mirror of https://github.com/k3s-io/k3s
179 lines
8.9 KiB
Go
179 lines
8.9 KiB
Go
// Copyright ©2015 The Gonum Authors. All rights reserved.
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// Use of this source code is governed by a BSD-style
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// license that can be found in the LICENSE file.
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// Package mat provides implementations of float64 and complex128 matrix
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// structures and linear algebra operations on them.
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//
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// Overview
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//
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// This section provides a quick overview of the mat package. The following
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// sections provide more in depth commentary.
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//
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// mat provides:
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// - Interfaces for Matrix classes (Matrix, Symmetric, Triangular)
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// - Concrete implementations (Dense, SymDense, TriDense)
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// - Methods and functions for using matrix data (Add, Trace, SymRankOne)
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// - Types for constructing and using matrix factorizations (QR, LU)
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// - The complementary types for complex matrices, CMatrix, CSymDense, etc.
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// In the documentation below, we use "matrix" as a short-hand for all of
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// the FooDense types implemented in this package. We use "Matrix" to
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// refer to the Matrix interface.
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//
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// A matrix may be constructed through the corresponding New function. If no
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// backing array is provided the matrix will be initialized to all zeros.
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// // Allocate a zeroed real matrix of size 3×5
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// zero := mat.NewDense(3, 5, nil)
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// If a backing data slice is provided, the matrix will have those elements.
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// All matrices are all stored in row-major format.
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// // Generate a 6×6 matrix of random values.
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// data := make([]float64, 36)
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// for i := range data {
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// data[i] = rand.NormFloat64()
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// }
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// a := mat.NewDense(6, 6, data)
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// Operations involving matrix data are implemented as functions when the values
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// of the matrix remain unchanged
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// tr := mat.Trace(a)
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// and are implemented as methods when the operation modifies the receiver.
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// zero.Copy(a)
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// Note that the input arguments to most functions and methods are interfaces
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// rather than concrete types `func Trace(Matrix)` rather than
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// `func Trace(*Dense)` allowing flexible use of internal and external
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// Matrix types.
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//
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// When a matrix is the destination or receiver for a function or method,
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// the operation will panic if the matrix is not the correct size.
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// An exception is if that destination is empty (see below).
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//
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// Empty matrix
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//
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// An empty matrix is one that has zero size. Empty matrices are used to allow
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// the destination of a matrix operation to assume the correct size automatically.
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// This operation will re-use the backing data, if available, or will allocate
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// new data if necessary. The IsEmpty method returns whether the given matrix
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// is empty. The zero-value of a matrix is empty, and is useful for easily
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// getting the result of matrix operations.
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// var c mat.Dense // construct a new zero-value matrix
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// c.Mul(a, a) // c is automatically adjusted to be the right size
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// The Reset method can be used to revert a matrix to an empty matrix.
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// Reset should not be used when multiple different matrices share the same backing
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// data slice. This can cause unexpected data modifications after being resized.
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// An empty matrix can not be sliced even if it does have an adequately sized
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// backing data slice, but can be expanded using its Grow method if it exists.
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//
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// The Matrix Interfaces
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//
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// The Matrix interface is the common link between the concrete types of real
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// matrices, The Matrix interface is defined by three functions: Dims, which
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// returns the dimensions of the Matrix, At, which returns the element in the
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// specified location, and T for returning a Transpose (discussed later). All of
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// the matrix types can perform these behaviors and so implement the interface.
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// Methods and functions are designed to use this interface, so in particular the method
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// func (m *Dense) Mul(a, b Matrix)
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// constructs a *Dense from the result of a multiplication with any Matrix types,
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// not just *Dense. Where more restrictive requirements must be met, there are also
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// additional interfaces like Symmetric and Triangular. For example, in
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// func (s *SymDense) AddSym(a, b Symmetric)
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// the Symmetric interface guarantees a symmetric result.
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//
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// The CMatrix interface plays the same role for complex matrices. The difference
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// is that the CMatrix type has the H method instead T, for returning the conjugate
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// transpose.
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//
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// (Conjugate) Transposes
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//
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// The T method is used for transposition on real matrices, and H is used for
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// conjugate transposition on complex matrices. For example, c.Mul(a.T(), b) computes
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// c = aᵀ * b. The mat types implement this method implicitly —
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// see the Transpose and Conjugate types for more details. Note that some
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// operations have a transpose as part of their definition, as in *SymDense.SymOuterK.
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//
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// Matrix Factorization
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//
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// Matrix factorizations, such as the LU decomposition, typically have their own
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// specific data storage, and so are each implemented as a specific type. The
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// factorization can be computed through a call to Factorize
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// var lu mat.LU
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// lu.Factorize(a)
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// The elements of the factorization can be extracted through methods on the
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// factorized type, i.e. *LU.UTo. The factorization types can also be used directly,
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// as in *Dense.SolveCholesky. Some factorizations can be updated directly,
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// without needing to update the original matrix and refactorize,
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// as in *LU.RankOne.
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//
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// BLAS and LAPACK
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//
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// BLAS and LAPACK are the standard APIs for linear algebra routines. Many
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// operations in mat are implemented using calls to the wrapper functions
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// in gonum/blas/blas64 and gonum/lapack/lapack64 and their complex equivalents.
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// By default, blas64 and lapack64 call the native Go implementations of the
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// routines. Alternatively, it is possible to use C-based implementations of the
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// APIs through the respective cgo packages and "Use" functions. The Go
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// implementation of LAPACK (used by default) makes calls
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// through blas64, so if a cgo BLAS implementation is registered, the lapack64
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// calls will be partially executed in Go and partially executed in C.
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//
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// Type Switching
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//
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// The Matrix abstraction enables efficiency as well as interoperability. Go's
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// type reflection capabilities are used to choose the most efficient routine
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// given the specific concrete types. For example, in
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// c.Mul(a, b)
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// if a and b both implement RawMatrixer, that is, they can be represented as a
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// blas64.General, blas64.Gemm (general matrix multiplication) is called, while
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// instead if b is a RawSymmetricer blas64.Symm is used (general-symmetric
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// multiplication), and if b is a *VecDense blas64.Gemv is used.
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//
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// There are many possible type combinations and special cases. No specific guarantees
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// are made about the performance of any method, and in particular, note that an
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// abstract matrix type may be copied into a concrete type of the corresponding
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// value. If there are specific special cases that are needed, please submit a
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// pull-request or file an issue.
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//
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// Invariants
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//
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// Matrix input arguments to functions are never directly modified. If an operation
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// changes Matrix data, the mutated matrix will be the receiver of a method, or
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// will be the first argument to a method or function.
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//
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// For convenience, a matrix may be used as both a receiver and as an input, e.g.
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// a.Pow(a, 6)
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// v.SolveVec(a.T(), v)
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// though in many cases this will cause an allocation (see Element Aliasing).
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// An exception to this rule is Copy, which does not allow a.Copy(a.T()).
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//
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// Element Aliasing
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//
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// Most methods in mat modify receiver data. It is forbidden for the modified
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// data region of the receiver to overlap the used data area of the input
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// arguments. The exception to this rule is when the method receiver is equal to one
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// of the input arguments, as in the a.Pow(a, 6) call above, or its implicit transpose.
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//
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// This prohibition is to help avoid subtle mistakes when the method needs to read
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// from and write to the same data region. There are ways to make mistakes using the
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// mat API, and mat functions will detect and complain about those.
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// There are many ways to make mistakes by excursion from the mat API via
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// interaction with raw matrix values.
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//
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// If you need to read the rest of this section to understand the behavior of
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// your program, you are being clever. Don't be clever. If you must be clever,
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// blas64 and lapack64 may be used to call the behavior directly.
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//
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// mat will use the following rules to detect overlap between the receiver and one
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// of the inputs:
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// - the input implements one of the Raw methods, and
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// - the address ranges of the backing data slices overlap, and
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// - the strides differ or there is an overlap in the used data elements.
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// If such an overlap is detected, the method will panic.
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//
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// The following cases will not panic:
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// - the data slices do not overlap,
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// - there is pointer identity between the receiver and input values after
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// the value has been untransposed if necessary.
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//
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// mat will not attempt to detect element overlap if the input does not implement a
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// Raw method. Method behavior is undefined if there is undetected overlap.
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//
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package mat // import "gonum.org/v1/gonum/mat"
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