Linear Algebra

The Linear Algebra module provides a high-level interface for linear algebra and numerical mathematics.

Factories

Name	Description
array	Converts a container to an array.
cholesky	Computes the Cholesky decomposition of a RealMatrix A.. By default, the standard algorithm is used; to use the robust algorithm instead, provide the option 'algorithm' with value 'robust'.
eigensolver	Computes the eigenvalues and eigenvectors of a matrix.
filled	Creates a container with the specified shape and type, filled with the specified value.
filled_like	Return a container with the same shape and type as a given container, filled with a prescribed value.
identity	Creates an identity matrix (or array; possibly rectangular) with the specified shape and type.
identity_like	Return the identity with the same shape and type as a given container.
lu	Computes the LU decomposition of a ComplexMatrix A.. By default, partial pivoting is used and the resulting factorization is PA = LU, where P is a permutation matrix, L is lower triangular with unit diagonal elements, and U is upper triangular. To use full pivoting instead, provide the option 'pivoting' with value 'full'. The resulting factorization is PAQ = LU, where P and Q are permutation matrices.
matrix	Converts a container to a matrix with the same elements.
ones	Creates a container with the specified shape and type, filled with ones.
ones_like	Return a container of ones with the same shape and type as a given container.
qr	Computes the QR decomposition of a RealMatrix A.. By default, no pivoting is used and the resulting factorization is A = QR, where Q is a unitary matrix and R is upper triangular. To use partial (column) pivoting, provide the option 'pivoting' with value 'partial'. The resulting factorization is AP = QR, where P is a permutation matrix. To use full pivoting, provide the option 'pivoting' with value 'full'.
random	Creates a container with the specified shape and type, filled with random values.
svd	Computes the singular value decomposition (SVD) of an n-by-p RealMatrix A as a product A=USV^T where U is a n-by-m unitary, V is a p-by-m unitary, $m = min(n,p)$, and S is an m-by-m real positive diagonal matrix; the diagonal entries of S are known as the singular values of A and the columns of U and V are known as the left and right singular vectors of A respectively. Singular values are always sorted in decreasing order.. By default, the Jacobi algorithm is used; to use the block divide and conquer (BDC) algorithm instead, provide the option 'algorithm' with value 'bdc'.
uninitialized	Creates a container with the specified shape and type.
uninitialized_like	Return a container of uninitialized values with the same shape and type as a given container.
vector	Converts a list to a column-vector.
zeros	Creates a container with the specified shape and type, filled with zeros.
zeros_like	Return a container of zeros with the same shape and type as a given container.

Types

Name	Description
BoolArray	A BoolArray.
ComplexArray	A ComplexArray.
ComplexMatrix	A ComplexMatrix.
ComplexSparseMatrix	A ComplexSparseMatrix.
IntegerArray	A IntegerArray.
RealArray	A RealArray.
RealMatrix	A RealMatrix.
RealSparseMatrix	A RealSparseMatrix.

Module Functions

Name	Description
ComplexBDCSVD	Constructs an SVD decomposition from a matrix.
ComplexColPivHouseholderQR	Constructs a ComplexColPivHouseholderQR from a matrix.
ComplexEigenSolver	Constructs a ComplexEigenSolver from a ComplexMatrix.
ComplexFullPivHouseholderQR	Constructs a ComplexFullPivHouseholderQR from a matrix.
ComplexFullPivLU	Constructs a LU decomposition from a matrix.
ComplexGeneralizedEigenSolver	Constructs a ComplexGeneralizedEigenSolver from a ComplexMatrix.
ComplexGeneralizedSelfAdjointEigenSolver	Constructs a ComplexGeneralizedSelfAdjointEigenSolver from a pair of ComplexMatrix.
ComplexHouseholderQR	Constructs a ComplexHouseholderQR from a matrix.
ComplexJacobiSVD	Constructs an SVD decomposition from a matrix.
ComplexLDLT	Constructs a ComplexLDLT from a matrix.
ComplexLLT	Constructs a ComplexLLT from a matrix.
ComplexPartialPivLU	Constructs a LU decomposition from a matrix.
ComplexSelfAdjointEigenSolver	Constructs a ComplexSelfAdjointEigenSolver from a self-adjoint ComplexMatrix.
RealBDCSVD	Constructs an SVD decomposition from a matrix.
RealColPivHouseholderQR	Constructs a RealColPivHouseholderQR from a matrix.
RealEigenSolver	Constructs a RealEigenSolver from a RealMatrix.
RealFullPivHouseholderQR	Constructs a RealFullPivHouseholderQR from a matrix.
RealFullPivLU	Constructs a LU decomposition from a matrix.
RealGeneralizedEigenSolver	Constructs a RealGeneralizedEigenSolver from a RealMatrix.
RealGeneralizedSelfAdjointEigenSolver	Constructs a RealGeneralizedSelfAdjointEigenSolver from a pair of RealMatrix.
RealHouseholderQR	Constructs a RealHouseholderQR from a matrix.
RealJacobiSVD	Constructs an SVD decomposition from a matrix.
RealLDLT	Constructs a RealLDLT from a matrix.
RealLLT	Constructs a RealLLT from a matrix.
RealPartialPivLU	Constructs a LU decomposition from a matrix.
RealSelfAdjointEigenSolver	Constructs a RealSelfAdjointEigenSolver from a self-adjoint RealMatrix.
abs	Calculates the absolute values of the ComplexArray arr element-wise.
abs2	Calculates the absolute values squared of the IntegerArray arr element-wise.
acos	Computes the inverse cosine (arccosine) of the ComplexArray arr element-wise.
acosh	Computes the inverse hyperbolic cosine of the RealArray arr element-wise.
asin	Computes the inverse sine (arcsine) of the ComplexArray arr element-wise.
asinh	Computes the inverse hyperbolic sine of the ComplexArray arr element-wise.
atan	Computes the inverse tangent (arctangent) of the RealArray arr element-wise.
atanh	Computes the inverse hyperbolic tangent of the ComplexArray arr element-wise.
betainc	Computes the regularized incomplete beta function of the RealArray arr element-wise.
ceil	Computes the smallest integer greater than or equal to each element of the RealArray arr element-wise.
concatenate	Creates a IntegerArray by concatenating several IntegerArray. The number of rows and columns must be consistent across all inputs IntegerArray.
concatenate_columns	Creates a ComplexArray by concatenating several ComplexArray along the column dimension (i.e. horizontally). The number of rows must be the same for all inputs ComplexArray.
concatenate_rows	Creates a BoolArray by concatenating several BoolArray along the row dimension (i.e. vertically). The number of columns must be the same for all inputs BoolArray.
conjugate	Conjugates the input RealMatrix in-place.
conjugated	Returns the complex conjugate of the ComplexArray.
cos	Computes the cosine of the ComplexArray arr element-wise.
cosh	Computes the hyperbolic cosine of the RealArray arr element-wise.
cube	Computes the cube (third power) of the RealArray arr element-wise.
cumprod	Returns the cumulative product of the elements in the RealMatrix.
cumsum	Returns the cumulative sum of the elements in the RealArray.
digamma	Computes the digamma function (derivative of the logarithm of the gamma function) of the RealArray arr element-wise.
eigenvalues	Returns the eigenvalues of the matrix.
eigenvectors	Returns the eigenvectors of the matrix.
erf	Computes the Gauss error function of the RealArray arr element-wise.
erfc	Computes the complementary error function of the RealArray arr element-wise.
exp	Computes the exponential function of the RealArray arr element-wise.
expm1	Computes exp(x) - 1 of the RealArray arr element-wise.
fix	Rounds the RealArray arr towards zero element-wise.
floor	Computes the largest integer less than or equal to each element of the RealArray arr element-wise.
igamma	Calculates the igamma of the RealArray arr0 and arr1 element-wise.
igammac	Calculates the igammac of the RealArray arr0 and arr1 element-wise.
imag	Returns the imaginary component of a RealMatrix.
index_max	Returns the maximum value of the elements in the RealArray.
index_min	Returns the minimum value of the elements in the RealArray.
inverse	Computes the multiplicative inverse (reciprocal) of the ComplexArray arr element-wise.
inverse_normal_cdf	Computes the inverse of the cumulative distribution function of the standard normal distribution (probit function) of the RealArray arr element-wise.
inverse_sqrt	Computes the reciprocal square root of the ComplexArray arr element-wise.
isfinite	Checks if each element of the RealArray arr is a finite number.
isinf	Checks if each element of the RealArray arr is infinite.
isnan	Checks if each element of the RealArray arr is NaN (not a number).
kron	Computes the Kronecker product between two arrays or matrices. It is defined as...
lgamma	Computes the natural logarithm of the absolute value of the gamma function of the RealArray arr element-wise.
linspaced	Sets all elements of the RealMatrix lhs to linearly spaced values.
log	Computes the natural logarithm (base e) of the RealArray arr element-wise.
log10	Computes the base-10 logarithm of the RealArray arr element-wise.
log1p	Computes the natural logarithm of (1 + x) of the ComplexArray arr element-wise.
log2	Computes the base-2 logarithm of the RealArray arr element-wise.
logical_and	Returns the logical AND of the BoolArray lhs and rhs.
logical_not	Returns the logical NOT of the BoolArray lhs.
logical_or	Returns the logical OR of the BoolArray lhs and rhs.
logical_xor	Returns the logical XOR of the BoolArray lhs and rhs.
logistic	Computes the logistic sigmoid function of the ComplexArray arr element-wise.
map	Applies a function to each element of the array.
matrix_d	Returns the factor D of the LDL^T decomposition.
matrix_l	Returns the factor L of the Cholesky decomposition of the matrix.
matrix_p	Returns the factor P of the LDL^T decomposition.
matrix_q	Returns the permutation matrix corresponding to the column interchanges in the LU factorization of the matrix.
matrix_r	Returns the upper triangular matrix R in the QR decomposition.
matrix_u	Returns the upper triangular matrix U in the LU decomposition.
matrix_v	Returns the unitary matrix V (right singular vectors) in the SVD decomposition.
max	Returns the maximum value of the elements in the RealMatrix along the given dimensions.
mean	Returns the mean value of the elements in the ComplexArray.
min	Returns the minimum value of the elements in the RealArray along the given dimensions.
modulus	Computes the remainder after division of a by m, where a is the dividend and m is the divisor. Note that nonzero results have the same sign as the divisor. This function is equivalent to a - m * floor(a / m).
nearest_integer	Computes the nearest integer to each element of the RealArray arr element-wise, rounding halfway cases to even.
phase	Computes the phase angle (argument) of the ComplexArray arr element-wise.
polygamma	Computes the polygamma function (the n-th derivative of the digamma function) of the RealArray arr element-wise.
pow	The **= operator exponentiates a ComplexArray by another one element-wise.
product	Returns the product of the elements in the ComplexMatrix.
rank	Returns the rank of the matrix leveraging the SVD decomposition.
reduce	Reduces the elements of the ComplexMatrix to a single value using a custom function fun along the given dimensions.
remainder	Computes the remainder after division of a by m, where a is the dividend and m is the divisor. Note that nonzero results have the same sign as the dividend. This function is equivalent to a - m * fix(a / m).
round	Computes the nearest integer to each element of the RealArray arr element-wise, rounding away from zero in halfway cases.
sign	Computes the sign (-1, 0, or 1) of the ComplexArray arr element-wise.
sin	Computes the sine of the RealArray arr element-wise.
singular_values	Returns the singular values of the matrix.
sinh	Computes the hyperbolic sine of the RealArray arr element-wise.
sqrt	Computes the square root of the RealArray arr element-wise.
square	Computes the square of the ComplexArray arr element-wise.
string	Converts a ComplexMatrix to its string representation.
sum	Returns the sum value of the elements in the IntegerArray along the given dimensions.
tan	Computes the tangent of the RealArray arr element-wise.
tanh	Computes the hyperbolic tangent of the RealArray arr element-wise.
zeta	Computes the Hurwitz zeta function of the RealArray arr0 and arr1 element-wise.

Module Symbols

Name	Description
!	Returns the logical NOT of the BoolArray lhs.
!=	Checks if two BoolArray lhs and rhs are element-wise not-equal.
%	Computes the remainder after division of a by m, where a is the dividend and m is the divisor. Note that nonzero results have the same sign as the divisor. This function is equivalent to a - m * floor(a / m).
*	Scalar-matrix multiplication operator.
**	The **= operator exponentiates a IntegerArray by another one element-wise.
*=	The * operator multiplies in-place two IntegerArray element-wise.
+	The + operator adds two IntegerArray element-wise.
++	Increments each element of the IntegerArray lhs.
+=	The += operator adds in-place two IntegerArray element-wise.
-	Negates each element of the RealMatrix lhs.
--	Decrements each element of the IntegerArray lhs.
-=	The -= operator subtracts in-place two RealArray element-wise.
/	The / operator divides two IntegerArray element-wise.
/=	The / operator divides in-place two ComplexArray element-wise.
<	Less than operator comparing two RealArray.
<=	Less or equal operator comparing two RealArray.
=	Sets the BoolArray lhs to the values in the BoolArray rhs.
==	Checks if two BoolArray lhs and rhs are element-wise equal.
>	Greater than operator comparing two RealArray.
>=	Greater or equal operator comparing two RealArray.
^	Returns the logical XOR of the BoolArray lhs and rhs.