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279 lines
6.7 KiB
279 lines
6.7 KiB
/**
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* Matrix Algebra over an 8-bit Galois Field
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*
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* Copyright 2015, Klaus Post
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* Copyright 2015, Backblaze, Inc.
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*/
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package reedsolomon
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import (
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"errors"
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"fmt"
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"strconv"
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"strings"
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)
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// byte[row][col]
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type matrix [][]byte
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// newMatrix returns a matrix of zeros.
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func newMatrix(rows, cols int) (matrix, error) {
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if rows <= 0 {
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return nil, errInvalidRowSize
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}
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if cols <= 0 {
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return nil, errInvalidColSize
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}
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m := matrix(make([][]byte, rows))
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for i := range m {
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m[i] = make([]byte, cols)
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}
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return m, nil
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}
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// NewMatrixData initializes a matrix with the given row-major data.
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// Note that data is not copied from input.
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func newMatrixData(data [][]byte) (matrix, error) {
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m := matrix(data)
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err := m.Check()
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if err != nil {
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return nil, err
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}
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return m, nil
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}
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// IdentityMatrix returns an identity matrix of the given size.
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func identityMatrix(size int) (matrix, error) {
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m, err := newMatrix(size, size)
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if err != nil {
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return nil, err
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}
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for i := range m {
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m[i][i] = 1
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}
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return m, nil
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}
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// errInvalidRowSize will be returned if attempting to create a matrix with negative or zero row number.
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var errInvalidRowSize = errors.New("invalid row size")
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// errInvalidColSize will be returned if attempting to create a matrix with negative or zero column number.
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var errInvalidColSize = errors.New("invalid column size")
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// errColSizeMismatch is returned if the size of matrix columns mismatch.
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var errColSizeMismatch = errors.New("column size is not the same for all rows")
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func (m matrix) Check() error {
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rows := len(m)
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if rows <= 0 {
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return errInvalidRowSize
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}
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cols := len(m[0])
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if cols <= 0 {
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return errInvalidColSize
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}
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for _, col := range m {
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if len(col) != cols {
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return errColSizeMismatch
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}
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}
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return nil
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}
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// String returns a human-readable string of the matrix contents.
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//
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// Example: [[1, 2], [3, 4]]
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func (m matrix) String() string {
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var rowOut []string
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for _, row := range m {
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var colOut []string
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for _, col := range row {
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colOut = append(colOut, strconv.Itoa(int(col)))
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}
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rowOut = append(rowOut, "["+strings.Join(colOut, ", ")+"]")
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}
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return "[" + strings.Join(rowOut, ", ") + "]"
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}
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// Multiply multiplies this matrix (the one on the left) by another
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// matrix (the one on the right) and returns a new matrix with the result.
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func (m matrix) Multiply(right matrix) (matrix, error) {
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if len(m[0]) != len(right) {
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return nil, fmt.Errorf("columns on left (%d) is different than rows on right (%d)", len(m[0]), len(right))
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}
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result, _ := newMatrix(len(m), len(right[0]))
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for r, row := range result {
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for c := range row {
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var value byte
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for i := range m[0] {
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value ^= galMultiply(m[r][i], right[i][c])
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}
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result[r][c] = value
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}
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}
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return result, nil
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}
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// Augment returns the concatenation of this matrix and the matrix on the right.
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func (m matrix) Augment(right matrix) (matrix, error) {
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if len(m) != len(right) {
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return nil, errMatrixSize
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}
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result, _ := newMatrix(len(m), len(m[0])+len(right[0]))
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for r, row := range m {
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for c := range row {
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result[r][c] = m[r][c]
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}
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cols := len(m[0])
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for c := range right[0] {
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result[r][cols+c] = right[r][c]
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}
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}
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return result, nil
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}
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// errMatrixSize is returned if matrix dimensions are doesn't match.
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var errMatrixSize = errors.New("matrix sizes does not match")
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func (m matrix) SameSize(n matrix) error {
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if len(m) != len(n) {
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return errMatrixSize
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}
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for i := range m {
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if len(m[i]) != len(n[i]) {
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return errMatrixSize
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}
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}
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return nil
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}
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// Returns a part of this matrix. Data is copied.
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func (m matrix) SubMatrix(rmin, cmin, rmax, cmax int) (matrix, error) {
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result, err := newMatrix(rmax-rmin, cmax-cmin)
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if err != nil {
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return nil, err
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}
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// OPTME: If used heavily, use copy function to copy slice
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for r := rmin; r < rmax; r++ {
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for c := cmin; c < cmax; c++ {
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result[r-rmin][c-cmin] = m[r][c]
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}
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}
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return result, nil
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}
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// SwapRows Exchanges two rows in the matrix.
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func (m matrix) SwapRows(r1, r2 int) error {
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if r1 < 0 || len(m) <= r1 || r2 < 0 || len(m) <= r2 {
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return errInvalidRowSize
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}
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m[r2], m[r1] = m[r1], m[r2]
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return nil
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}
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// IsSquare will return true if the matrix is square
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// and nil if the matrix is square
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func (m matrix) IsSquare() bool {
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return len(m) == len(m[0])
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}
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// errSingular is returned if the matrix is singular and cannot be inversed
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var errSingular = errors.New("matrix is singular")
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// errNotSquare is returned if attempting to inverse a non-square matrix.
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var errNotSquare = errors.New("only square matrices can be inverted")
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// Invert returns the inverse of this matrix.
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// Returns ErrSingular when the matrix is singular and doesn't have an inverse.
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// The matrix must be square, otherwise ErrNotSquare is returned.
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func (m matrix) Invert() (matrix, error) {
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if !m.IsSquare() {
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return nil, errNotSquare
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}
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size := len(m)
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work, _ := identityMatrix(size)
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work, _ = m.Augment(work)
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err := work.gaussianElimination()
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if err != nil {
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return nil, err
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}
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return work.SubMatrix(0, size, size, size*2)
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}
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func (m matrix) gaussianElimination() error {
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rows := len(m)
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columns := len(m[0])
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// Clear out the part below the main diagonal and scale the main
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// diagonal to be 1.
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for r := 0; r < rows; r++ {
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// If the element on the diagonal is 0, find a row below
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// that has a non-zero and swap them.
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if m[r][r] == 0 {
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for rowBelow := r + 1; rowBelow < rows; rowBelow++ {
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if m[rowBelow][r] != 0 {
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m.SwapRows(r, rowBelow)
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break
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}
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}
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}
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// If we couldn't find one, the matrix is singular.
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if m[r][r] == 0 {
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return errSingular
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}
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// Scale to 1.
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if m[r][r] != 1 {
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scale := galDivide(1, m[r][r])
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for c := 0; c < columns; c++ {
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m[r][c] = galMultiply(m[r][c], scale)
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}
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}
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// Make everything below the 1 be a 0 by subtracting
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// a multiple of it. (Subtraction and addition are
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// both exclusive or in the Galois field.)
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for rowBelow := r + 1; rowBelow < rows; rowBelow++ {
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if m[rowBelow][r] != 0 {
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scale := m[rowBelow][r]
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for c := 0; c < columns; c++ {
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m[rowBelow][c] ^= galMultiply(scale, m[r][c])
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}
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}
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}
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}
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// Now clear the part above the main diagonal.
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for d := 0; d < rows; d++ {
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for rowAbove := 0; rowAbove < d; rowAbove++ {
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if m[rowAbove][d] != 0 {
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scale := m[rowAbove][d]
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for c := 0; c < columns; c++ {
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m[rowAbove][c] ^= galMultiply(scale, m[d][c])
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}
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}
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}
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}
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return nil
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}
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// Create a Vandermonde matrix, which is guaranteed to have the
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// property that any subset of rows that forms a square matrix
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// is invertible.
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func vandermonde(rows, cols int) (matrix, error) {
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result, err := newMatrix(rows, cols)
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if err != nil {
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return nil, err
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}
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for r, row := range result {
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for c := range row {
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result[r][c] = galExp(byte(r), c)
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}
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}
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return result, nil
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}
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