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