MatrixUtils (Apache Commons Math 3.3 API)

org.apache.commons.math3.linear

## Class MatrixUtils

- java.lang.Object
- - org.apache.commons.math3.linear.MatrixUtils

-     public class MatrixUtils
      extends Object

  A collection of static methods that operate on or return matrices.
  Version:
    $Id: MatrixUtils.java 1533638 2013-10-18 21:19:18Z tn $

- - ### Field Summary

    Fields
    Modifier and Type | Field and Description
    --- | ---
    `static RealMatrixFormat` | `DEFAULT_FORMAT` The default format for `RealMatrix` objects.
    `static RealMatrixFormat` | `OCTAVE_FORMAT` A format for `RealMatrix` objects compatible with octave.

  - ### Method Summary

    Methods
    Modifier and Type | Method and Description
    --- | ---
    `static Array2DRowRealMatrix` | `bigFractionMatrixToRealMatrix(FieldMatrix<BigFraction> m)` Convert a `FieldMatrix`/`BigFraction` matrix to a `RealMatrix`.
    `static RealMatrix` | `blockInverse(RealMatrix m, int splitIndex)` Computes the inverse of the given matrix by splitting it into 4 sub-matrices.
    `static void` | `checkAdditionCompatible(AnyMatrix left, AnyMatrix right)` Check if matrices are addition compatible.
    `static void` | `checkColumnIndex(AnyMatrix m, int column)` Check if a column index is valid.
    `static void` | `checkMatrixIndex(AnyMatrix m, int row, int column)` Check if matrix indices are valid.
    `static void` | `checkMultiplicationCompatible(AnyMatrix left, AnyMatrix right)` Check if matrices are multiplication compatible
    `static void` | `checkRowIndex(AnyMatrix m, int row)` Check if a row index is valid.
    `static void` | `checkSubMatrixIndex(AnyMatrix m, int[] selectedRows, int[] selectedColumns)` Check if submatrix ranges indices are valid.
    `static void` | `checkSubMatrixIndex(AnyMatrix m, int startRow, int endRow, int startColumn, int endColumn)` Check if submatrix ranges indices are valid.
    `static void` | `checkSubtractionCompatible(AnyMatrix left, AnyMatrix right)` Check if matrices are subtraction compatible
    `static void` | `checkSymmetric(RealMatrix matrix, double eps)` Checks whether a matrix is symmetric.
    `static <T extends FieldElement<T>> FieldMatrix<T>` | `createColumnFieldMatrix(T[] columnData)` Creates a column `FieldMatrix` using the data from the input array.
    `static RealMatrix` | `createColumnRealMatrix(double[] columnData)` Creates a column `RealMatrix` using the data from the input array.
    `static <T extends FieldElement<T>> FieldMatrix<T>` | `createFieldDiagonalMatrix(T[] diagonal)` Returns a diagonal matrix with specified elements.
    `static <T extends FieldElement<T>> FieldMatrix<T>` | `createFieldIdentityMatrix(Field<T> field, int dimension)` Returns `dimension x dimension` identity matrix.
    `static <T extends FieldElement<T>> FieldMatrix<T>` | `createFieldMatrix(Field<T> field, int rows, int columns)` Returns a `FieldMatrix` with specified dimensions.
    `static <T extends FieldElement<T>> FieldMatrix<T>` | `createFieldMatrix(T[][] data)` Returns a `FieldMatrix` whose entries are the the values in the the input array.
    `static <T extends FieldElement<T>> FieldVector<T>` | `createFieldVector(T[] data)` Creates a `FieldVector` using the data from the input array.
    `static RealMatrix` | `createRealDiagonalMatrix(double[] diagonal)` Returns a diagonal matrix with specified elements.
    `static RealMatrix` | `createRealIdentityMatrix(int dimension)` Returns `dimension x dimension` identity matrix.
    `static RealMatrix` | `createRealMatrix(double[][] data)` Returns a `RealMatrix` whose entries are the the values in the the input array.
    `static RealMatrix` | `createRealMatrix(int rows, int columns)` Returns a `RealMatrix` with specified dimensions.
    `static RealVector` | `createRealVector(double[] data)` Creates a `RealVector` using the data from the input array.
    `static <T extends FieldElement<T>> FieldMatrix<T>` | `createRowFieldMatrix(T[] rowData)` Create a row `FieldMatrix` using the data from the input array.
    `static RealMatrix` | `createRowRealMatrix(double[] rowData)` Create a row `RealMatrix` using the data from the input array.
    `static void` | `deserializeRealMatrix(Object instance, String fieldName, ObjectInputStream ois)` Deserialize a `RealMatrix` field in a class.
    `static void` | `deserializeRealVector(Object instance, String fieldName, ObjectInputStream ois)` Deserialize a `RealVector` field in a class.
    `static Array2DRowRealMatrix` | `fractionMatrixToRealMatrix(FieldMatrix<Fraction> m)` Convert a `FieldMatrix`/`Fraction` matrix to a `RealMatrix`.
    `static RealMatrix` | `inverse(RealMatrix matrix)` Computes the inverse of the given matrix.
    `static RealMatrix` | `inverse(RealMatrix matrix, double threshold)` Computes the inverse of the given matrix.
    `static boolean` | `isSymmetric(RealMatrix matrix, double eps)` Checks whether a matrix is symmetric.
    `static void` | `serializeRealMatrix(RealMatrix matrix, ObjectOutputStream oos)` Serialize a `RealMatrix`.
    `static void` | `serializeRealVector(RealVector vector, ObjectOutputStream oos)` Serialize a `RealVector`.
    `static void` | `solveLowerTriangularSystem(RealMatrix rm, RealVector b)` Solve a system of composed of a Lower Triangular Matrix `RealMatrix`.
    `static void` | `solveUpperTriangularSystem(RealMatrix rm, RealVector b)` Solver a system composed of an Upper Triangular Matrix `RealMatrix`.

    - ### Methods inherited from class java.lang.Object

      `clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait`

- - ### Field Detail

    - #### DEFAULT_FORMAT

          public static final RealMatrixFormat DEFAULT_FORMAT

      The default format for `RealMatrix` objects.
      Since:
        3.1

    - #### OCTAVE_FORMAT

          public static final RealMatrixFormat OCTAVE_FORMAT

      A format for `RealMatrix` objects compatible with octave.
      Since:
        3.1

  - ### Method Detail

    - #### createRealMatrix

          public static RealMatrix createRealMatrix(int rows,
                                    int columns)

      Returns a `RealMatrix` with specified dimensions.

      The type of matrix returned depends on the dimension. Below 212 elements (i.e. 4096 elements or 64×64 for a square matrix) which can be stored in a 32kB array, a `Array2DRowRealMatrix` instance is built. Above this threshold a `BlockRealMatrix` instance is built.

      The matrix elements are all set to 0.0.

      Parameters:
        `rows` - number of rows of the matrix
        `columns` - number of columns of the matrix
      Returns:
        RealMatrix with specified dimensions
      See Also:
        `createRealMatrix(double[][])`

    - #### createFieldMatrix

          public static <T extends FieldElement<T>> FieldMatrix<T> createFieldMatrix(Field<T> field,
                                                                     int rows,
                                                                     int columns)

      Returns a `FieldMatrix` with specified dimensions.

      The type of matrix returned depends on the dimension. Below 212 elements (i.e. 4096 elements or 64×64 for a square matrix), a `FieldMatrix` instance is built. Above this threshold a `BlockFieldMatrix` instance is built.

      The matrix elements are all set to field.getZero().

      Type Parameters:
        `T` - the type of the field elements
      Parameters:
        `field` - field to which the matrix elements belong
        `rows` - number of rows of the matrix
        `columns` - number of columns of the matrix
      Returns:
        FieldMatrix with specified dimensions
      Since:
        2.0
      See Also:
        `createFieldMatrix(FieldElement[][])`

    - #### createRealMatrix

          public static RealMatrix createRealMatrix(double[][] data)
                                             throws NullArgumentException,
                                                    DimensionMismatchException,
                                                    NoDataException

      Returns a `RealMatrix` whose entries are the the values in the the input array.

      The type of matrix returned depends on the dimension. Below 212 elements (i.e. 4096 elements or 64×64 for a square matrix) which can be stored in a 32kB array, a `Array2DRowRealMatrix` instance is built. Above this threshold a `BlockRealMatrix` instance is built.

      The input array is copied, not referenced.

      Parameters:
        `data` - input array
      Returns:
        RealMatrix containing the values of the array
      Throws:
        `DimensionMismatchException` - if `data` is not rectangular (not all rows have the same length).
        `NoDataException` - if a row or column is empty.
        `NullArgumentException` - if either `data` or `data[0]` is `null`.
        `DimensionMismatchException` - if `data` is not rectangular.
      See Also:
        `createRealMatrix(int, int)`

    - #### createFieldMatrix

          public static <T extends FieldElement<T>> FieldMatrix<T> createFieldMatrix(T[][] data)
                                                                          throws DimensionMismatchException,
                                                                                 NoDataException,
                                                                                 NullArgumentException

      Returns a `FieldMatrix` whose entries are the the values in the the input array.

      The type of matrix returned depends on the dimension. Below 212 elements (i.e. 4096 elements or 64×64 for a square matrix), a `FieldMatrix` instance is built. Above this threshold a `BlockFieldMatrix` instance is built.

      The input array is copied, not referenced.

      Type Parameters:
        `T` - the type of the field elements
      Parameters:
        `data` - input array
      Returns:
        a matrix containing the values of the array.
      Throws:
        `DimensionMismatchException` - if `data` is not rectangular (not all rows have the same length).
        `NoDataException` - if a row or column is empty.
        `NullArgumentException` - if either `data` or `data[0]` is `null`.
      Since:
        2.0
      See Also:
        `createFieldMatrix(Field, int, int)`

    - #### createRealIdentityMatrix

          public static RealMatrix createRealIdentityMatrix(int dimension)

      Returns `dimension x dimension` identity matrix.
      Parameters:
        `dimension` - dimension of identity matrix to generate
      Returns:
        identity matrix
      Throws:
        `IllegalArgumentException` - if dimension is not positive
      Since:
        1.1

    - #### createFieldIdentityMatrix

          public static <T extends FieldElement<T>> FieldMatrix<T> createFieldIdentityMatrix(Field<T> field,
                                                                             int dimension)

      Returns `dimension x dimension` identity matrix.
      Type Parameters:
        `T` - the type of the field elements
      Parameters:
        `field` - field to which the elements belong
        `dimension` - dimension of identity matrix to generate
      Returns:
        identity matrix
      Throws:
        `IllegalArgumentException` - if dimension is not positive
      Since:
        2.0

    - #### createRealDiagonalMatrix

          public static RealMatrix createRealDiagonalMatrix(double[] diagonal)

      Returns a diagonal matrix with specified elements.
      Parameters:
        `diagonal` - diagonal elements of the matrix (the array elements will be copied)
      Returns:
        diagonal matrix
      Since:
        2.0

    - #### createFieldDiagonalMatrix

          public static <T extends FieldElement<T>> FieldMatrix<T> createFieldDiagonalMatrix(T[] diagonal)

      Returns a diagonal matrix with specified elements.
      Type Parameters:
        `T` - the type of the field elements
      Parameters:
        `diagonal` - diagonal elements of the matrix (the array elements will be copied)
      Returns:
        diagonal matrix
      Since:
        2.0

    - #### createRealVector

          public static RealVector createRealVector(double[] data)
                                             throws NoDataException,
                                                    NullArgumentException

      Creates a `RealVector` using the data from the input array.
      Parameters:
        `data` - the input data
      Returns:
        a data.length RealVector
      Throws:
        `NoDataException` - if `data` is empty.
        `NullArgumentException` - if `data` is `null`.

    - #### createFieldVector

          public static <T extends FieldElement<T>> FieldVector<T> createFieldVector(T[] data)
                                                                          throws NoDataException,
                                                                                 NullArgumentException,
                                                                                 ZeroException

      Creates a `FieldVector` using the data from the input array.
      Type Parameters:
        `T` - the type of the field elements
      Parameters:
        `data` - the input data
      Returns:
        a data.length FieldVector
      Throws:
        `NoDataException` - if `data` is empty.
        `NullArgumentException` - if `data` is `null`.
        `ZeroException` - if `data` has 0 elements

    - #### createRowRealMatrix

          public static RealMatrix createRowRealMatrix(double[] rowData)
                                                throws NoDataException,
                                                       NullArgumentException

      Create a row `RealMatrix` using the data from the input array.
      Parameters:
        `rowData` - the input row data
      Returns:
        a 1 x rowData.length RealMatrix
      Throws:
        `NoDataException` - if `rowData` is empty.
        `NullArgumentException` - if `rowData` is `null`.

    - #### createRowFieldMatrix

          public static <T extends FieldElement<T>> FieldMatrix<T> createRowFieldMatrix(T[] rowData)
                                                                             throws NoDataException,
                                                                                    NullArgumentException

      Create a row `FieldMatrix` using the data from the input array.
      Type Parameters:
        `T` - the type of the field elements
      Parameters:
        `rowData` - the input row data
      Returns:
        a 1 x rowData.length FieldMatrix
      Throws:
        `NoDataException` - if `rowData` is empty.
        `NullArgumentException` - if `rowData` is `null`.

    - #### createColumnRealMatrix

          public static RealMatrix createColumnRealMatrix(double[] columnData)
                                                   throws NoDataException,
                                                          NullArgumentException

      Creates a column `RealMatrix` using the data from the input array.
      Parameters:
        `columnData` - the input column data
      Returns:
        a columnData x 1 RealMatrix
      Throws:
        `NoDataException` - if `columnData` is empty.
        `NullArgumentException` - if `columnData` is `null`.

    - #### createColumnFieldMatrix

          public static <T extends FieldElement<T>> FieldMatrix<T> createColumnFieldMatrix(T[] columnData)
                                                                                throws NoDataException,
                                                                                       NullArgumentException

      Creates a column `FieldMatrix` using the data from the input array.
      Type Parameters:
        `T` - the type of the field elements
      Parameters:
        `columnData` - the input column data
      Returns:
        a columnData x 1 FieldMatrix
      Throws:
        `NoDataException` - if `data` is empty.
        `NullArgumentException` - if `columnData` is `null`.

    - #### checkSymmetric

          public static void checkSymmetric(RealMatrix matrix,
                            double eps)

      Checks whether a matrix is symmetric.
      Parameters:
        `matrix` - Matrix to check.
        `eps` - Relative tolerance.
      Throws:
        `NonSquareMatrixException` - if the matrix is not square.
        `NonSymmetricMatrixException` - if the matrix is not symmetric.
      Since:
        3.1

    - #### isSymmetric

          public static boolean isSymmetric(RealMatrix matrix,
                            double eps)

      Checks whether a matrix is symmetric.
      Parameters:
        `matrix` - Matrix to check.
        `eps` - Relative tolerance.
      Returns:
        `true` if `matrix` is symmetric.
      Since:
        3.1

    - #### checkMatrixIndex

          public static void checkMatrixIndex(AnyMatrix m,
                              int row,
                              int column)
                                       throws OutOfRangeException

      Check if matrix indices are valid.
      Parameters:
        `m` - Matrix.
        `row` - Row index to check.
        `column` - Column index to check.
      Throws:
        `OutOfRangeException` - if `row` or `column` is not a valid index.

    - #### checkRowIndex

          public static void checkRowIndex(AnyMatrix m,
                           int row)
                                    throws OutOfRangeException

      Check if a row index is valid.
      Parameters:
        `m` - Matrix.
        `row` - Row index to check.
      Throws:
        `OutOfRangeException` - if `row` is not a valid index.

    - #### checkColumnIndex

          public static void checkColumnIndex(AnyMatrix m,
                              int column)
                                       throws OutOfRangeException

      Check if a column index is valid.
      Parameters:
        `m` - Matrix.
        `column` - Column index to check.
      Throws:
        `OutOfRangeException` - if `column` is not a valid index.

    - #### checkSubMatrixIndex

          public static void checkSubMatrixIndex(AnyMatrix m,
                                 int startRow,
                                 int endRow,
                                 int startColumn,
                                 int endColumn)
                                          throws NumberIsTooSmallException,
                                                 OutOfRangeException

      Check if submatrix ranges indices are valid. Rows and columns are indicated counting from 0 to `n - 1`.
      Parameters:
        `m` - Matrix.
        `startRow` - Initial row index.
        `endRow` - Final row index.
        `startColumn` - Initial column index.
        `endColumn` - Final column index.
      Throws:
        `OutOfRangeException` - if the indices are invalid.
        `NumberIsTooSmallException` - if `endRow < startRow` or `endColumn < startColumn`.

    - #### checkSubMatrixIndex

          public static void checkSubMatrixIndex(AnyMatrix m,
                                 int[] selectedRows,
                                 int[] selectedColumns)
                                          throws NoDataException,
                                                 NullArgumentException,
                                                 OutOfRangeException

      Check if submatrix ranges indices are valid. Rows and columns are indicated counting from 0 to n-1.
      Parameters:
        `m` - Matrix.
        `selectedRows` - Array of row indices.
        `selectedColumns` - Array of column indices.
      Throws:
        `NullArgumentException` - if `selectedRows` or `selectedColumns` are `null`.
        `NoDataException` - if the row or column selections are empty (zero length).
        `OutOfRangeException` - if row or column selections are not valid.

    - #### checkAdditionCompatible

          public static void checkAdditionCompatible(AnyMatrix left,
                                     AnyMatrix right)
                                              throws MatrixDimensionMismatchException

      Check if matrices are addition compatible.
      Parameters:
        `left` - Left hand side matrix.
        `right` - Right hand side matrix.
      Throws:
        `MatrixDimensionMismatchException` - if the matrices are not addition compatible.

    - #### checkSubtractionCompatible

          public static void checkSubtractionCompatible(AnyMatrix left,
                                        AnyMatrix right)
                                                 throws MatrixDimensionMismatchException

      Check if matrices are subtraction compatible
      Parameters:
        `left` - Left hand side matrix.
        `right` - Right hand side matrix.
      Throws:
        `MatrixDimensionMismatchException` - if the matrices are not addition compatible.

    - #### checkMultiplicationCompatible

          public static void checkMultiplicationCompatible(AnyMatrix left,
                                           AnyMatrix right)
                                                    throws DimensionMismatchException

      Check if matrices are multiplication compatible
      Parameters:
        `left` - Left hand side matrix.
        `right` - Right hand side matrix.
      Throws:
        `DimensionMismatchException` - if matrices are not multiplication compatible.

    - #### fractionMatrixToRealMatrix

          public static Array2DRowRealMatrix fractionMatrixToRealMatrix(FieldMatrix<Fraction> m)

      Convert a `FieldMatrix`/`Fraction` matrix to a `RealMatrix`.
      Parameters:
        `m` - Matrix to convert.
      Returns:
        the converted matrix.

    - #### bigFractionMatrixToRealMatrix

          public static Array2DRowRealMatrix bigFractionMatrixToRealMatrix(FieldMatrix<BigFraction> m)

      Convert a `FieldMatrix`/`BigFraction` matrix to a `RealMatrix`.
      Parameters:
        `m` - Matrix to convert.
      Returns:
        the converted matrix.

    - #### serializeRealVector

          public static void serializeRealVector(RealVector vector,
                                 ObjectOutputStream oos)
                                          throws IOException

      Serialize a `RealVector`.

      This method is intended to be called from within a private `writeObject` method (after a call to `oos.defaultWriteObject()`) in a class that has a `RealVector` field, which should be declared `transient`. This way, the default handling does not serialize the vector (the `RealVector` interface is not serializable by default) but this method does serialize it specifically.

      The following example shows how a simple class with a name and a real vector should be written:

           public class NamedVector implements Serializable {

               private final String name;
               private final transient RealVector coefficients;

               // omitted constructors, getters ...

               private void writeObject(ObjectOutputStream oos) throws IOException {
                   oos.defaultWriteObject();  // takes care of name field
                   MatrixUtils.serializeRealVector(coefficients, oos);
               }

               private void readObject(ObjectInputStream ois) throws ClassNotFoundException, IOException {
                   ois.defaultReadObject();  // takes care of name field
                   MatrixUtils.deserializeRealVector(this, "coefficients", ois);
               }

           }

      Parameters:
        `vector` - real vector to serialize
        `oos` - stream where the real vector should be written
      Throws:
        `IOException` - if object cannot be written to stream
      See Also:
        `deserializeRealVector(Object, String, ObjectInputStream)`

    - #### deserializeRealVector

          public static void deserializeRealVector(Object instance,
                                   String fieldName,
                                   ObjectInputStream ois)
                                            throws ClassNotFoundException,
                                                   IOException

      Deserialize a `RealVector` field in a class.

      This method is intended to be called from within a private `readObject` method (after a call to `ois.defaultReadObject()`) in a class that has a `RealVector` field, which should be declared `transient`. This way, the default handling does not deserialize the vector (the `RealVector` interface is not serializable by default) but this method does deserialize it specifically.

      Parameters:
        `instance` - instance in which the field must be set up
        `fieldName` - name of the field within the class (may be private and final)
        `ois` - stream from which the real vector should be read
      Throws:
        `ClassNotFoundException` - if a class in the stream cannot be found
        `IOException` - if object cannot be read from the stream
      See Also:
        `serializeRealVector(RealVector, ObjectOutputStream)`

    - #### serializeRealMatrix

          public static void serializeRealMatrix(RealMatrix matrix,
                                 ObjectOutputStream oos)
                                          throws IOException

      Serialize a `RealMatrix`.

      This method is intended to be called from within a private `writeObject` method (after a call to `oos.defaultWriteObject()`) in a class that has a `RealMatrix` field, which should be declared `transient`. This way, the default handling does not serialize the matrix (the `RealMatrix` interface is not serializable by default) but this method does serialize it specifically.

      The following example shows how a simple class with a name and a real matrix should be written:

           public class NamedMatrix implements Serializable {

               private final String name;
               private final transient RealMatrix coefficients;

               // omitted constructors, getters ...

               private void writeObject(ObjectOutputStream oos) throws IOException {
                   oos.defaultWriteObject();  // takes care of name field
                   MatrixUtils.serializeRealMatrix(coefficients, oos);
               }

               private void readObject(ObjectInputStream ois) throws ClassNotFoundException, IOException {
                   ois.defaultReadObject();  // takes care of name field
                   MatrixUtils.deserializeRealMatrix(this, "coefficients", ois);
               }

           }

      Parameters:
        `matrix` - real matrix to serialize
        `oos` - stream where the real matrix should be written
      Throws:
        `IOException` - if object cannot be written to stream
      See Also:
        `deserializeRealMatrix(Object, String, ObjectInputStream)`

    - #### deserializeRealMatrix

          public static void deserializeRealMatrix(Object instance,
                                   String fieldName,
                                   ObjectInputStream ois)
                                            throws ClassNotFoundException,
                                                   IOException

      Deserialize a `RealMatrix` field in a class.

      This method is intended to be called from within a private `readObject` method (after a call to `ois.defaultReadObject()`) in a class that has a `RealMatrix` field, which should be declared `transient`. This way, the default handling does not deserialize the matrix (the `RealMatrix` interface is not serializable by default) but this method does deserialize it specifically.

      Parameters:
        `instance` - instance in which the field must be set up
        `fieldName` - name of the field within the class (may be private and final)
        `ois` - stream from which the real matrix should be read
      Throws:
        `ClassNotFoundException` - if a class in the stream cannot be found
        `IOException` - if object cannot be read from the stream
      See Also:
        `serializeRealMatrix(RealMatrix, ObjectOutputStream)`

    - #### solveLowerTriangularSystem

          public static void solveLowerTriangularSystem(RealMatrix rm,
                                        RealVector b)
                                                 throws DimensionMismatchException,
                                                        MathArithmeticException,
                                                        NonSquareMatrixException

      Solve a system of composed of a Lower Triangular Matrix `RealMatrix`.

      This method is called to solve systems of equations which are of the lower triangular form. The matrix `RealMatrix` is assumed, though not checked, to be in lower triangular form. The vector `RealVector` is overwritten with the solution. The matrix is checked that it is square and its dimensions match the length of the vector.

      Parameters:
        `rm` - RealMatrix which is lower triangular
        `b` - RealVector this is overwritten
      Throws:
        `DimensionMismatchException` - if the matrix and vector are not conformable
        `NonSquareMatrixException` - if the matrix `rm` is not square
        `MathArithmeticException` - if the absolute value of one of the diagonal coefficient of `rm` is lower than `Precision.SAFE_MIN`

    - #### solveUpperTriangularSystem

          public static void solveUpperTriangularSystem(RealMatrix rm,
                                        RealVector b)
                                                 throws DimensionMismatchException,
                                                        MathArithmeticException,
                                                        NonSquareMatrixException

      Solver a system composed of an Upper Triangular Matrix `RealMatrix`.

      This method is called to solve systems of equations which are of the lower triangular form. The matrix `RealMatrix` is assumed, though not checked, to be in upper triangular form. The vector `RealVector` is overwritten with the solution. The matrix is checked that it is square and its dimensions match the length of the vector.

      Parameters:
        `rm` - RealMatrix which is upper triangular
        `b` - RealVector this is overwritten
      Throws:
        `DimensionMismatchException` - if the matrix and vector are not conformable
        `NonSquareMatrixException` - if the matrix `rm` is not square
        `MathArithmeticException` - if the absolute value of one of the diagonal coefficient of `rm` is lower than `Precision.SAFE_MIN`

    - #### blockInverse

          public static RealMatrix blockInverse(RealMatrix m,
                                int splitIndex)

      Computes the inverse of the given matrix by splitting it into 4 sub-matrices.
      Parameters:
        `m` - Matrix whose inverse must be computed.
        `splitIndex` - Index that determines the "split" line and column. The element corresponding to this index will part of the upper-left sub-matrix.
      Returns:
        the inverse of `m`.
      Throws:
        `NonSquareMatrixException` - if `m` is not square.

    - #### inverse

          public static RealMatrix inverse(RealMatrix matrix)
                                    throws NullArgumentException,
                                           SingularMatrixException,
                                           NonSquareMatrixException

      Computes the inverse of the given matrix.

      By default, the inverse of the matrix is computed using the QR-decomposition, unless a more efficient method can be determined for the input matrix.

      Note: this method will use a singularity threshold of 0, use `inverse(RealMatrix, double)` if a different threshold is needed.

      Parameters:
        `matrix` - Matrix whose inverse shall be computed
      Returns:
        the inverse of `matrix`
      Throws:
        `NullArgumentException` - if `matrix` is `null`
        `SingularMatrixException` - if m is singular
        `NonSquareMatrixException` - if matrix is not square
      Since:
        3.3

    - #### inverse

          public static RealMatrix inverse(RealMatrix matrix,
                           double threshold)
                                    throws NullArgumentException,
                                           SingularMatrixException,
                                           NonSquareMatrixException

      Computes the inverse of the given matrix.

      By default, the inverse of the matrix is computed using the QR-decomposition, unless a more efficient method can be determined for the input matrix.

      Parameters:
        `matrix` - Matrix whose inverse shall be computed
        `threshold` - Singularity threshold
      Returns:
        the inverse of `m`
      Throws:
        `NullArgumentException` - if `matrix` is `null`
        `SingularMatrixException` - if matrix is singular
        `NonSquareMatrixException` - if matrix is not square
      Since:
        3.3

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