Introduction

 
Creation of Sparse Matrices

      Construction of Initialized Sparse Matrices
            SparseMatrix(R, m, n, Q) : Rng, RngIntElt, RngIntElt, [ <RngIntElt, RngIntElt, RngElt> ] -> MtrxSprs
            SparseMatrix(R, m, n) : Rng, RngIntElt, RngIntElt -> MtrxSprs
            SparseMatrix(m, n) : RngIntElt, RngIntElt -> MtrxSprs

      Construction of Trivial Sparse Matrices
            SparseMatrix(R) : Rng -> MtrxSprs
            Example SMat_Create (H28E1)

      Construction of Structured Matrices
            IdentitySparseMatrix(R, n) : Rng, RngElt -> MtrxSprs
            ScalarSparseMatrix(n, s) : RngIntElt, RngElt -> MtrxSprs
            ScalarSparseMatrix(R, n, s) : Rng, RngIntElt, RngElt -> MtrxSprs
            DiagonalSparseMatrix(R, n, Q) : Rng, RngIntElt, [ RngElt ] -> MtrxSprs
            DiagonalSparseMatrix(R, Q) : Rng, [ RngElt ] -> MtrxSprs
            DiagonalSparseMatrix(Q) : [ RngElt ] -> MtrxSprs
            SparseMonomialMatrix(g) : GrpPermElt -> MtrxSprs

      Parents of Sparse Matrices
            SparseMatrixStructure(R) : [ Rng ] -> MtrxSprsStr

 
Accessing Sparse Matrices

      Elementary Properties
            BaseRing(A) : MtrxSprs -> Rng
            NumberOfRows(A) : MtrxSprs -> RngIntElt
            NumberOfColumns(A) : MtrxSprs -> RngIntElt
            ElementToSequence(A) : MtrxSprs -> [ <RngIntElt, RngIntElt, RngElt> ]
            NumberOfNonZeroEntries(A) : MtrxSprs -> RngIntElt
            Density(A) : MtrxSprs -> FldRe
            Support(A, i) : MtrxSprs, RngIntElt -> [RngIntElt]
            Support(A) : MtrxSprs -> [ <RngIntElt, RngIntElt, RngElt> ]

      Weights
            RowWeight(A, i) : MtrxSprs, RngIntElt -> RngIntElt
            RowWeights(A) : MtrxSprs -> [RngIntElt]
            ColumnWeight(A, j) : MtrxSprs, RngIntElt -> RngIntElt
            ColumnWeights(A) : MtrxSprs -> [RngIntElt]

 
Accessing or Modifying Entries
      A[i] : MtrxSprs, RngIntElt -> ModTupRngElt
      A[i, j] : MtrxSprs, RngIntElt, RngIntElt -> RngElt
      A[i, j] := x : MtrxSprs, RngIntElt, RngIntElt, RngElt ->
      SetEntry(~A, i, j, x) : MtrxSprs, RngIntElt, RngIntElt, RngElt ->
      Example SMat_Indexing (H28E2)

      Extracting and Inserting Blocks
            Submatrix(A, i, j, p, q) : MtrxSprs, RngIntElt, RngIntElt, RngIntElt, RngIntElt -> MtrxSprs
            SubmatrixRange(A, i, j, r, s) : MtrxSprs, RngIntElt, RngIntElt, RngIntElt, RngIntElt -> MtrxSprs
            Submatrix(A, I, J) : MtrxSprs, [RngIntElt], [RngIntElt] -> MtrxSprs
            InsertBlock(A, B, i, j) : MtrxSprs, MtrxSprs, RngIntElt, RngIntElt -> MtrxSprs
            RowSubmatrix(A, i, k) : MtrxSprs, RngIntElt, RngIntElt -> MtrxSprs
            RowSubmatrix(A, i) : MtrxSprs, RngIntElt -> MtrxSprs
            RowSubmatrixRange(A, i, j) : MtrxSprs, RngIntElt, RngIntElt -> MtrxSprs
            ColumnSubmatrix(A, i, k) : MtrxSprs, RngIntElt, RngIntElt -> MtrxSprs
            ColumnSubmatrix(A, i) : MtrxSprs, RngIntElt -> MtrxSprs
            ColumnSubmatrixRange(A, i, j) : MtrxSprs, RngIntElt, RngIntElt -> MtrxSprs

      Row and Column Operations
            SwapRows(A, i, j) : MtrxSprs, RngIntElt, RngIntElt -> MtrxSprs
            SwapColumns(A, i, j) : MtrxSprs, RngIntElt, RngIntElt -> MtrxSprs
            ReverseRows(A) : MtrxSprs -> MtrxSprs
            ReverseColumns(A) : MtrxSprs -> MtrxSprs
            AddRow(A, c, i, j) : MtrxSprs, RngElt, RngIntElt, RngIntElt -> MtrxSprs
            AddColumn(A, c, i, j) : MtrxSprs, RngElt, RngIntElt, RngIntElt -> MtrxSprs
            MultiplyRow(A, c, i) : MtrxSprs, RngElt, RngIntElt -> MtrxSprs
            MultiplyColumn(A, c, i) : MtrxSprs, RngElt, RngIntElt -> MtrxSprs
            RemoveRow(A, i) : MtrxSprs, RngIntElt -> MtrxSprs
            RemoveColumn(A, j) : MtrxSprs, RngIntElt -> MtrxSprs
            RemoveRowColumn(A, i, j) : MtrxSprs, RngIntElt -> MtrxSprs
            RemoveZeroRows(A) : MtrxSprs -> MtrxSprs

 
Building Block Matrices
      HorizontalJoin(A, B) : MtrxSprs, MtrxSprs -> MtrxSprs
      VerticalJoin(A, B) : MtrxSprs, MtrxSprs -> MtrxSprs
      DiagonalJoin(A, B) : MtrxSprs, MtrxSprs -> MtrxSprs

 
Conversion to and from Dense Matrices
      Matrix(A) : MtrxSprs -> Mtrx
      SparseMatrix(A) : Mtrx -> MtrxSprs

 
Changing Ring
      ChangeRing(A, R) : MtrxSprs, Rng -> MtrxSprs

 
Predicates
      A eq B : MtrxSprs, MtrxSprs -> BoolElt
      IsZero(A) : MtrxSprs -> BoolElt
      IsOne(A) : MtrxSprs -> BoolElt
      IsMinusOne(A) : MtrxSprs -> BoolElt
      IsScalar(A) : MtrxSprs -> BoolElt
      IsDiagonal(A) : MtrxSprs -> BoolElt
      IsSymmetric(A) : MtrxSprs -> BoolElt
      IsUpperTriangular(A) : MtrxSprs -> BoolElt
      IsLowerTriangular(A) : MtrxSprs -> BoolElt

 
Elementary Arithmetic
      A + B : MtrxSprs, MtrxSprs -> MtrxSprs
      A - B : MtrxSprs, MtrxSprs -> MtrxSprs
      A * B : MtrxSprs, MtrxSprs -> MtrxSprs
      x * A : RngElt, MtrxSprs -> MtrxSprs
      -A : MtrxSprs -> MtrxSprs
      A ^-1 : MtrxSprs, RngIntElt -> MtrxSprs
      A ^ n : MtrxSprs, RngIntElt -> MtrxSprs
      Transpose(A) : MtrxSprs -> MtrxSprs

 
Multiplying Vectors or Matrices by Sparse Matrices
      v * A : ModTupRng, MtrxSprs -> ModTupRng
      MultiplyByTranspose(v, A) : ModTupRng, MtrxSprs -> ModTupRng

 
Non-trivial Properties

      Nullspace and Rowspace
            Nullspace(A) : MtrxSprs -> ModTupRng
            NullspaceMatrix(A) : MtrxSprs -> Mtrx
            NullspaceOfTranspose(A) : MtrxSprs -> ModTupRng
            Rowspace(A) : MtrxSprs -> ModTupRng

      Rank
            Rank(A) : MtrxSprs -> RngIntElt

 
Determinant and Other Properties
      Determinant(A: parameters) : MtrxSprs -> RngElt

 
Minimal and Characteristic Polynomials and Eigenvalues
      MinimalPolynomial(A: parameters) : MtrxSprs -> RngUPolElt
      CharacteristicPolynomial(A: parameters) : MtrxSprs -> RngUPolElt
      MinimalAndCharacteristicPolynomials(A: parameters) : MtrxSprs -> RngUPolElt, RngUPolElt
      FactoredMinimalPolynomial(A: parameters) : MtrxSprs -> [ <RngUPolElt, RngIntElt>]
      FactoredCharacteristicPolynomial(A: parameters) : MtrxSprs -> [ <RngUPolElt, RngIntElt>]
      FactoredMinimalAndCharacteristicPolynomials(A: parameters) : MtrxSprs -> [<RngUPolElt, RngIntElt>], [<RngUPolElt, RngIntElt>]
      Eigenvalues(A) : MtrxSprs -> { <FldElt, RngIntElt> }
      Eigenspace(A, e) : AlgMatElt, FldElt -> ModTup

      Elementary Divisors (Smith Form)
            ElementaryDivisors(A) : MtrxSprs -> [RngElt]

      Verbosity
            SetVerbose("SparseMatrix", v) : MonStgElt, RngIntElt ->

 
Linear Systems (Structured Gaussian Elimination)
      ModularSolution(A, M) : MtrxSprs, RngIntElt -> ModTupRng
      Example SMat_DiscreteLog (H28E3)

 
Bibliography

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