(* ---------------------------------------------------------------------------- * $Id: LINALGRN.md,v 1.3 1995/11/05 09:21:58 kredel Exp $ * ---------------------------------------------------------------------------- * This file is part of MAS. * ---------------------------------------------------------------------------- * Copyright (c) 1989 - 1992 Universitaet Passau * ---------------------------------------------------------------------------- * $Log: LINALGRN.md,v $ * Revision 1.3 1995/11/05 09:21:58 kredel * Fixed comments. * * Revision 1.2 1992/02/12 17:33:10 pesch * Moved CONST definition to the right place * * Revision 1.1 1992/01/22 15:12:33 kredel * Initial revision * * ---------------------------------------------------------------------------- *) DEFINITION MODULE LINALGRN; (* MAS Linear Algebra Rational Number Definition Module. *) (*------------------------------------------------------------------------- Definition Module: Linear Algebra Rational Number Programmierpraktikum SS 1990: JUERGEN MUELLER, Heinz Kredel Contents: Standartoperations for matrix and vector manipulations, Gaussian elimination, LU-Decomposition, Solve, Inversion, Nullspace, Determinant. Stand : 27.09.90, Juergen Mueller from then date of the file, Heinz Kredel Remarks: A vector is represented as a list of elements. The elements may be integers or rational numbers. A matix is represented as a list of vectors. In most circumstances these vectors are interpreted row vectors, but they can also be interpreted as column vectors. --------------------------------------------------------------------------*) FROM MASSTOR IMPORT LIST; (* --------------- arbitrary matrices ------------- *)CONSTrcsid = "$Id: LINALGRN.md,v 1.3 1995/11/05 09:21:58 kredel Exp $";CONSTcopyright = "Copyright (c) 1989 - 1992 Universitaet Passau";PROCEDURE MDIM(M : LIST): LIST; (*Matrix dimension. M is a matrix. MDIM returns max( row, column) of M. *)PROCEDURE MGET(M, k, l : LIST): LIST; (*Matrix get. M is a matrix. k, l are integers, 0 le k le rows(M), 0 le l le columns(M). MGET returns the element M(k,l) of matrix M. *)PROCEDURE MSET(M, k, l, x : LIST): LIST; (*Matrix set. M is a matrix. k, l are integers, 0 le k le rows(M), 0 le l le columns(M). MSET sets the element M(k,l) to x. The new matrix is returned. *)PROCEDURE VDELEL(V, i : LIST): LIST; (*Vector delete element. V is a vector. The i-th element of V is deleted. 0 le i le length(V). *)PROCEDURE MDELCOL(M, i : LIST): LIST; (*Matrix delete column. M is a vector of row vectors. In each row the i-th element is deleted, 0 le i le columns(M). The new matrix is returned. *)PROCEDURE MMINOR(M, i, j : LIST): LIST; (*Matrix minor. M is a vector of row vectors. The i-th column, 0 le i le rows(M), and in each remaining row the j-th element, 0 le j le columns(M), is deleted. *)PROCEDURE MTRANS(M : LIST): LIST; (*Matrix transpose. M is a matrix. The transposed matrix is returned. *)PROCEDURE VEL(a, n : LIST): LIST; (*Vector elements. A vector of length n with elements a is returned. *)PROCEDURE MFILL(M, m, n: LIST): LIST; (*Matrix fill. M is an upper triangular matrix. A (m x n) matrix with zeros in the lower triangular part is returned. *)PROCEDURE MRANG(U: LIST): LIST; (*Matrix rang. U is an upper triangular matrix from a LU-decomposition. The rang of U is returned. *) (* --------------- rational number matrices ------------- *)PROCEDURE RNMHILBERT(m, n : LIST): LIST; (*Rational number matrix Hilbert. m, n integer. A (m x n) rational number Hilbert matrix is returned. *)PROCEDURE RNUM(m, n : LIST): LIST; (*Rational number unit matrix. m, n integer. A (m x n) rational number unit matrix is returned. *)PROCEDURE RNVWRITE(A, s : LIST); (*Rational number vector write. A is a rational number vector. A is written to the output stream. The rational numbers are written as rational numbers if s = -1, as decimal approximations, with s decimal digits if s >= 0 or in floating point format if s < -1. *)PROCEDURE RNVREAD(): LIST; (*Rational number vector read. A rational number vector is read from the input stream, and returned. *)PROCEDURE RNMWRITE(A, s : LIST); (*Rational number matrix write. A is a rational number matrix. A is written to the output stream. The rational numbers are written as rational numbers if s = -1, as decimal approximations, with s decimal digits if s >= 0 or in floating point format if s < -1. *)PROCEDURE RNMREAD(): LIST; (*Rational number matrix read. A rational number matrix is read from the input stream, and returned. *)PROCEDURE RNVFIV(A : LIST): LIST; (*Rational number vector from integer vector. A is an integer vector. A rational number vector with denominators 1 and nominators equal to the elements of A is returned. *)PROCEDURE RNMFIM(M : LIST): LIST; (*Rational number matrix from integer matrix. A is an integer matrix. A rational number matrix with denominators 1 and nominators equal to the elements of A is returned. *)PROCEDURE RNVDIF(A, B : LIST): LIST; (*Rational number vector difference. A and B are rational number vectors. The rational number vector C = A - B is returned. *)PROCEDURE RNVQ(A, B : LIST): LIST; (*Rational number vector quotient. A and B are rational number vectors. The rational number vector C = A / FIRST(B) is returned. *)PROCEDURE RNVQF(A : LIST): LIST; (*Rational number vector quotient. A is a rational number vector. The rational number vector C = A / FIRST(A) is returned. *)PROCEDURE RNVVSUM(A, B : LIST): LIST; (*Rational number vector vector sum. A and B are rational number vectors. A rational number vector C = A + B is returned. *)PROCEDURE RNVSVSUM(A, B : LIST): LIST; (*Rational number vector scalar sum. A and B are rational number vectors. A rational number vector C = A + FIRST(B) is returned. *)PROCEDURE RNVSSUM(A : LIST): LIST; (*Rational number vector scalar sum. A is a rational number vector. A rational number C = a1 + a2 + ... + an is returned. *)PROCEDURE RNVSVPROD(A, B : LIST): LIST; (*Rational number vector scalar vector product. A and B are rational number vectors. A rational number vector C = (a1*FIRST(B), ..., an*FIRST(B)) is returned. *)PROCEDURE RNVVPROD(A, B : LIST): LIST; (*Rational number vector vector product. A and B are rational number vectors. A rational number vector C = (a1*b1, ..., an*bn) is returned. *)PROCEDURE RNVSPROD(A, B : LIST): LIST; (*Rational number vector scalar product. A and B are rational number vectors. A rational number C = a1*b1 + ... + an*bn is returned. *)PROCEDURE RNVMAX(M : LIST): LIST; (*Rational number vector maximum norm. M is a rational number vector. A rational number a = maximum absolute value M(i) is returned. *)PROCEDURE RNVLC(a, A, b, B : LIST): LIST; (*Rational number vector linear combination. A and B are rational number vectors. a and b are rational numbers. A rational number vector C = a*A + b*B is returned. *)PROCEDURE RNSVPROD(a, A : LIST): LIST; (*Rational number vector product with scalar. A is a rational number vector. a is a rational number. A rational number vector C = a*A is returned. *)PROCEDURE RNMSUM(A, B : LIST): LIST; (*Rational number matrix sum. A and B are rational number matrices. A rational number matrix C = A + B is returned. *)PROCEDURE RNMDIF(A, B : LIST): LIST; (*Rational number matrix difference. A and B are rational number matrices. A rational number matrix C = A - B is returned. *)PROCEDURE RNMPROD(A, B : LIST): LIST; (*Rational number matrix product. A and B are rational number matrices. A rational number matrix C = A * B is returned, if the number of coloums of A is equal to the number of rows of B, otherwise the empty matrix is returned. *)PROCEDURE RNSMPROD(A, B : LIST): LIST; (*Rational number scalar and matrix product. A is a rational number matrix. B is a rational number. A rational number matrix C = A * B is returned. *)PROCEDURE RNMMAX(M : LIST): LIST; (*Rational number matrix maximum norm. M is a rational number matrix. A rational number a = maximum absolute value M(i,j) is returned. *)PROCEDURE RNMGE(M : LIST): LIST; (*Rational number matrix Gaussian elimination. M is a (n x m) rational number matrix. A (n x m) rational number matrix resulting from Gaussian elimination is returned. RNMGELUD is called. *)PROCEDURE RNMDET(M : LIST): LIST; (*Rational number matrix determinant, using Gaussian elimination. M is a rational number matrix. The determinant of M is returned. *)PROCEDURE RNMDETL(M : LIST): LIST; (*Rational number matrix determinant, using Laplace expansion. M is a rational number matrix. The determinant of M is returned. *)PROCEDURE RNMGELUD(M : LIST;VARL, U: LIST); (*Rational number matrix Gaussian elimination LU-decomposition. M is a rational number matrix represented rowwise. L is a lower triangular rational number matrix represented columnwise. U is an upper triangular rational number matrix represented rowwise. M = L * U for appropriate modifications of L and U. The pivot operations are also recorded in L. *)PROCEDURE RNMLT(L, b : LIST): LIST; (*Rational matrix lower triangular matrix transformation. L is a lower triangular rational number matrix represented columnwise as generated by RNMGELUD. b is a rational number vector. A rational number vector u = L * b is returned, such that if M * x = b and M = L * U, then U * x = u. *)PROCEDURE RNMUT(U, b : LIST): LIST; (*Rational matrix upper triangular matrix transformation. U is an upper triangular rational number matrix represented rowwise as generated by RNMGELUD. b is a rational number vector b = L * b' as generated by RNMLT. A rational number vector x, such that U * x = b is returned. If no such x exists, then an empty vector is returned. If more than one such x exists, then for free x(i), x(i) = 0 is taken. *)PROCEDURE RNMSDS(L, U, b : LIST): LIST; (*Rational number matrix solve decomposed system. L is a lower triangular rational number matrix represented columnwise, U is an upper triangular rational number matrix represented rowwise. L and U as generated by RNMGELUD. If M = L * U, then a rational number vector x, such that M * x = b is returned. If no such x exists, then an empty vector is returned. If more than one such x exists, then for free x(i), x(i) = 0 is taken. *)PROCEDURE RNMINV(A : LIST): LIST; (*Rational number matrix inversion. A is a rational number matrix represented rowwise. If it exists, the inverse matrix of A is returned, otherwise an empty matrix is returned. *)PROCEDURE RNMUNS(U : LIST): LIST; (*Rational number matrix upper triangular matrix solution null space. U is an upper triangular rational number matrix represented rowwise as generated by RNMGELUD. A matrix X of linear independent rational number vectors x is returned, such that for each x in X, U * x = 0 holds. If only x = 0 satisfies the condition U * x = 0, then the matrix X is empty. *)ENDLINALGRN. (* -EOF- *)