1 // Copyright ©2015 The Gonum Authors. All rights reserved.
2 // Use of this source code is governed by a BSD-style
3 // license that can be found in the LICENSE file.
5 // Package mat provides implementations of float64 and complex128 matrix
6 // structures and linear algebra operations on them.
10 // This section provides a quick overview of the mat package. The following
11 // sections provide more in depth commentary.
14 // - Interfaces for Matrix classes (Matrix, Symmetric, Triangular)
15 // - Concrete implementations (Dense, SymDense, TriDense)
16 // - Methods and functions for using matrix data (Add, Trace, SymRankOne)
17 // - Types for constructing and using matrix factorizations (QR, LU)
18 // - The complementary types for complex matrices, CMatrix, CSymDense, etc.
20 // A matrix may be constructed through the corresponding New function. If no
21 // backing array is provided the matrix will be initialized to all zeros.
22 // // Allocate a zeroed real matrix of size 3×5
23 // zero := mat.NewDense(3, 5, nil)
24 // If a backing data slice is provided, the matrix will have those elements.
25 // Matrices are all stored in row-major format.
26 // // Generate a 6×6 matrix of random values.
27 // data := make([]float64, 36)
28 // for i := range data {
29 // data[i] = rand.NormFloat64()
31 // a := mat.NewDense(6, 6, data)
32 // Operations involving matrix data are implemented as functions when the values
33 // of the matrix remain unchanged
35 // and are implemented as methods when the operation modifies the receiver.
38 // Receivers must be the correct size for the matrix operations, otherwise the
39 // operation will panic. As a special case for convenience, a zero-value matrix
40 // will be modified to have the correct size, allocating data if necessary.
41 // var c mat.Dense // construct a new zero-sized matrix
42 // c.Mul(a, a) // c is automatically adjusted to be 6×6
44 // Zero-value of a matrix
46 // A zero-value matrix is either the Go language definition of a zero-value or
47 // is a zero-sized matrix with zero-length stride. Matrix implementations may have
48 // a Reset method to revert the receiver into a zero-valued matrix and an IsZero
49 // method that returns whether the matrix is zero-valued.
50 // So the following will all result in a zero-value matrix.
52 // - a := NewDense(0, 0, make([]float64, 0, 100))
54 // A zero-value matrix can not be sliced even if it does have an adequately sized
55 // backing data slice, but can be expanded using its Grow method if it exists.
57 // The Matrix Interfaces
59 // The Matrix interface is the common link between the concrete types of real
60 // matrices, The Matrix interface is defined by three functions: Dims, which
61 // returns the dimensions of the Matrix, At, which returns the element in the
62 // specified location, and T for returning a Transpose (discussed later). All of
63 // the concrete types can perform these behaviors and so implement the interface.
64 // Methods and functions are designed to use this interface, so in particular the method
65 // func (m *Dense) Mul(a, b Matrix)
66 // constructs a *Dense from the result of a multiplication with any Matrix types,
67 // not just *Dense. Where more restrictive requirements must be met, there are also the
68 // Symmetric and Triangular interfaces. For example, in
69 // func (s *SymDense) AddSym(a, b Symmetric)
70 // the Symmetric interface guarantees a symmetric result.
72 // The CMatrix interface plays the same role for complex matrices. The difference
73 // is that the CMatrix type has the H method instead T, for returning the conjugate
76 // (Conjugate) Transposes
78 // The T method is used for transposition on real matrices, and H is used for
79 // conjugate transposition on complex matrices. For example, c.Mul(a.T(), b) computes
80 // c = a^T * b. The mat types implement this method implicitly —
81 // see the Transpose and Conjugate types for more details. Note that some
82 // operations have a transpose as part of their definition, as in *SymDense.SymOuterK.
84 // Matrix Factorization
86 // Matrix factorizations, such as the LU decomposition, typically have their own
87 // specific data storage, and so are each implemented as a specific type. The
88 // factorization can be computed through a call to Factorize
91 // The elements of the factorization can be extracted through methods on the
92 // factorized type, i.e. *LU.UTo. The factorization types can also be used directly,
93 // as in *Dense.SolveCholesky. Some factorizations can be updated directly,
94 // without needing to update the original matrix and refactorize,
99 // BLAS and LAPACK are the standard APIs for linear algebra routines. Many
100 // operations in mat are implemented using calls to the wrapper functions
101 // in gonum/blas/blas64 and gonum/lapack/lapack64 and their complex equivalents.
102 // By default, blas64 and lapack64 call the native Go implementations of the
103 // routines. Alternatively, it is possible to use C-based implementations of the
104 // APIs through the respective cgo packages and "Use" functions. The Go
105 // implementation of LAPACK (used by default) makes calls
106 // through blas64, so if a cgo BLAS implementation is registered, the lapack64
107 // calls will be partially executed in Go and partially executed in C.
111 // The Matrix abstraction enables efficiency as well as interoperability. Go's
112 // type reflection capabilities are used to choose the most efficient routine
113 // given the specific concrete types. For example, in
115 // if a and b both implement RawMatrixer, that is, they can be represented as a
116 // blas64.General, blas64.Gemm (general matrix multiplication) is called, while
117 // instead if b is a RawSymmetricer blas64.Symm is used (general-symmetric
118 // multiplication), and if b is a *VecDense blas64.Gemv is used.
120 // There are many possible type combinations and special cases. No specific guarantees
121 // are made about the performance of any method, and in particular, note that an
122 // abstract matrix type may be copied into a concrete type of the corresponding
123 // value. If there are specific special cases that are needed, please submit a
124 // pull-request or file an issue.
128 // Matrix input arguments to functions are never directly modified. If an operation
129 // changes Matrix data, the mutated matrix will be the receiver of a function.
131 // For convenience, a matrix may be used as both a receiver and as an input, e.g.
133 // v.SolveVec(a.T(), v)
134 // though in many cases this will cause an allocation (see Element Aliasing).
135 // An exception to this rule is Copy, which does not allow a.Copy(a.T()).
139 // Most methods in mat modify receiver data. It is forbidden for the modified
140 // data region of the receiver to overlap the used data area of the input
141 // arguments. The exception to this rule is when the method receiver is equal to one
142 // of the input arguments, as in the a.Pow(a, 6) call above, or its implicit transpose.
144 // This prohibition is to help avoid subtle mistakes when the method needs to read
145 // from and write to the same data region. There are ways to make mistakes using the
146 // mat API, and mat functions will detect and complain about those.
147 // There are many ways to make mistakes by excursion from the mat API via
148 // interaction with raw matrix values.
150 // If you need to read the rest of this section to understand the behavior of
151 // your program, you are being clever. Don't be clever. If you must be clever,
152 // blas64 and lapack64 may be used to call the behavior directly.
154 // mat will use the following rules to detect overlap between the receiver and one
156 // - the input implements one of the Raw methods, and
157 // - the address ranges of the backing data slices overlap, and
158 // - the strides differ or there is an overlap in the used data elements.
159 // If such an overlap is detected, the method will panic.
161 // The following cases will not panic:
162 // - the data slices do not overlap,
163 // - there is pointer identity between the receiver and input values after
164 // the value has been untransposed if necessary.
166 // mat will not attempt to detect element overlap if the input does not implement a
167 // Raw method. Method behavior is undefined if there is undetected overlap.
169 package mat // import "gonum.org/v1/gonum/mat"
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