NArray::ShapeError) 


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# File 'lib/numo/linalg.rb', line 1878

def ldl(a, uplo: 'U', hermitian: true)
  raise Numo::NArray::ShapeError, 'input array a must be 2-dimensional' if a.ndim != 2
  raise Numo::NArray::ShapeError, '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'

  complex = bchr =~ /c|z/
  fnc = complex && hermitian ? :"#{bchr}hetrf" : :"#{bchr}sytrf"
  lud = a.dup
  ipiv, info = Numo::Linalg::Lapack.send(fnc, lud, uplo: uplo)

  raise LapackError, "the #{info.abs}-th argument of #{fnc} had illegal value" if info.negative?

  if info.positive?
    warn("the factorization has been completed, but the D[#{info - 1}, #{info - 1}] is " \
         'exactly zero, indicating that the block diagonal matrix is singular.')
  end

  _lud_permutation(lud, ipiv, uplo: uplo, hermitian: hermitian)
end

.logm(a) ⇒ Numo::NArray

Computes the matrix logarithm using its eigenvalue decomposition.

Parameters:

  • a (Numo::NArray) —

    The n-by-n square matrix.

Returns:

  • (Numo::NArray) —

    The matrix logarithm of ‘a`.

Raises:

  • (Numo::NArray::ShapeError) 


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# File 'lib/numo/linalg.rb', line 2044

def logm(a)
  raise Numo::NArray::ShapeError, 'input array a must be 2-dimensional' if a.ndim != 2
  raise Numo::NArray::ShapeError, 'input array a must be square' if a.shape[0] != a.shape[1]

  ev, vl, = eig(a, left: true, right: false)
  v = vl.transpose.conj
  inv_v = Numo::Linalg.inv(v)
  log_ev = Numo::NMath.log(ev)

  inv_v.dot(log_ev.diag).dot(v)
end

.lstsq(a, b, driver: 'lsd', rcond: nil) ⇒ Array<Numo::NArray, Float/Complex, Integer, Numo::NArray>

Computes the least-squares solution to a linear matrix equation.

Parameters:

  • a (Numo::NArray) —

    The m-by-n input matrix.

  • b (Numo::NArray) —

    The m-dimensional right-hand side vector or the m-by-nrhs right-hand side matrix.

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

    The LAPACK driver to be used (This argument is ignored, ‘lsd’ is always used).

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

    The threshold value for small singular values of ‘a`.

Returns:

  • (Array<Numo::NArray, Float/Complex, Integer, Numo::NArray>) —

    The least-squares solution matrix / vector ‘x`, the sum of squared residuals, the effective rank of `a`, and the singular values of `a`.

Raises:

  • (Numo::NArray::ShapeError) 


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# File 'lib/numo/linalg.rb', line 1960

def lstsq(a, b, driver: 'lsd', rcond: nil) # rubocop:disable Lint/UnusedMethodArgument, Metrics/AbcSize
  raise Numo::NArray::ShapeError, 'input array a must be 2-dimensional' if a.ndim != 2
  raise Numo::NArray::ShapeError, "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)
  raise ArgumentError, "invalid array type: #{a.class}" if bchr == 'n'

  m, n = a.shape
  if m < n
    if b.ndim == 1
      x = Numo::DFloat.zeros(n)
      x[0...b.size] = b
    else
      x = Numo::DFloat.zeros(n, b.shape[1])
      x[0...b.shape[0], 0...b.shape[1]] = b
    end
  else
    x = b.dup
  end

  fnc = :"#{bchr}gelsd"
  s, rank, info = Numo::Linalg::Lapack.send(fnc, a.dup, x, rcond: rcond)

  raise LapackError, "the #{info.abs}-th argument of #{fnc} had illegal value" if info.negative?
  raise LapackError, 'the algorithm for computing the SVD failed to converge' if info.positive?

  resids = x.class[]
  if m > n
    if rank == n
      resids = if b.ndim == 1
                 (x[n..].abs**2).sum(axis: 0)
               else
                 (x[n..-1, true].abs**2).sum(axis: 0)
               end
    end
    x = if b.ndim == 1
          x[false, 0...n]
        else
          x[false, 0...n, true]
        end
  end

  [x, resids, rank, s]
end

.lu(a, permute_l: false) ⇒ Array<Numo::NArray>

Computes the LU decomposition of a matrix using partial pivoting.

Examples:

require 'numo/linalg'

a = Numo::DFloat.new(3, 4).rand - 0.5
pm, l, u = Numo::Linalg.lu(a)
error = (pm.dot(l).dot(u) - a).abs.max
pp error
# => 5.551115123125783e-17

l, u = Numo::Linalg.lu(a, permute_l: true)
error = (l.dot(u) - a).abs.max
pp error
# => 5.551115123125783e-17

Parameters:

  • a (Numo::NArray) —

    The m-by-n matrix to be decomposed.

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

    If true, returns ‘L` with the permutation applied.

Returns:

  • (Array<Numo::NArray>) —

    if ‘permute_l` is `false`, the permutation matrix `P`, lower-triangular matrix `L`, and upper-triangular matrix `U` ([P, L, U]). if `permute_l` is `true`, the permuted lower-triangular matrix `L` and upper-triangular matrix `U` ([L, U]).

Raises:

  • (Numo::NArray::ShapeError) 


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# File 'lib/numo/linalg.rb', line 1367

def lu(a, permute_l: false)
  raise Numo::NArray::ShapeError, 'input array a must be 2-dimensional' if a.ndim != 2

  m, n = a.shape
  k = [m, n].min
  lu, piv = lu_fact(a)
  l = lu.tril.tap { |nary| nary[nary.diag_indices] = 1 }[true, 0...k].dup
  u = lu.triu[0...k, 0...n].dup
  perm = a.class.eye(m).tap do |nary|
    piv.to_a.each_with_index { |i, j| nary[true, [i - 1, j]] = nary[true, [j, i - 1]].dup }
  end

  permute_l ? [perm.dot(l), u] : [perm, l, u]
end

.lu_fact(a) ⇒ Array<Numo::NArray>

Computes the LU decomposition of a matrix using partial pivoting.

Parameters:

  • a (Numo::NArray) —

    The m-by-n matrix to be decomposed.

Returns:

  • (Array<Numo::NArray>) —

    The LU decomposition and pivot indices ([lu, piv]).

Raises:

  • (Numo::NArray::ShapeError) 


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# File 'lib/numo/linalg.rb', line 1386

def lu_fact(a)
  raise Numo::NArray::ShapeError, 'input array a must be 2-dimensional' if a.ndim != 2

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

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

  raise LapackError, "the #{info.abs}-th argument of getrf had illegal value" if info.negative?

  if info.positive?
    warn("the factorization has been completed, but the factor U[#{info - 1}, #{info - 1}] is " \
         'exactly zero, indicating that the matrix is singular.')
  end

  [lu, piv]
end

.lu_inv(lu, ipiv) ⇒ Numo::NArray

Computes the inverse of a matrix using its LU decomposition.

Parameters:

  • lu (Numo::NArray) —

    The LU decomposition of the n-by-n matrix ‘A`.

  • ipiv (Numo::Int32) —

    The pivot indices from ‘lu_fact`.

Returns:

  • (Numo::NArray) —

    The inverse of the matrix ‘A`.

Raises:

  • (ArgumentError) 


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# File 'lib/numo/linalg.rb', line 2179

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

  fnc = :"#{bchr}getri"
  inv, info = Numo::Linalg::Lapack.send(fnc, lu.dup, ipiv)

  raise LapackError, "the #{info.abs}-th argument of #{fnc} had illegal value" if info.negative?
  raise LapackError, 'the matrix is singular and its inverse could not be computed' if info.positive?

  inv
end

.lu_solve(lu, ipiv, b, trans: 'N') ⇒ Numo::NArray

Solves linear equation ‘A * x = b` or `A * X = B` for `x` using the LU decomposition of `A`.

Examples:

require 'numo/linalg'

a = Numo::DFloat.new(3, 3).rand
b = Numo::DFloat.eye(3)
lu, ipiv = Numo::Linalg.lu_fact(a)
x = Numo::Linalg.lu_solve(lu, ipiv, b)

puts (b - a.dot(x)).abs.max
=> 2.220446049250313e-16

Parameters:

  • lu (Numo::NArray) —

    The LU decomposition of the n-by-n matrix ‘A`.

  • ipiv (Numo::Int32/Int64) —

    The pivot indices from ‘lu_fact`.

  • b (Numo::NArray) —

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

  • trans (String) (defaults to: 'N') —

    The type of system to be solved.

    • ‘N’: solve ‘A * x = b` (No transpose),

    • ‘T’: solve ‘A^T * x = b` (Transpose),

    • ‘C’: solve ‘A^H * x = b` (Conjugate transpose).

Returns:

  • (Numo::NArray) —

    The solusion vector / matrix ‘X`.

Raises:

  • (Numo::NArray::ShapeError) 


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# File 'lib/numo/linalg.rb', line 1426

def lu_solve(lu, ipiv, b, trans: 'N')
  raise Numo::NArray::ShapeError, 'input array lu must be 2-dimensional' if lu.ndim != 2
  raise Numo::NArray::ShapeError, 'input array lu must be square' if lu.shape[0] != lu.shape[1]
  raise Numo::NArray::ShapeError, "incompatible dimensions: lu.shape[0] = #{lu.shape[0]} != b.shape[0] = #{b.shape[0]}" if lu.shape[0] != b.shape[0]
  raise ArgumentError, 'trans must be "N", "T", or "C"' unless %w[N T C].include?(trans)

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

  getrs = :"#{bchr}getrs"
  x, info = Numo::Linalg::Lapack.send(getrs, lu, ipiv, b.dup)

  raise LapackError, "the #{info.abs}-th argument of getrs had illegal value" if info.negative?

  x
end

.matmul(a, b) ⇒ Numo::NArray

Computes the matrix multiplication of two arrays.

Examples:

require 'numo/linalg'

a = Numo::DFloat[[1, 0], [0, 1]]
b = Numo::DFloat[[4, 1], [2, 2]]
pp Numo::Linalg.matmul(a, b)
# =>
# Numo::DFloat#shape=[2,2]
# [[4, 1],
#  [2, 2]]

Parameters:

  • a (Numo::NArray) —

    The first array.

  • b (Numo::NArray) —

    The second array.

Returns:

  • (Numo::NArray) —

    The matrix product of ‘a` and `b`.



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# File 'lib/numo/linalg.rb', line 1162

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

.matrix_balance(a, permute: true, scale: true, separate: false) ⇒ Array<Numo::NArray, Numo::NArray>

Computes a diagonal similarity transformation that balances a square matrix.

Examples:

require 'numo/linalg'

a = Numo::DFloat[[1, 0, 0], [1, 2, 0], [1, 2, 3]]
b, h = Numo::Linalg.matrix_balance(a)
pp b
# =>
# Numo::DFloat#shape=[3,3]
# [[3, 2, 1],
#  [0, 2, 1],
#  [0, 0, 1]]
pp h
# =>
# Numo::DFloat#shape=[3,3]
# [[0, 0, 1],
#  [0, 1, 0],
#  [1, 0, 0]]
pp (Numo::Linalg.inv(h).dot(a).dot(h) - b).abs.max
# => 0.0

Parameters:

  • a (Numo::NArray) —

    The n-by-n square matrix.

  • permute (Boolean) (defaults to: true) —

    The flag indicating whether to permute the matrix.

  • scale (Boolean) (defaults to: true) —

    The flag indicating whether to scale the matrix.

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

    The flag indicating whether to return scaling factors and permutation indices separately.

Returns:

  • (Array<Numo::NArray, Numo::NArray>) —

    if ‘separate` is `false`, the balanced matrix and the similarity transformation matrix `H` ([b, h]). if `separate` is `true`, the balanced matrix, the scaling factors, and the permutation indices ([b, scaler, perm]).

Raises:

  • (Numo::NArray::ShapeError) 


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# File 'lib/numo/linalg.rb', line 1539

def matrix_balance(a, permute: true, scale: true, separate: false) # rubocop:disable Metrics/AbcSize, Metrics/CyclomaticComplexity, Metrics/MethodLength, Metrics/PerceivedComplexity
  raise Numo::NArray::ShapeError, 'input array a must be 2-dimensional' if a.ndim != 2

  n = a.shape[0]
  raise ArgumentError, 'input array a must be square' if a.shape[1] != n

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

  job = if permute && scale
          'B'
        elsif permute && !scale
          'P'
        elsif !permute && scale
          'S'
        else
          'N'
        end
  fnc = :"#{bchr}gebal"
  b, lo, hi, prm_scl, info = Numo::Linalg::Lapack.send(fnc, a.dup, job: job)

  raise LapackError, "the #{info.abs}-th argument of #{fnc} had illegal value" if info.negative?

  # convert from Fortran style index to Ruby style index.
  lo -= 1
  hi -= 1
  iprm_scl = Numo::Int32.cast(prm_scl) - 1

  # extract scaling factors
  scaler = prm_scl.class.ones(n)
  scaler[lo...(hi + 1)] = prm_scl[lo...(hi + 1)]

  # extract permutation indices
  perm = Numo::Int32.new(n).seq
  if hi < n - 1
    iprm_scl[(hi + 1)...n].to_a.reverse.each.with_index(1) do |s, i|
      j = n - i
      next if s == j

      tmp_ls, tmp_lj = perm[[s, j]].to_a
      tmp_rj, tmp_rs = perm[[j, s]].to_a
      perm[[s, j]] = [tmp_rj, tmp_rs]
      perm[[j, s]] = [tmp_ls, tmp_lj]
    end
  end
  if lo > 0 # rubocop:disable Style/NumericPredicate
    iprm_scl[0...lo].to_a.each_with_index do |s, j|
      next if s == j

      tmp_ls, tmp_lj = perm[[s, j]].to_a
      tmp_rj, tmp_rs = perm[[j, s]].to_a
      perm[[s, j]] = [tmp_rj, tmp_rs]
      perm[[j, s]] = [tmp_ls, tmp_lj]
    end
  end

  return [b, scaler, perm] if separate

  # construct inverse permutation matrix
  inv_perm = Numo::Int32.zeros(n)
  inv_perm[perm] = Numo::Int32.new(n).seq
  h = scaler.diag[inv_perm, true].dup

  [b, h]
end

.matrix_power(a, n) ⇒ Numo::NArray

Computes the matrix ‘a` raised to the power of `n`.

Examples:

require 'numo/linalg'

a = Numo::DFloat[[1, 2], [3, 4]]
pp Numo::Linalg.matrix_power(a, 3)
# =>
# Numo::DFloat#shape=[2,2]
# [[37,  54],
#  [81, 118]]

Parameters:

  • a (Numo::NArray) —

    The square matrix.

  • n (Integer) —

    The exponent.

Returns:

  • (Numo::NArray) —

    The matrix ‘a` raised to the power of `n`.

Raises:

  • (Numo::NArray::ShapeError) 


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# File 'lib/numo/linalg.rb', line 1181

def matrix_power(a, n)
  raise Numo::NArray::ShapeError, 'input array a must be 2-dimensional' if a.ndim != 2
  raise Numo::NArray::ShapeError, 'input array a must be square' if a.shape[0] != a.shape[1]
  raise ArgumentError, "exponent n must be an integer: #{n}" unless n.is_a?(Integer)

  if n.zero?
    a.class.eye(a.shape[0])
  elsif n.positive?
    r = a.dup
    (n - 1).times { r = Numo::Linalg.matmul(r, a) }
    r
  else
    inv_a = inv(a)
    r = inv_a.dup
    (-n - 1).times { r = Numo::Linalg.matmul(r, inv_a) }
    r
  end
end

.matrix_rank(a, tol: nil, driver: 'svd') ⇒ Integer

Computes the rank of a matrix using SVD.

Parameters:

  • a (Numo::NArray) —

    The input matrix.

  • tol (Float) (defaults to: nil) —

    The threshold value for small singular values of ‘a`.

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

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

Returns:

  • (Integer) —

    The rank of the matrix.



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# File 'lib/numo/linalg.rb', line 1944

def matrix_rank(a, tol: nil, driver: 'svd')
  return a.ne(0).any? ? 1 : 0 if a.ndim < 2

  s = svdvals(a, driver: driver)
  tol ||= s.max(axis: -1, keepdims: true) * (a.shape[-2..].max * s.class::EPSILON)
  s.gt(tol).count(axis: -1)
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/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/linalg.rb', line 133

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::Linalg::Blas.send(:"#{blas_char(a)}nrm2", a) if ord.nil? || ord == 2
           when 2
             if ord.nil? || ord == 'fro'
               Numo::Linalg::Lapack.send(:"#{blas_char(a)}lange", a, norm: 'F')
             elsif ord.is_a?(Numeric)
               if ord == 1
                 Numo::Linalg::Lapack.send(:"#{blas_char(a)}lange", a, norm: '1')
               elsif !ord.infinite?.nil? && ord.infinite?.positive?
                 Numo::Linalg::Lapack.send(:"#{blas_char(a)}lange", a, norm: 'I')
               end
             end
           else
             if ord.nil?
               b = a.flatten.dup
               Numo::Linalg::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 [1, 2].include?(axis.size)
  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::Linalg::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::Linalg::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::Linalg::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

.null_space(a, rcond: nil) ⇒ Numo::NArray

Computes an orthonormal basis for the null space of ‘A` using SVD.

Examples:

require 'numo/linalg'

a = Numo::DFloat.new(3, 5).rand - 0.5
n = Numo::Linalg.null_space(a)
pp n
# =>
# Numo::DFloat#shape=[5,2]
# [[0.214096, -0.404277],
#  [-0.482225, -0.51557],
#  [-0.584394, -0.246804],
#  [0.596612, -0.351468],
#  [-0.155434, 0.621535]]
pp n.transpose.dot(n)
# =>
# Numo::DFloat#shape=[2,2]
# [[1, 1.31078e-16],
#  [1.31078e-16, 1]]

Parameters:

  • a (Numo::NArray) —

    The m-by-n input matrix.

  • 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 orthonormal basis for the null space of ‘a`.

Raises:

  • (Numo::NArray::ShapeError) 


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# File 'lib/numo/linalg.rb', line 1333

def null_space(a, rcond: nil)
  raise Numo::NArray::ShapeError, 'input array a must be 2-dimensional' if a.ndim != 2

  s, _u, vt = svd(a, driver: 'sdd', job: 'A')
  tol = if rcond.nil? || rcond.negative?
          a.shape.max * s.class::EPSILON
        else
          rcond
        end
  rank = s.gt(tol * s.max).count
  vt[rank...vt.shape[0], true].conj.transpose.dup
end

.orth(a, rcond: nil) ⇒ Numo::NArray

Computes an orthonormal basis for the range of ‘A` using SVD.

Examples:

require 'numo/linalg'

a = Numo::DFloat[[1, 2, 3], [2, 4, 6], [-1, 1, -1]]
u = Numo::Linalg.orth(a)
pp u
# =>
# Numo::DFloat#shape=[3,2]
# [[-0.446229, -0.0296535],
#  [-0.892459, -0.059307],
#  [0.0663073, -0.997799]]
pp u.transpose.dot(u)
# =>
# Numo::DFloat#shape=[2,2]
# [[1, -1.97749e-16],
#  [-1.97749e-16, 1]]

Parameters:

  • a (Numo::NArray) —

    The m-by-n input matrix.

  • 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 orthonormal basis for the range of ‘a`.

Raises:

  • (Numo::NArray::ShapeError) 


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# File 'lib/numo/linalg.rb', line 1296

def orth(a, rcond: nil)
  raise Numo::NArray::ShapeError, 'input array a must be 2-dimensional' if a.ndim != 2

  s, u, = svd(a, driver: 'sdd', job: 'S')
  tol = if rcond.nil? || rcond.negative?
          a.shape.max * s.class::EPSILON
        else
          rcond
        end
  rank = s.gt(tol * s.max).count
  u[true, 0...rank].dup
end

.orthogonal_procrustes(a, b) ⇒ Array<Numo::NArray, Float>

Computes the orthogonal Procrustes problem.

en.wikipedia.org/wiki/Orthogonal_Procrustes_problem

Examples:

require 'numo/linalg'

a = Numo::DFloat[[2, 0, 1], [-2, 0, 0]]
b = a.fliplr
r, scale = Numo::Linalg.orthogonal_procrustes(a, b)

pp b
# =>
# Numo::DFloat(view)#shape=[2,3]
# [[1, 0, 2],
#  [0, 0, -2]]
pp a.dot(r)
# =>
# Numo::DFloat#shape=[2,3]
# [[1, 0, 2],
#  [1.58669e-16, 0, -2]]
pp (b - a.dot(r)).abs.max
# =>
# 2.220446049250313e-16

Parameters:

  • a (Numo::NArray) —

    The first input matrix.

  • b (Numo::NArray) —

    The second input matrix.

Returns:

  • (Array<Numo::NArray, Float>) —

    The orthogonal matrix ‘R` and the scale factor `scale`.

Raises:

  • (Numo::NArray::ShapeError) 


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# File 'lib/numo/linalg.rb', line 1497

def orthogonal_procrustes(a, b)
  raise Numo::NArray::ShapeError, 'input array a must be 2-dimensional' if a.ndim != 2
  raise Numo::NArray::ShapeError, 'input array b must be 2-dimensional' if b.ndim != 2
  raise Numo::NArray::ShapeError, "incompatible dimensions: a.shape = #{a.shape} != b.shape = #{b.shape}" if a.shape != b.shape

  m = b.transpose.dot(a.conj).transpose
  s, u, vt = svd(m, driver: 'svd', job: 'S')
  r = u.dot(vt)
  scale = s.sum
  [r, scale]
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/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/linalg.rb', line 504

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.dup
end

.polar(a, side: 'right') ⇒ Array<Numo::NArray>

Computes the polar decomposition of a matrix.

en.wikipedia.org/wiki/Polar_decomposition

Examples:

require 'numo/linalg'

a = Numo::DFloat[[0.5, 1, 2], [1.5, 3, 4]]
u, p = Numo::Linalg.polar(a)
pp u.dot(p)
# =>
# Numo::DFloat#shape=[2,3]
# [[0.5, 1, 2],
#  [1.5, 3, 4]]
pp u.dot(u.transpose)
# =>
# Numo::DFloat#shape=[2,2]
# [[1, -1.68043e-16],
#  [-1.68043e-16, 1]]

u, p = Numo::Linalg.polar(a, side: 'left')
pp p.dot(u)
# =>
# Numo::DFloat#shape=[2,3]
# [[0.5, 1, 2],
#  [1.5, 3, 4]]

Parameters:

  • a (Numo::NArray) —

    The m-by-n matrix to be decomposed.

  • side (String) (defaults to: 'right') —

    The side of polar decomposition (‘right’ or ‘left’).

Returns:

  • (Array<Numo::NArray>) —

    The unitary matrix ‘U` and the positive-semidefinite Hermitian matrix `P` such that `A = U * P` if side=’right’, or ‘A = P * U` if side=’left’.

Raises:

  • (Numo::NArray::ShapeError) 


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# File 'lib/numo/linalg.rb', line 544

def polar(a, side: 'right')
  raise Numo::NArray::ShapeError, 'input array a must be 2-dimensional' if a.ndim != 2
  raise ArugumentError, "invalid side: #{side}" unless %w[left right].include?(side)

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

  s, w, vh = svd(a, driver: 'svd', job: 'S')
  u = w.dot(vh)
  p_mat = if side == 'right'
            vh.transpose.conj.dot(s.diag).dot(vh)
          else
            w.dot(s.diag).dot(w.transpose.conj)
          end
  [u, p_mat]
end

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

Computes the QR decomposition of a matrix.

Examples:

require '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:

  • (Numo::NArray::ShapeError) 


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# File 'lib/numo/linalg.rb', line 597

def qr(a, mode: 'reduce')
  raise Numo::NArray::ShapeError, '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::Linalg::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::Linalg::Lapack.send(org_ung_qr, qr[true, 0...m], tau)[0]
      elsif mode == 'economic'
        Numo::Linalg::Lapack.send(org_ung_qr, qr, tau)[0]
      else
        qqr = a.class.zeros(m, m)
        qqr[0...m, 0...n] = qr
        Numo::Linalg::Lapack.send(org_ung_qr, qqr, tau)[0]
      end

  [q, r]
end

.qz(a, b) ⇒ Array<Numo::NArray, Numo::NArray, Numo::NArray, Numo::NArray>

Computes the QZ decomposition (generalized Schur decomposition) of a pair of square matrices.

The QZ decomposition is given by ‘A = Q * AA * Z^H` and `B = Q * BB * Z^H`, where `A` and `B` are the input matrices, `Q` and `Z` are unitary matrices, and `AA` and `BB` are upper triangular matrices (or quasi-upper triangular matrices in real case).

Examples:

require 'numo/linalg'

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

aa, bb, q, z = Numo::Linalg.qz(a, b)

pp (a - q.dot(aa).dot(z.transpose)).abs.max
# => 1.7763568394002505e-15
pp (b - q.dot(bb).dot(z.transpose)).abs.max
# => 1.1102230246251565e-15

Parameters:

  • a (Numo::NArray) —

    The n-by-n square matrix.

  • b (Numo::NArray) —

    The n-by-n square matrix.

Returns:

  • (Array<Numo::NArray, Numo::NArray, Numo::NArray, Numo::NArray>) —

    The matrices ‘AA`, `BB`, `Q`, and `Z` such that `A = Q * AA * Z^H` and `B = Q * BB * Z^H`.

Raises:

  • (Numo::NArray::ShapeError) 


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# File 'lib/numo/linalg.rb', line 727

def qz(a, b) # rubocop:disable Metrics/AbcSize
  raise Numo::NArray::ShapeError, 'input array a must be 2-dimensional' if a.ndim != 2
  raise Numo::NArray::ShapeError, 'input array b must be 2-dimensional' if b.ndim != 2
  raise Numo::NArray::ShapeError, 'input array a must be square' if a.shape[0] != a.shape[1]
  raise Numo::NArray::ShapeError, 'input array b must be square' if b.shape[0] != b.shape[1]
  raise Numo::NArray::ShapeError, "incompatible dimensions: a.shape = #{a.shape} != b.shape = #{b.shape}" if a.shape != b.shape

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

  fnc = :"#{bchr}gges"
  if %w[d s].include?(bchr)
    aa, bb, _ar, _ai, _beta, q, z, _sdim, info = Numo::Linalg::Lapack.send(fnc, a.dup, b.dup)
  else
    aa, bb, _alpha, _beta, q, z, _sdim, info = Numo::Linalg::Lapack.send(fnc, a.dup, b.dup)
  end

  n = a.shape[0]
  raise LapackError, "the #{-info}-th argument of #{fnc} had illegal value" if info.negative?
  raise LapackError, 'something other than QZ iteration failed.' if info == n + 1
  raise LapackError, "reordering failed in #{bchr}tgsen" if info == n + 3

  if info == n + 2
    raise LapackError, 'after reordering, roundoff changed values of some eigenvalues ' \
                       'so that leading eigenvalues in the Generalized Schur form no ' \
                       'longer satisfy the sorting condition.'
  end

  if info.positive? && info <= n
    warn('the QZ iteration failed. (a, b) are not in Schur form, ' \
         "but alpha[i] and beta[i] for i = #{info},...,n are correct.")
  end

  [aa, bb, q, z]
end

.rq(a, mode: 'full') ⇒ Array<Numo::NArray>/Numo::NArray

Computes the RQ decomposition of a matrix.

Examples:

require 'numo/linalg'

a = Numo::DFloat.new(2, 3).rand
r, q = Numo::Linalg.rq(a)
pp r
# =>
# Numo::DFloat#shape=[2,3]
# [[0, -0.381748, -0.79309],
#  [0, 0, -0.41502]]
pp q
# =>
# Numo::DFloat#shape=[3,3]
# [[0.227957, 0.874475, -0.428169],
#  [0.844617, -0.396377, -0.359872],
#  [-0.484416, -0.279603, -0.828953]]
puts (a - r.dot(q)).abs.max
# => 5.551115123125783e-17

r, q = Numo::Linalg.rq(a, mode: 'economic')
pp r
# =>
# Numo::DFloat#shape=[2,2]
# [[-0.381748, -0.79309],
#  [0, -0.41502]]
pp q
# =>
# Numo::DFloat#shape=[2,3]
# [[0.844617, -0.396377, -0.359872],
#  [-0.484416, -0.279603, -0.828953]]
puts (a - r.dot(q)).abs.max
# => 5.551115123125783e-17

Parameters:

  • a (Numo::NArray) —

    The m-by-n matrix to be decomposed.

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

    The mode of decomposition.

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

    • “r” – returns only R,

    • “economic” – returns both R [m, k] and Q [k, n], where k = min(m, n).

Returns:

  • (Array<Numo::NArray>/Numo::NArray) —

    if mode=‘full’ or ‘economic’, returns [R, Q]. if mode=‘r’, returns R.

Raises:

  • (Numo::NArray::ShapeError) 


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# File 'lib/numo/linalg.rb', line 672

def rq(a, mode: 'full') # rubocop:disable  Metrics/AbcSize
  raise Numo::NArray::ShapeError, 'input array a must be 2-dimensional' if a.ndim != 2
  raise ArgumentError, "invalid mode: #{mode}" unless %w[full r economic].include?(mode)

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

  fnc = :"#{bchr}gerqf"
  rq, tau, info = Numo::Linalg::Lapack.send(fnc, a.dup)
  raise LapackError, "the #{-info}-th argument of #{fnc} had illegal value" if info.negative?

  m, n = rq.shape
  r = rq.triu(n - m).dup
  r = r[true, (n - m)...n].dup if mode == 'economic' && n > m

  return r if mode == 'r'

  fnc = %w[d s].include?(bchr) ? :"#{bchr}orgrq" : :"#{bchr}ungrq"
  tmp = if n < m
          rq[(m - n)...m, 0...n].dup
        elsif mode == 'economic'
          rq.dup
        else
          rq.class.zeros(n, n).tap { |mat| mat[(n - m)...n, true] = rq }
        end

  q, info = Numo::Linalg::Lapack.send(fnc, tmp, tau)
  raise LapackError, "the #{-info}-th argument of #{fnc} had illegal value" if info.negative?

  [r, q]
end

.schur(a, sort: nil) ⇒ Array<Numo::NArray, Numo::NArray, Integer>

Computes the Schur decomposition of a square matrix. The Schur decomposition is given by ‘A = Z * T * Z^H`, where `A` is the input matrix, `Z` is a unitary matrix, and `T` is an upper triangular matrix (or quasi-upper triangular matrix in real case).

Examples:

require 'numo/linalg'

a = Numo::DFloat[[0, 2, 3], [4, 5, 6], [7, 8, 9]]
t, z, sdim = Numo::Linalg.schur(a)
pp t
# =>
# Numo::DFloat#shape=[3,3]
# [[16.0104, 4.81155, 0.920982],
#  [0, -1.91242, 0.0274406],
#  [0, 0, -0.0979794]]
pp z
# =>
# Numo::DFloat#shape=[3,3]
# [[-0.219668, -0.94667, 0.235716],
#  [-0.527141, -0.0881306, -0.845195],
#  [-0.820895, 0.309918, 0.479669]]
pp sdim
# => 0
pp (a - z.dot(t).dot(z.transpose)).abs.max
# => 1.0658141036401503e-14

Parameters:

  • a (Numo::NArray) —

    The n-by-n square matrix.

  • sort (String/Nil) (defaults to: nil) —

    The option for sorting eigenvalues (‘lhp’, ‘rhp’, ‘iuc’, ‘ouc’, or nil).

    • ‘lhp’: eigenvalue.real < 0

    • ‘rhp’: eigenvalue.real >= 0

    • ‘iuc’: eigenvalue.abs <= 1

    • ‘ouc’: eigenvalue.abs > 1

Returns:

  • (Array<Numo::NArray, Numo::NArray, Integer>) —

    The Schur form ‘T`, the unitary matrix `Z`, and the number of eigenvalues for which the sorting condition is true.

Raises:

  • (Numo::NArray::ShapeError) 


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

def schur(a, sort: nil)
  raise Numo::NArray::ShapeError, 'input array a must be 2-dimensional' if a.ndim != 2
  raise Numo::NArray::ShapeError, 'input array a must be square' if a.shape[0] != a.shape[1]
  raise ArgumentError, "invalid sort: #{sort}" unless sort.nil? || %w[lhp rhp iuc ouc].include?(sort)

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

  fnc = :"#{bchr}gees"
  if %w[d s].include?(bchr)
    b, _wr, _wi, v, sdim, info = Numo::Linalg::Lapack.send(fnc, a.dup, jobvs: 'V', sort: sort)
  else
    b, _w, v, sdim, info = Numo::Linalg::Lapack.send(fnc, a.dup, jobvs: 'V', sort: sort)
  end

  n = a.shape[0]
  raise LapackError, "the #{-info}-th argument of #{fnc} had illegal value" if info.negative?
  raise LapackError, 'the QR algorithm failed to compute all the eigenvalues.' if info.positive? && info <= n
  raise LapackError, 'the eigenvalues could not be reordered.' if info == n + 1

  if info == n + 2
    raise LapackError, 'after reordering, roundoff changed values of some eigenvalues ' \
                       'so that leading eigenvalues in the Schur form no longer satisfy ' \
                       'the sorting condition.'
  end

  [b, v, sdim]
end

.signm(a) ⇒ Numo::NArray

Computes the matrix sign function using its inverse and square root matrices.

Parameters:

  • a (Numo::NArray) —

    The n-by-n square matrix.

Returns:

  • (Numo::NArray) —

    The matrix sign function of ‘a`.

Raises:

  • (Numo::NArray::ShapeError) 


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# File 'lib/numo/linalg.rb', line 2165

def signm(a)
  raise Numo::NArray::ShapeError, 'input array a must be 2-dimensional' if a.ndim != 2
  raise Numo::NArray::ShapeError, 'input array a must be square' if a.shape[0] != a.shape[1]

  a_sqrt = sqrtm(a.dot(a))
  a_inv = Numo::Linalg.inv(a)
  a_inv.dot(a_sqrt)
end

.sinhm(a) ⇒ Numo::NArray

Computes the matrix hyperbolic sine using the matrix exponential.

Parameters:

  • a (Numo::NArray) —

    The n-by-n square matrix.

Returns:

  • (Numo::NArray) —

    The matrix hyperbolic sine of ‘a`.

Raises:

  • (Numo::NArray::ShapeError) 


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# File 'lib/numo/linalg.rb', line 2114

def sinhm(a)
  raise Numo::NArray::ShapeError, 'input array a must be 2-dimensional' if a.ndim != 2
  raise Numo::NArray::ShapeError, 'input array a must be square' if a.shape[0] != a.shape[1]

  0.5 * (expm(a) - expm(-a))
end

.sinm(a) ⇒ Numo::NArray

Computes the matrix sine using the matrix exponential.

Parameters:

  • a (Numo::NArray) —

    The n-by-n square matrix.

Returns:

  • (Numo::NArray) —

    The matrix sine of ‘a`.

Raises:

  • (Numo::NArray::ShapeError) 


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# File 'lib/numo/linalg.rb', line 2060

def sinm(a)
  raise Numo::NArray::ShapeError, 'input array a must be 2-dimensional' if a.ndim != 2
  raise Numo::NArray::ShapeError, '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'

  b = a * 1.0i
  if %w[z c].include?(bchr)
    -0.5i * (expm(b) - expm(-b))
  else
    expm(b).imag
  end
end

.slogdet(a) ⇒ Array<Float/Complex>

Computes the sign and natural logarithm of the determinant of a matrix.

Parameters:

  • a (Numo::NArray) —

    The n-by-n square matrix.

Returns:

  • (Array<Float/Complex>) —

    The sign and natural logarithm of the determinant of ‘a` ([sign, logdet]).



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# File 'lib/numo/linalg.rb', line 1924

def slogdet(a)
  lu, ipiv = lu_fact(a)
  dg = lu.diagonal
  return 0, (-Float::INFINITY) if dg.eq(0).any?

  idx = ipiv.class.new(ipiv.shape[-1]).seq(1)
  n_nonzero = ipiv.ne(idx).count(axis: -1)
  sign = ((-1.0)**(n_nonzero % 2)) * (dg / dg.abs).prod

  logdet = Numo::NMath.log(dg.abs).sum(axis: -1)

  [sign, logdet]
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/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:

  • (Numo::NArray::ShapeError) 


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# File 'lib/numo/linalg.rb', line 909

def solve(a, b, driver: 'gen', uplo: 'U') # rubocop:disable Lint/UnusedMethodArgument
  raise Numo::NArray::ShapeError, 'input array a must be 2-dimensional' if a.ndim != 2
  raise Numo::NArray::ShapeError, '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"
  _lu, x, _ipiv, info = Numo::Linalg::Lapack.send(gesv, a.dup, b.dup)
  raise LapackError, "the #{-info}-th argument of getrf had illegal value" if info.negative?

  if info.positive?
    warn('the factorization has been completed, but the factor is singular, ' \
         'so the solution could not be computed.')
  end

  x
end

.solve_banded(l_u, ab, b) ⇒ Numo::NArray

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

Examples:

require 'numo/linalg'

# Banded matrix A:
# [4 2 1 0 0]
# [1 5 2 1 0]
# [0 1 6 2 1]
# [0 0 1 7 2]
# [0 0 0 1 8]
#
ab = Numo::DFloat[[0, 0, 1, 1, 1],
                  [0, 2, 2, 2, 2],
                  [4, 5, 6, 7, 8],
                  [1, 1, 1, 1, 0]]

b = Numo::DFloat[1, 2, 3, 4, 5]

x = Numo::Linalg.solve_banded([1, 2], ab, b)
pp x
# =>
# Numo::DFloat#shape=[5]
# [0.0832055, 0.211273, 0.244632, 0.371166, 0.578604]

a = ab[2, true].diag + ab[0, 2...].diag(2) + ab[1, 1...].diag(1) + ab[3, 0...-1].diag(-1)
pp a
# =>
# Numo::DFloat#shape=[5,5]
# [[4, 2, 1, 0, 0],
#  [1, 5, 2, 1, 0],
#  [0, 1, 6, 2, 1],
#  [0, 0, 1, 7, 2],
#  [0, 0, 0, 1, 8]]
pp a.dot(x)
# =>
# Numo::DFloat#shape=[5]
# [1, 2, 3, 4, 5]

Parameters:

  • l_u (Array<Integer>) —

    The number of sub-diagonals and super-diagonals of the banded matrix ‘A`.

  • ab (Numo::NArray) —

    The (l + u + 1)-by-n array representing the banded matrix ‘A`.

  • b (Numo::NArray) —

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

Returns:

  • (Numo::NArray) —

    The solution vector / matrix ‘X`.

Raises:

  • (Numo::NArray::ShapeError) 


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# File 'lib/numo/linalg.rb', line 1010

def solve_banded(l_u, ab, b)
  raise Numo::NArray::ShapeError, 'input array a must be 2-dimensional' if ab.ndim != 2

  unless l_u.is_a?(Array) && l_u.size == 2 && l_u.all?(Integer)
    raise ArgumentError, 'l_u must be an array of two integers'
  end

  kl, ku = l_u
  raise Numo::NArray::ShapeError, "ab.shape[0] must be equal to l + u + 1: #{ab.shape[0]} != #{kl} + #{ku} + 1" if ab.shape[0] != kl + ku + 1

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

  gbsv = :"#{bchr}gbsv"

  tmp = ab.class.zeros((2 * kl) + ku + 1, ab.shape[1])
  tmp[kl..., true] = ab
  _, x, _, info = Numo::Linalg::Lapack.send(gbsv, tmp, b.dup, kl: kl, ku: ku)
  raise LapackError, "wrong value is given to the #{-info}-th argument of #{gbsv} used internally" if info.negative?
  raise LapackError, 'the leading minor of the matrix is not positive definite' if info.positive?

  x
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/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:

  • (Numo::NArray::ShapeError) 


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# File 'lib/numo/linalg.rb', line 952

def solve_triangular(a, b, lower: false)
  raise Numo::NArray::ShapeError, 'input array a must be 2-dimensional' if a.ndim != 2
  raise Numo::NArray::ShapeError, '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::Linalg::Lapack.send(trtrs, a, b.dup, uplo: uplo)
  raise LapackError, "wrong value is given to the #{info}-th argument of #{trtrs} used internally" if info.negative?

  x
end

.solveh_banded(ab, b, lower: false) ⇒ Numo::NArray

Solves linear equation ‘A * x = b` or `A * X = B` for `x` assuming `A` is a symmetric/Hermitian positive-definite banded matrix.

Examples:

require 'numo/linalg'

# Banded matrix A:
# [4 2 1 0 0]
# [2 5 2 1 0]
# [1 2 6 2 1]
# [0 1 2 7 2]
# [0 0 1 2 8]
#
# The banded representation ab for lower-banded form is:
ab = Numo::DFloat[[4, 5, 6, 7, 8],
                  [2, 2, 2, 2, 0],
                  [1, 1, 1, 0, 0]]
# The banded representation ab for upper-banded form is:
# ab = Numo::DFloat[[0, 0, 1, 1, 1],
#                   [0, 2, 2, 2, 2],
#                   [4, 5, 6, 7, 8]]
b = Numo::DFloat[1, 2, 3, 4, 5]

x = Numo::Linalg.solveh_banded(ab, b, lower: true)
pp x
# =>
# Numo::DFloat#shape=[5]
# [0.0903361, 0.210084, 0.218487, 0.331933, 0.514706]

a = ab[0, true].diag + ab[1, 0...-1].diag(1) + ab[2, 0...-2].diag(2) +
 ab[1, 0...-1].diag(-1) + ab[2, 0...-2].diag(-2)
pp a.dot(x)
# => Numo::DFloat#shape=[5]
# [1, 2, 3, 4, 5]

Parameters:

  • ab (Numo::NArray) —

    The m-by-n array representing the banded matrix ‘A`.

  • b (Numo::NArray) —

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

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

    The flag indicating whether to be in the lower-banded form.

Returns:

  • (Numo::NArray) —

    The solusion vector / matrix ‘X`.

Raises:

  • (Numo::NArray::ShapeError) 


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# File 'lib/numo/linalg.rb', line 1073

def solveh_banded(ab, b, lower: false)
  raise Numo::NArray::ShapeError, 'input array a must be 2-dimensional' if ab.ndim != 2

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

  pbsv = :"#{bchr}pbsv"
  uplo = lower ? 'L' : 'U'
  x, info = Numo::Linalg::Lapack.send(pbsv, ab.dup, b.dup, uplo: uplo)
  raise LapackError, "wrong value is given to the #{info}-th argument of #{pbsv} used internally" if info.negative?
  raise LapackError, 'the leading minor of the matrix is not positive definite' if info.positive?

  x
end

.sqrtm(a) ⇒ Numo::NArray

Computes the square root of a matrix using its eigenvalue decomposition.

Parameters:

  • a (Numo::NArray) —

    The n-by-n square matrix.

Returns:

  • (Numo::NArray) —

    The matrix square root of ‘a`.

Raises:

  • (Numo::NArray::ShapeError) 


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# File 'lib/numo/linalg.rb', line 2149

def sqrtm(a)
  raise Numo::NArray::ShapeError, 'input array a must be 2-dimensional' if a.ndim != 2
  raise Numo::NArray::ShapeError, 'input array a must be square' if a.shape[0] != a.shape[1]

  ev, vl, = eig(a, left: true, right: false)
  v = vl.transpose.conj
  inv_v = Numo::Linalg.inv(v)
  sqrt_ev = Numo::NMath.sqrt(ev)

  inv_v.dot(sqrt_ev.diag).dot(v)
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/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/linalg.rb', line 1122

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

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

  [s, u, vt]
end

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

Computes the singular values of a matrix.

Examples:

require 'numo/linalg'

a = Numo::DFloat[[1, 2, 3], [2, 4, 6], [-1, 1, -1]]
pp Numo::Linalg.svdvals(a)
# => Numo::DFloat#shape=[3]
# [8.38434, 1.64402, 5.41675e-17]

Parameters:

  • a (Numo::NArray) —

    Matrix to be decomposed.

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

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

Returns:

  • (Numo::NArray) —

    The singular values of ‘a`.

Raises:

  • (ArgumentError) 


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# File 'lib/numo/linalg.rb', line 1213

def svdvals(a, driver: 'sdd')
  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::Linalg::Lapack.send(gesdd, a.dup, jobz: 'N')
  when 'svd'
    gesvd = :"#{bchr}gesvd"
    s, _u, _vt, info = Numo::Linalg::Lapack.send(gesvd, a.dup, jobu: 'N', jobvt: 'N')
  else
    raise ArgumentError, "invalid driver: #{driver}"
  end

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

  s
end

.tanhm(a) ⇒ Numo::NArray

Computes the matrix hyperbolic tangent.

Parameters:

  • a (Numo::NArray) —

    The n-by-n square matrix.

Returns:

  • (Numo::NArray) —

    The matrix hyperbolic tangent of ‘a`.

Raises:

  • (Numo::NArray::ShapeError) 


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# File 'lib/numo/linalg.rb', line 2136

def tanhm(a)
  raise Numo::NArray::ShapeError, 'input array a must be 2-dimensional' if a.ndim != 2
  raise Numo::NArray::ShapeError, 'input array a must be square' if a.shape[0] != a.shape[1]

  a_sinh = sinhm(a)
  a_cosh = coshm(a)
  a_sinh.dot(Numo::Linalg.inv(a_cosh))
end

.tanm(a) ⇒ Numo::NArray

Computes the matrix tangent.

Parameters:

  • a (Numo::NArray) —

    The n-by-n square matrix.

Returns:

  • (Numo::NArray) —

    The matrix tangent of ‘a`.

Raises:

  • (Numo::NArray::ShapeError) 


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# File 'lib/numo/linalg.rb', line 2098

def tanm(a)
  raise Numo::NArray::ShapeError, 'input array a must be 2-dimensional' if a.ndim != 2
  raise Numo::NArray::ShapeError, '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'

  a_sin = sinm(a)
  a_cos = cosm(a)
  a_sin.dot(Numo::Linalg.inv(a_cos))
end