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This is maxima.info, produced by makeinfo version 4.1 from maxima.texi. This is a Texinfo Maxima Manual Copyright 1994,2001 William F. Schelter START-INFO-DIR-ENTRY * Maxima: (maxima). A computer algebra system. END-INFO-DIR-ENTRY File: maxima.info, Node: Definitions for Matrices and Linear Algebra, Prev: Introduction to Matrices and Linear Algebra, Up: Matrices and Linear Algebra Definitions for Matrices and Linear Algebra =========================================== - Function: ADDCOL (M,list1,list2,...,listn) appends the column(s) given by the one or more lists (or matrices) onto the matrix M. - Function: ADDROW (M,list1,list2,...,listn) appends the row(s) given by the one or more lists (or matrices) onto the matrix M. - Function: ADJOINT (matrix) computes the adjoint of a matrix. - Function: AUGCOEFMATRIX ([eq1, ...], [var1, ...]) the augmented coefficient matrix for the variables var1,... of the system of linear equations eq1,.... This is the coefficient matrix with a column adjoined for the constant terms in each equation (i.e. those not dependent upon var1,...). Do EXAMPLE(AUGCOEFMATRIX); for an example. - Function: CHARPOLY (M, var) computes the characteristic polynomial for Matrix M with respect to var. That is, DETERMINANT(M - DIAGMATRIX(LENGTH(M),var)). For examples of this command, do EXAMPLE(CHARPOLY); . - Function: COEFMATRIX ([eq1, ...], [var1, ...]) the coefficient matrix for the variables var1,... of the system of linear equations eq1,... - Function: COL (M,i) gives a matrix of the ith column of the matrix M. - Function: COLUMNVECTOR (X) a function in the EIGEN package. Do LOAD(EIGEN) to use it. COLUMNVECTOR takes a LIST as its argument and returns a column vector the components of which are the elements of the list. The first element is the first component,...etc...(This is useful if you want to use parts of the outputs of the functions in this package in matrix calculations.) - Function: CONJUGATE (X) a function in the EIGEN package on the SHARE directory. It returns the complex conjugate of its argument. This package may be loaded by LOAD(EIGEN); . For a complete description of this package, do PRINTFILE("eigen.usg"); . - Function: COPYMATRIX (M) creates a copy of the matrix M. This is the only way to make a copy aside from recreating M elementwise. Copying a matrix may be useful when SETELMX is used. - Function: DETERMINANT (M) computes the determinant of M by a method similar to Gaussian elimination. The form of the result depends upon the setting of the switch RATMX. There is a special routine for dealing with sparse determininants which can be used by setting the switches RATMX:TRUE and SPARSE:TRUE. - Variable: DETOUT default: [FALSE] if TRUE will cause the determinant of a matrix whose inverse is computed to be kept outside of the inverse. For this switch to have an effect DOALLMXOPS and DOSCMXOPS should be FALSE (see their descriptions). Alternatively this switch can be given to EV which causes the other two to be set correctly. - Function: DIAGMATRIX (n, x) returns a diagonal matrix of size n by n with the diagonal elements all x. An identity matrix is created by DIAGMATRIX(n,1), or one may use IDENT(n). - Variable: DOALLMXOPS default: [TRUE] if TRUE all operations relating to matrices are carried out. If it is FALSE then the setting of the individual DOT switches govern which operations are performed. - Variable: DOMXEXPT default: [TRUE] if TRUE, %E^MATRIX([1,2],[3,4]) ==> MATRIX([%E,%E^2],[%E^3,%E^4]) In general, this transformation affects expressions of the form <base>^<power> where <base> is an expression assumed scalar or constant, and <power> is a list or matrix. This transformation is turned off if this switch is set to FALSE. - Variable: DOMXMXOPS default: [TRUE] if TRUE then all matrix-matrix or matrix-list operations are carried out (but not scalar-matrix operations); if this switch is FALSE they are not. - Variable: DOMXNCTIMES default: [FALSE] Causes non-commutative products of matrices to be carried out. - Variable: DONTFACTOR default: [] may be set to a list of variables with respect to which factoring is not to occur. (It is initially empty). Factoring also will not take place with respect to any variables which are less important (using the variable ordering assumed for CRE form) than those on the DONTFACTOR list. - Variable: DOSCMXOPS default: [FALSE] if TRUE then scalar-matrix operations are performed. - Variable: DOSCMXPLUS default: [FALSE] if TRUE will cause SCALAR + MATRIX to give a matrix answer. This switch is not subsumed under DOALLMXOPS. - Variable: DOT0NSCSIMP default: [TRUE] Causes a non-commutative product of zero and a nonscalar term to be simplified to a commutative product. - Variable: DOT0SIMP default: [TRUE] Causes a non-commutative product of zero and a scalar term to be simplified to a commutative product. - Variable: DOT1SIMP default: [TRUE] Causes a non-commutative product of one and another term to be simplified to a commutative product. - Variable: DOTASSOC default: [TRUE] when TRUE causes (A.B).C to simplify to A.(B.C) - Variable: DOTCONSTRULES default: [TRUE] Causes a non-commutative product of a constant and another term to be simplified to a commutative product. Turning on this flag effectively turns on DOT0SIMP, DOT0NSCSIMP, and DOT1SIMP as well. - Variable: DOTDISTRIB default: [FALSE] if TRUE will cause A.(B+C) to simplify to A.B+A.C - Variable: DOTEXPTSIMP default: [TRUE] when TRUE causes A.A to simplify to A^^2 - Variable: DOTIDENT default: [1] The value to be returned by X^^0. - Variable: DOTSCRULES default: [FALSE] when TRUE will cause A.SC or SC.A to simplify to SC*A and A.(SC*B) to simplify to SC*(A.B) - Function: ECHELON (M) produces the echelon form of the matrix M. That is, M with elementary row operations performed on it such that the first non-zero element in each row in the resulting matrix is a one and the column elements under the first one in each row are all zero. [2 1 - A -5 B ] (D2) [ ] [A B C ] (C3) ECHELON(D2); [ A - 1 5 B ] [1 - ----- - --- ] [ 2 2 ] (D3) [ ] [ 2 C + 5 A B ] [0 1 ------------] [ 2 ] [ 2 B + A - A] - Function: EIGENVALUES (mat) There is a package on the SHARE; directory which contains functions for computing EIGENVALUES and EIGENVECTORS and related matrix computations. For information on it do PRINTFILE(EIGEN,USAGE,SHARE); . EIGENVALUES(mat) takes a MATRIX as its argument and returns a list of lists the first sublist of which is the list of eigenvalues of the matrix and the other sublist of which is the list of the multiplicities of the eigenvalues in the corresponding order. [ The MACSYMA function SOLVE is used here to find the roots of the characteristic polynomial of the matrix. Sometimes SOLVE may not be able to find the roots of the polynomial;in that case nothing in this package except CONJUGATE, INNERPRODUCT, UNITVECTOR, COLUMNVECTOR and GRAMSCHMIDT will work unless you know the eigenvalues. In some cases SOLVE may generate very messy eigenvalues. You may want to simplify the answers yourself before you go on. There are provisions for this and they will be explained below. ( This usually happens when SOLVE returns a not-so-obviously real expression for an eigenvalue which is supposed to be real...)] The EIGENVALUES command is available directly from MACSYMA. To use the other functions you must have loaded in the EIGEN package, either by a previous call to EIGENVALUES, or by doing LOADFILE("eigen"); . - Function: EIGENVECTORS (MAT) takes a MATRIX as its argument and returns a list of lists the first sublist of which is the output of the EIGENVALUES command and the other sublists of which are the eigenvectors of the matrix corresponding to those eigenvalues respectively. This function will work directly from MACSYMA, but if you wish to take advantage of the flags for controlling it (see below), you must first load in the EIGEN package from the SHARE; directory. You may do that by LOADFILE("eigen");. The flags that affect this function are: NONDIAGONALIZABLE[FALSE] will be set to TRUE or FALSE depending on whether the matrix is nondiagonalizable or diagonalizable after an EIGENVECTORS command is executed. HERMITIANMATRIX[FALSE] If set to TRUE will cause the degenerate eigenvectors of the hermitian matrix to be orthogonalized using the Gram-Schmidt algorithm. KNOWNEIGVALS[FALSE] If set to TRUE the EIGEN package will assume the eigenvalues of the matrix are known to the user and stored under the global name LISTEIGVALS. LISTEIGVALS should be set to a list similar to the output of the EIGENVALUES command. ( The MACSYMA function ALGSYS is used here to solve for the eigenvectors. Sometimes if the eigenvalues are messy, ALGSYS may not be able to produce a solution. In that case you are advised to try to simplify the eigenvalues by first finding them using EIGENVALUES command and then using whatever marvelous tricks you might have to reduce them to something simpler. You can then use the KNOWNEIGVALS flag to proceed further. ) - Function: EMATRIX (m, n, x, i, j) will create an m by n matrix all of whose elements are zero except for the i,j element which is x. - Function: ENTERMATRIX (m, n) allows one to enter a matrix element by element with MACSYMA requesting values for each of the m*n entries. (C1) ENTERMATRIX(3,3); Is the matrix 1. Diagonal 2. Symmetric 3. Antisymmetric 4. General Answer 1, 2, 3 or 4 1; Row 1 Column 1: A; Row 2 Column 2: B; Row 3 Column 3: C; Matrix entered. [ A 0 0 ] [ ] (D1) [ 0 B 0 ] [ ] [ 0 0 C ] - Function: GENMATRIX (array, i2, j2, i1, j1) generates a matrix from the array using array(i1,j1) for the first (upper-left) element and array(i2,j2) for the last (lower-right) element of the matrix. If j1=i1 then j1 may be omitted. If j1=i1=1 then i1 and j1 may both be omitted. If a selected element of the array doesn't exist a symbolic one will be used. (C1) H[I,J]:=1/(I+J-1)$ (C2) GENMATRIX(H,3,3); [ 1 1] [1 - -] [ 2 3] [ ] [1 1 1] (D2) [- - -] [2 3 4] [ ] [1 1 1] [- - -] [3 4 5] - Function: GRAMSCHMIDT (X) a function in the EIGEN package. Do LOAD(EIGEN) to use it. GRAMSCHMIDT takes a LIST of lists the sublists of which are of equal length and not necessarily orthogonal (with respect to the innerproduct defined above) as its argument and returns a similar list each sublist of which is orthogonal to all others. (Returned results may contain integers that are factored. This is due to the fact that the MACSYMA function FACTOR is used to simplify each substage of the Gram-Schmidt algorithm. This prevents the expressions from getting very messy and helps to reduce the sizes of the numbers that are produced along the way.) - Function: HACH (a,b,m,n,l) An implementation of Hacijan's linear programming algorithm is available by doing BATCH("kach.mc"$. Details of use are available by doing BATCH("kach.dem"); - Function: IDENT (n) produces an n by n identity matrix. - Function: INNERPRODUCT (X,Y) a function in the EIGEN package. Do LOAD(EIGEN) to use it. INNERPRODUCT takes two LISTS of equal length as its arguments and returns their inner (scalar) product defined by (Complex Conjugate of X).Y (The "dot" operation is the same as the usual one defined for vectors). - Function: INVERT (matrix) finds the inverse of a matrix using the adjoint method. This allows a user to compute the inverse of a matrix with bfloat entries or polynomials with floating pt. coefficients without converting to cre-form. The DETERMINANT command is used to compute cofactors, so if RATMX is FALSE (the default) the inverse is computed without changing the representation of the elements. The current implementation is inefficient for matrices of high order. The DETOUT flag if true keeps the determinant factored out of the inverse. Note: the results are not automatically expanded. If the matrix originally had polynomial entries, better appearing output can be generated by EXPAND(INVERT(mat)),DETOUT. If it is desirable to then divide through by the determinant this can be accomplished by XTHRU(%) or alternatively from scratch by EXPAND(ADJOINT(mat))/EXPAND(DETERMINANT(mat)). INVERT(mat):=ADJOINT(mat)/DETERMINANT(mat). See also DESCRIBE("^^"); for another method of inverting a matrix. - Variable: LMXCHAR default: [[] - The character used to display the (left) delimiter of a matrix (see also RMXCHAR). - Function: MATRIX (row1, ..., rown) defines a rectangular matrix with the indicated rows. Each row has the form of a list of expressions, e.g. [A, X**2, Y, 0] is a list of 4 elements. There are a number of MACSYMA commands which deal with matrices, for example: DETERMINANT, CHARPOLY, GENMATRIX, ADDCOL, ADDROW, COPYMATRIX, TRANSPOSE, ECHELON, and RANK. There is also a package on the SHARE directory for computing EIGENVALUES. Try DESCRIBE on these for more information. Matrix multiplication is effected by using the dot operator, ".", which is also convenient if the user wishes to represent other non-commutative algebraic operations. The exponential of the "." operation is "^^" . Thus, for a matrix A, A.A = A^^2 and, if it exists, A^^-1 is the inverse of A. The operations +,-,*,** are all element-by-element operations; all operations are normally carried out in full, including the . (dot) operation. Many switches exist for controlling simplification rules involving dot and matrix-list operations. Options Relating to Matrices: LMXCHAR, RMXCHAR, RATMX, LISTARITH, DETOUT, DOALLMXOPS, DOMXEXPT DOMXMXOPS, DOSCMXOPS, DOSCMXPLUS, SCALARMATRIX, and SPARSE. Do DESCRIBE(option) for details on them. - Function: MATRIXMAP (fn, M) will map the function fn onto each element of the matrix M. - Function: MATRIXP (exp) is TRUE if exp is a matrix else FALSE. - Variable: MATRIX_ELEMENT_ADD default: [+] - May be set to "?"; may also be the name of a function, or a LAMBDA expression. In this way, a rich variety of algebraic structures may be simulated. For more details, do DEMO("matrix.dem1"); and DEMO("matrix.dem2");. - Variable: MATRIX_ELEMENT_MULT default: [*] - May be set to "."; may also be the name of a function, or a LAMBDA expression. In this way, a rich variety of algebraic structures may be simulated. For more details, do DEMO("matrix.dem1"); and DEMO("matrix.dem2"); - Variable: MATRIX_ELEMENT_TRANSPOSE default: [FALSE] - Other useful settings are TRANSPOSE and NONSCALARS; may also be the name of a function, or a LAMBDA expression. In this way, a rich variety of algebraic structures may be simulated. For more details, do DEMO("matrix.dem1"); and DEMO("matrix.dem2");. - Function: MATTRACE (M) computes the trace [sum of the elements on the main diagonal] of the square matrix M. It is used by NCHARPOLY, an alternative to MACSYMA's CHARPOLY. It is used by doing LOADFILE("nchrpl"); - Function: MINOR (M, i, j) computes the i,j minor of the matrix M. That is, M with row i and column j removed. - Function: NCEXPT (A,B) if an (non-commutative) exponential expression is too wide to be displayed as A^^B it will appear as NCEXPT(A,B). - Function: NCHARPOLY (M,var) finds the characteristic polynomial of the matrix M with respect to var. This is an alternative to MACSYMA's CHARPOLY. NCHARPOLY works by computing traces of powers of the given matrix, which are known to be equal to sums of powers of the roots of the characteristic polynomial. From these quantities the symmetric functions of the roots can be calculated, which are nothing more than the coefficients of the characteristic polynomial. CHARPOLY works by forming the determinant of VAR * IDENT [N] - A. Thus NCHARPOLY wins, for example, in the case of large dense matrices filled with integers, since it avoids polynomial arithmetic altogether. It may be used by doing LOADFILE("nchrpl"); - Function: NEWDET (M,n) also computes the determinant of M but uses the Johnson-Gentleman tree minor algorithm. M may be the name of a matrix or array. The argument n is the order; it is optional if M is a matrix. - declaration: NONSCALAR - makes ai behave as does a list or matrix with respect to the dot operator. - Function: NONSCALARP (exp) is TRUE if exp is a non-scalar, i.e. it contains atoms declared as non-scalars, lists, or matrices. - Function: PERMANENT (M,n) computes the permanent of the matrix M. A permanent is like a determinant but with no sign changes. - Function: RANK (M) computes the rank of the matrix M. That is, the order of the largest non-singular subdeterminant of M. Caveat: RANK may return the wrong answer if it cannot determine that a matrix element that is equivalent to zero is indeed so. - Variable: RATMX default: [FALSE] - if FALSE will cause determinant and matrix addition, subtraction, and multiplication to be performed in the representation of the matrix elements and will cause the result of matrix inversion to be left in general representation. If it is TRUE, the 4 operations mentioned above will be performed in CRE form and the result of matrix inverse will be in CRE form. Note that this may cause the elements to be expanded (depending on the setting of RATFAC) which might not always be desired. - Function: ROW (M, i) gives a matrix of the ith row of matrix M. - Variable: SCALARMATRIXP default: [TRUE] - if TRUE, then whenever a 1 x 1 matrix is produced as a result of computing the dot product of matrices it will be converted to a scalar, namely the only element of the matrix. If set to ALL, then this conversion occurs whenever a 1 x 1 matrix is simplified. If set to FALSE, no conversion will be done. - Function: SETELMX (x, i, j, M) changes the i,j element of M to x. The altered matrix is returned as the value. The notation M[i,j]:x may also be used, altering M in a similar manner, but returning x as the value. - Function: SIMILARITYTRANSFORM (MAT) a function in the EIGEN package. Do LOAD(EIGEN) to use it. SIMILARITYTRANSFORM takes a MATRIX as its argument and returns a list which is the output of the UNITEIGENVECTORS command. In addition if the flag NONDIAGONALIZABLE is FALSE two global matrices LEFTMATRIX and RIGHTMATRIX will be generated. These matrices have the property that LEFTMATRIX.MAT.RIGHTMATRIX is a diagonal matrix with the eigenvalues of MAT on the diagonal. If NONDIAGONALIZABLE is TRUE these two matrices will not be generated. If the flag HERMITIANMATRIX is TRUE then LEFTMATRIX is the complex conjugate of the transpose of RIGHTMATRIX. Otherwise LEFTMATRIX is the inverse of RIGHTMATRIX. RIGHTMATRIX is the matrix the columns of which are the unit eigenvectors of MAT. The other flags (see DESCRIBE(EIGENVALUES); and DESCRIBE(EIGENVECTORS);) have the same effects since SIMILARITYTRANSFORM calls the other functions in the package in order to be able to form RIGHTMATRIX. - Variable: SPARSE default: [FALSE] - if TRUE and if RATMX:TRUE then DETERMINANT will use special routines for computing sparse determinants. - Function: SUBMATRIX (m1, ..., M, n1, ...) creates a new matrix composed of the matrix M with rows mi deleted, and columns ni deleted. - Function: TRANSPOSE (M) produces the transpose of the matrix M. - Function: TRIANGULARIZE (M) produces the upper triangular form of the matrix M which needn't be square. - Function: UNITEIGENVECTORS (MAT) a function in the EIGEN package. Do LOAD(EIGEN) to use it. UNITEIGENVECTORS takes a MATRIX as its argument and returns a list of lists the first sublist of which is the output of the EIGENVALUES command and the other sublists of which are the unit eigenvectors of the matrix corresponding to those eigenvalues respectively. The flags mentioned in the description of the EIGENVECTORS command have the same effects in this one as well. In addition there is a flag which may be useful : KNOWNEIGVECTS[FALSE] - If set to TRUE the EIGEN package will assume that the eigenvectors of the matrix are known to the user and are stored under the global name LISTEIGVECTS. LISTEIGVECTS should be set to a list similar to the output of the EIGENVECTORS command. (If KNOWNEIGVECTS is set to TRUE and the list of eigenvectors is given the setting of the flag NONDIAGONALIZABLE may not be correct. If that is the case please set it to the correct value. The author assumes that the user knows what he is doing and will not try to diagonalize a matrix the eigenvectors of which do not span the vector space of the appropriate dimension...) - Function: UNITVECTOR (X) a function in the EIGEN package. Do LOAD(EIGEN) to use it. UNITVECTOR takes a LIST as its argument and returns a unit list. (i.e. a list with unit magnitude). - Function: VECTORSIMP (vectorexpression) This function employs additional simplifications, together with various optional expansions according to the settings of the following global flags: EXPANDALL, EXPANDDOT, EXPANDDOTPLUS, EXPANDCROSS, EXPANDCROSSPLUS, EXPANDCROSSCROSS, EXPANDGRAD, EXPANDGRADPLUS, EXPANDGRADPROD, EXPANDDIV, EXPANDDIVPLUS, EXPANDDIVPROD, EXPANDCURL, EXPANDCURLPLUS, EXPANDCURLCURL, EXPANDLAPLACIAN, EXPANDLAPLACIANPLUS, EXPANDLAPLACIANPROD. All these flags have default value FALSE. The PLUS suffix refers to employing additivity or distributivity. The PROD suffix refers to the expansion for an operand that is any kind of product. EXPANDCROSSCROSS refers to replacing p~(q~r) with (p.r)*q-(p.q)*r, and EXPANDCURLCURL refers to replacing CURL CURL p with GRAD DIV p + DIV GRAD p. EXPANDCROSS:TRUE has the same effect as EXPANDCROSSPLUS:EXPANDCROSSCROSS:TRUE, etc. Two other flags, EXPANDPLUS and EXPANDPROD, have the same effect as setting all similarly suffixed flags true. When TRUE, another flag named EXPANDLAPLACIANTODIVGRAD, replaces the LAPLACIAN operator with the composition DIV GRAD. All of these flags are initially FALSE. For convenience, all of these flags have been declared EVFLAG. For orthogonal curvilinear coordinates, the global variables COORDINATES[[X,Y,Z]], DIMENSION[3], SF[[1,1,1]], and SFPROD[1] are set by the function invocation - Variable: VECT_CROSS default:[FALSE] - If TRUE allows DIFF(X~Y,T) to work where ~ is defined in SHARE;VECT (where VECT_CROSS is set to TRUE, anyway.) - Function: ZEROMATRIX (m,n) takes integers m,n as arguments and returns an m by n matrix of 0's. - special symbol: "[" - [ and ] are the characters which MACSYMA uses to delimit a list. File: maxima.info, Node: Affine, Next: Tensor, Prev: Matrices and Linear Algebra, Up: Top Affine ****** * Menu: * Definitions for Affine:: File: maxima.info, Node: Definitions for Affine, Prev: Affine, Up: Affine Definitions for Affine ====================== - Function: FAST_LINSOLVE (eqns,variables) Solves the linear system of equations EQNS for the variables VARIABLES and returns a result suitable to SUBLIS. The function is faster than linsolve for system of equations which are sparse. - Function: GROBNER_BASIS (eqns) Takes as argument a macsyma list of equations and returns a grobner basis for them. The function POLYSIMP may now be used to simplify other functions relative to the equations. GROBNER_BASIS([3*X^2+1,Y*X])$ POLYSIMP(Y^2*X+X^3*9+2)==> -3*x+2 Polysimp(f)==> 0 if and only if f is in the ideal generated by the EQNS ie. if and only if f is a polynomial combination of the elements of EQNS. - Function: SET_UP_DOT_SIMPLIFICATIONS (eqns,[check-thru-degree]) The eqns are polynomial equations in non commutative variables. The value of CURRENT_VARIABLES is the list of variables used for computing degrees. The equations must be homogeneous, in order for the procedure to terminate. If you have checked overlapping simplifications in DOT_SIMPLIFICATIONS above the degree of f, then the following is true: DOTSIMP(f)==> 0 if and only if f is in the ideal generated by the EQNS ie. if and only if f is a polynomial combination of the elements of EQNS. The degree is that returned by NC_DEGREE. This in turn is influenced by the weights of individual variables. - Function: DECLARE_WEIGHT (var1,wt1,var2,wt2,...) Assigns VAR1 weight WT1, VAR2 weight wt2.. These are the weights used in computing NC_DEGREE. - Function: NC_DEGREE (poly) Degree of a non commutative polynomial. See DECLARE_WEIGHTS. - Function: DOTSIMP (f) ==> 0 if and only if f is in the ideal generated by the EQNS ie. if and only if f is a polynomial combination of the elements of EQNS. - Function: FAST_CENTRAL_ELEMENTS (variables,degree) if SET_UP_DOT_SIMPLIFICATIONS has been previously done, finds the central polynomials in the variables in the given degree, For example: set_up_dot_simplifications([y.x+x.y],3); fast_central_elements([x,y],2); [y.y,x.x]; - Function: CHECK_OVERLAPS (degree,add-to-simps) checks the overlaps thru degree, making sure that you have sufficient simplification rules in each degree, for dotsimp to work correctly. This process can be speeded up if you know before hand what the dimension of the space of monomials is. If it is of finite global dimension, then HILBERT should be used. If you don't know the monomial dimensions, do not specify a RANK_FUNCTIION. An optional third argument RESET, false says don't bother to query about resetting things. - Function: MONO (vari,n) VARI is a list of variables. Returns the list of independent monomials relative to the current dot_simplifications, in degree N - Function: MONOMIAL_DIMENSIONS (n) Compute the hilbert series through degreen n for the current algebra. - Function: EXTRACT_LINEAR_EQUATIONS (List_nc_polys,monoms) Makes a list of the coefficients of the polynomials in list_nc_polys of the monoms. MONOMS is a list of noncommutative monomials. The coefficients should be scalars. Use LIST_NC_MONOMIALS to build the list of monoms. - Function: LIST_NC_MONOMIALS (polys_or_list) returns a list of the non commutative monomials occurring in a polynomial or a collection of polynomials. - Function: PCOEFF (poly monom [variables-to-exclude-from-cof (list-variables monom)]) This function is called from lisp level, and uses internal poly format. CL-MAXIMA>>(setq me (st-rat #$x^2*u+y+1$)) (#:Y 1 1 0 (#:X 2 (#:U 1 1) 0 1)) CL-MAXIMA>>(pcoeff me (st-rat #$x^2$)) (#:U 1 1) Rule: if a variable appears in monom it must be to the exact power, and if it is in variables to exclude it may not appear unless it was in monom to the exact power. (pcoeff pol 1 ..) will exclude variables like substituting them to be zero. - Function: NEW-DISREP (poly) From lisp this returns the general maxima format for an arg which is in st-rat form: (displa(new-disrep (setq me (st-rat #$x^2*u+y+1$)))) 2 Y + U X + 1 - Function: CREATE_LIST (form,var1,list1,var2,list2,...) Create a list by evaluating FORM with VAR1 bound to each element of LIST1, and for each such binding bind VAR2 to each element of LIST2,... The number of elements in the result will be length(list1)*length(list2)*... Each VARn must actually be a symbol-it will not be evaluated. The LISTn args will be evaluated once at the beginning of the iteration. (C82) create_list1(x^i,i,[1,3,7]); (D82) [X,X^3,X^7] With a double iteration: (C79) create_list([i,j],i,[a,b],j,[e,f,h]); (D79) [[A,E],[A,F],[A,H],[B,E],[B,F],[B,H]] Instead of LISTn two args maybe supplied each of which should evaluate to a number. These will be the inclusive lower and upper bounds for the iteration. (C81) create_list([i,j],i,[1,2,3],j,1,i); (D81) [[1,1],[2,1],[2,2],[3,1],[3,2],[3,3]] Note that the limits or list for the j variable can depend on the current value of i. - Variable: ALL_DOTSIMP_DENOMS if its value is FALSE the denominators encountered in getting dotsimps will not be collected. To collect the denoms ALL_DOTSIMP_DENOMS:[]; and they will be nconc'd onto the end of the list. File: maxima.info, Node: Tensor, Next: Ctensor, Prev: Affine, Up: Top Tensor ****** * Menu: * Introduction to Tensor:: * Definitions for Tensor:: File: maxima.info, Node: Introduction to Tensor, Next: Definitions for Tensor, Prev: Tensor, Up: Tensor Introduction to Tensor ====================== - Indicial Tensor Manipulation package. It may be loaded by LOADFILE("itensr"); A manual for the Tensor packages is available in share/tensor.descr. A demo is available by DEMO("itenso.dem1"); (and additional demos are in ("itenso.dem2"), ("itenso.dem3") and following). - There are two tensor packages in MACSYMA, CTENSR and ITENSR. CTENSR is Component Tensor Manipulation, and may be accessed with LOAD(CTENSR); . ITENSR is Indicial Tensor Manipulation, and is loaded by doing LOAD(ITENSR); A manual for CTENSR AND ITENSR is available from the LCS Publications Office. Request MIT/LCS/TM-167. In addition, demos exist on the TENSOR; directory under the filenames CTENSO DEMO1, DEMO2, etc. and ITENSO DEMO1, DEMO2, etc. Do DEMO("ctenso.dem1"); or DEMO("itenso.dem2"); Send bugs or comments to RP or TENSOR. File: maxima.info, Node: Definitions for Tensor, Prev: Introduction to Tensor, Up: Tensor Definitions for Tensor ====================== - Function: CANFORM (exp) [Tensor Package] Simplifies exp by renaming dummy indices and reordering all indices as dictated by symmetry conditions imposed on them. If ALLSYM is TRUE then all indices are assumed symmetric, otherwise symmetry information provided by DECSYM declarations will be used. The dummy indices are renamed in the same manner as in the RENAME function. When CANFORM is applied to a large expression the calculation may take a considerable amount of time. This time can be shortened by calling RENAME on the expression first. Also see the example under DECSYM. Note: CANFORM may not be able to reduce an expression completely to its simplest form although it will always return a mathematically correct result. - Function: CANTEN (exp) [Tensor Package] Simplifies exp by renaming (see RENAME) and permuting dummy indices. CANTEN is restricted to sums of tensor products in which no derivatives are present. As such it is limited and should only be used if CANFORM is not capable of carrying out the required simplification. - Function: CARG (exp) returns the argument (phase angle) of exp. Due to the conventions and restrictions, principal value cannot be guaranteed unless exp is numeric. - Variable: COUNTER default: [1] determines the numerical suffix to be used in generating the next dummy index in the tensor package. The prefix is determined by the option DUMMYX[#]. - Function: DEFCON (tensor1,<tensor2,tensor3>) gives tensor1 the property that the contraction of a product of tensor1 and tensor2 results in tensor3 with the appropriate indices. If only one argument, tensor1, is given, then the contraction of the product of tensor1 with any indexed object having the appropriate indices (say tensor) will yield an indexed object with that name, i.e.tensor, and with a new set of indices reflecting the contractions performed. For example, if METRIC: G, then DEFCON(G) will implement the raising and lowering of indices through contraction with the metric tensor. More than one DEFCON can be given for the same indexed object; the latest one given which applies in a particular contraction will be used. CONTRACTIONS is a list of those indexed objects which have been given contraction properties with DEFCON. - Function: FLUSH (exp,tensor1,tensor2,...) Tensor Package - will set to zero, in exp, all occurrences of the tensori that have no derivative indices. - Function: FLUSHD (exp,tensor1,tensor2,...) Tensor Package - will set to zero, in exp, all occurrences of the tensori that have derivative indices. - Function: FLUSHND (exp,tensor,n) Tensor Package - will set to zero, in exp, all occurrences of the differentiated object tensor that have n or more derivative indices as the following example demonstrates. (C1) SHOW(A([I],[J,R],K,R)+A([I],[J,R,S],K,R,S)); J R S J R (D1) A + A I,K R S I,K R (C2) SHOW(FLUSHND(D1,A,3)); J R (D2) A I,K R - Function: KDELTA (L1,L2) is the generalized Kronecker delta function defined in the Tensor package with L1 the list of covariant indices and L2 the list of contravariant indices. KDELTA([i],[j]) returns the ordinary Kronecker delta. The command EV(EXP,KDELTA) causes the evaluation of an expression containing KDELTA([],[]) to the dimension of the manifold. - Function: LC (L) is the permutation (or Levi-Civita) tensor which yields 1 if the list L consists of an even permutation of integers, -1 if it consists of an odd permutation, and 0 if some indices in L are repeated. - Function: LORENTZ (exp) imposes the Lorentz condition by substituting 0 for all indexed objects in exp that have a derivative index identical to a contravariant index. - Function: MAKEBOX (exp) will display exp in the same manner as SHOW; however, any tensor d'Alembertian occurring in exp will be indicated using the symbol []. For example, []P([M],[N]) represents G([],[I,J])*P([M],[N],I,J). - Function: METRIC (G) specifies the metric by assigning the variable METRIC:G; in addition, the contraction properties of the metric G are set up by executing the commands DEFCON(G), DEFCON(G,G,KDELTA). The variable METRIC, default: [], is bound to the metric, assigned by the METRIC(g) command. - Function: NTERMSG () gives the user a quick picture of the "size" of the Einstein tensor. It returns a list of pairs whose second elements give the number of terms in the components specified by the first elements. - Function: NTERMSRCI () returns a list of pairs, whose second elements give the number of terms in the RICCI component specified by the first elements. In this way, it is possible to quickly find the non-zero expressions and attempt simplification. - Function: NZETA (Z) returns the complex value of the Plasma Dispersion Function for complex Z. NZETAR(Z) ==> REALPART(NZETA(Z)) NZETAI(Z) returns IMAGPART(NZETA(Z)). This function is related to the complex error function by NZETA(Z) = %I*SQRT(%PI)*EXP(-Z^2)*(1-ERF(-%I*Z)). - Function: RAISERIEMANN (dis) returns the contravariant components of the Riemann curvature tensor as array elements UR[I,J,K,L]. These are displayed if dis is TRUE. - Variable: RATEINSTEIN default: [] - if TRUE rational simplification will be performed on the non-zero components of Einstein tensors; if FACRAT:TRUE then the components will also be factored. - Variable: RATRIEMAN - This switch has been renamed RATRIEMANN. - Variable: RATRIEMANN default: [] - one of the switches which controls simplification of Riemann tensors; if TRUE, then rational simplification will be done; if FACRAT:TRUE then each of the components will also be factored. - Function: REMCON (tensor1,tensor2,...) removes all the contraction properties from the tensori. REMCON(ALL) removes all contraction properties from all indexed objects. - Function: RICCICOM (dis) Tensor package) This function first computes the covariant components LR[i,j] of the Ricci tensor (LR is a mnemonic for "lower Ricci"). Then the mixed Ricci tensor is computed using the contravariant metric tensor. If the value of the argument to RICCICOM is TRUE, then these mixed components, RICCI[i,j] (the index i is covariant and the index j is contravariant), will be displayed directly. Otherwise, RICCICOM(FALSE) will simply compute the entries of the array RICCI[i,j] without displaying the results. - Function: RINVARIANT () Tensor package) forms the invariant obtained by contracting the tensors R[i,j,k,l]*UR[i,j,k,l]. This object is not automatically simplified since it can be very large. - Function: SCURVATURE () returns the scalar curvature (obtained by contracting the Ricci tensor) of the Riemannian manifold with the given metric. - Function: SETUP () this has been renamed to TSETUP(); Sets up a metric for Tensor calculations. - Function: WEYL (dis) computes the Weyl conformal tensor. If the argument dis is TRUE, the non-zero components W[I,J,K,L] will be displayed to the user. Otherwise, these components will simply be computed and stored. If the switch RATWEYL is set to TRUE, then the components will be rationally simplified; if FACRAT is TRUE then the results will be factored as well. File: maxima.info, Node: Ctensor, Next: Series, Prev: Tensor, Up: Top Ctensor ******* * Menu: * Introduction to Ctensor:: * Definitions for Ctensor:: File: maxima.info, Node: Introduction to Ctensor, Next: Definitions for Ctensor, Prev: Ctensor, Up: Ctensor Introduction to Ctensor ======================= - Component Tensor Manipulation Package. To use the CTENSR package, type TSETUP(); which automatically loads it from within MACSYMA (if it is not already loaded) and then prompts the user to input his coordinate system. The user is first asked to specify the dimension of the manifold. If the dimension is 2, 3 or 4 then the list of coordinates defaults to [X,Y], [X,Y,Z] or [X,Y,Z,T] respectively. These names may be changed by assigning a new list of coordinates to the variable OMEGA (described below) and the user is queried about this. ** Care must be taken to avoid the coordinate names conflicting with other object definitions **. Next, the user enters the metric either directly or from a file by specifying its ordinal position. As an example of a file of common metrics, see TENSOR;METRIC FILE. The metric is stored in the matrix LG. Finally, the metric inverse is computed and stored in the matrix UG. One has the option of carrying out all calculations in a power series. A sample protocol is begun below for the static, spherically symmetric metric (standard coordinates) which will be applied to the problem of deriving Einstein's vacuum equations (which lead to the Schwarzschild solution) as an example. Many of the functions in CTENSR will be displayed for the standard metric as examples. (C2) TSETUP(); Enter the dimension of the coordinate system: 4; Do you wish to change the coordinate names? N; Do you want to 1. Enter a new metric? 2. Enter a metric from a file? 3. Approximate a metric with a Taylor series? Enter 1, 2 or 3 1; Is the matrix 1. Diagonal 2. Symmetric 3. Antisymmetric 4. General Answer 1, 2, 3 or 4 1; Row 1 Column 1: A; Row 2 Column 2: X^2; Row 3 Column 3: X^2*SIN(Y)^2; Row 4 Column 4: -D; Matrix entered. Enter functional dependencies with the DEPENDS function or 'N' if none DEPENDS([A,D],X); Do you wish to see the metric? Y; [ A 0 0 0 ] [ ] [ 2 ] [ 0 X 0 0 ] [ ] [ 2 2 ] [ 0 0 X SIN (Y) 0 ] [ ] [ 0 0 0 - D ] Do you wish to see the metric inverse? N; File: maxima.info, Node: Definitions for Ctensor, Prev: Introduction to Ctensor, Up: Ctensor Definitions for Ctensor ======================= - Function: CHR1 ([i,j,k]) yields the Christoffel symbol of the first kind via the definition (g + g - g )/2 . ik,j jk,i ij,k To evaluate the Christoffel symbols for a particular metric, the variable METRIC must be assigned a name as in the example under CHR2. - Function: CHR2 ([i,j],[k]) yields the Christoffel symbol of the second kind defined by the relation ks CHR2([i,j],[k]) = g (g + g - g )/2 is,j js,i ij,s - Function: CHRISTOF (arg) A function in the CTENSR (Component Tensor Manipulation) package. It computes the Christoffel symbols of both kinds. The arg determines which results are to be immediately displayed. The Christoffel symbols of the first and second kinds are stored in the arrays LCS[i,j,k] and MCS[i,j,k] respectively and defined to be symmetric in the first two indices. If the argument to CHRISTOF is LCS or MCS then the unique non-zero values of LCS[i,j,k] or MCS[i,j,k], respectively, will be displayed. If the argument is ALL then the unique non-zero values of LCS[i,j,k] and MCS[i,j,k] will be displayed. If the argument is FALSE then the display of the elements will not occur. The array elements MCS[i,j,k] are defined in such a manner that the final index is contravariant. - Function: COVDIFF (exp,v1,v2,...) yields the covariant derivative of exp with respect to the variables vi in terms of the Christoffel symbols of the second kind (CHR2). In order to evaluate these, one should use EV(exp,CHR2). - Function: CURVATURE ([i,j,k],[h]) Indicial Tensor Package) yields the Riemann curvature tensor in terms of the Christoffel symbols of the second kind (CHR2). The following notation is used: h h h %1 h CURVATURE = - CHR2 - CHR2 CHR2 + CHR2 i j k i k,j %1 j i k i j,k h %1 + CHR2 CHR2 %1 k i j - Variable: DIAGMETRIC default:[] - An option in the CTENSR (Component Tensor Manipulation) package. If DIAGMETRIC is TRUE special routines compute all geometrical objects (which contain the metric tensor explicitly) by taking into consideration the diagonality of the metric. Reduced run times will, of course, result. Note: this option is set automatically by TSETUP if a diagonal metric is specified. - Variable: DIM default:[4] - An option in the CTENSR (Component Tensor Manipulation) package. DIM is the dimension of the manifold with the default 4. The command DIM:N; will reset the dimension to any other integral value. - Function: EINSTEIN (dis) A function in the CTENSR (Component Tensor Manipulation) package. EINSTEIN computes the mixed Einstein tensor after the Christoffel symbols and Ricci tensor have been obtained (with the functions CHRISTOF and RICCICOM). If the argument dis is TRUE, then the non-zero values of the mixed Einstein tensor G[i,j] will be displayed where j is the contravariant index. RATEINSTEIN[TRUE] if TRUE will cause the rational simplification on these components. If RATFAC[FALSE] is TRUE then the components will also be factored. - Function: LRICCICOM (dis) A function in the CTENSR (Component Tensor Manipulation) package. LRICCICOM computes the covariant (symmetric) components LR[i,j] of the Ricci tensor. If the argument dis is TRUE, then the non-zero components are displayed. - Function: MOTION (dis) A function in the CTENSR (Component Tensor Manipulation) package. MOTION computes the geodesic equations of motion for a given metric. They are stored in the array EM[i]. If the argument dis is TRUE then these equations are displayed. - Variable: OMEGA default:[] - An option in the CTENSR (Component Tensor Manipulation) package. OMEGA assigns a list of coordinates to the variable. While normally defined when the function TSETUP is called, one may redefine the coordinates with the assignment OMEGA:[j1,j2,...jn] where the j's are the new coordinate names. A call to OMEGA will return the coordinate name list. Also see DESCRIBE(TSETUP); . - Function: RIEMANN (dis) A function in the CTENSR (Component Tensor Manipulation) Package. RIEMANN computes the Riemann curvature tensor from the given metric and the corresponding Christoffel symbols. If dis is TRUE, the non-zero components R[i,j,k,l] will be displayed. All the indicated indices are covariant. As with the Einstein tensor, various switches set by the user control the simplification of the components of the Riemann tensor. If RATRIEMAN[TRUE] is TRUE then rational simplification will be done. If RATFAC[FALSE] is TRUE then each of the components will also be factored. - Function: TRANSFORM - The TRANSFORM command in the CTENSR package has been renamed to TTRANSFORM. - Function: TSETUP () A function in the CTENSR (Component Tensor Manipulation) package which automatically loads the CTENSR package from within MACSYMA (if it is not already loaded) and then prompts the user to make use of it. Do DESCRIBE(CTENSR); for more details. - Function: TTRANSFORM (matrix) A function in the CTENSR (Component Tensor Manipulation) package which will perform a coordinate transformation upon an arbitrary square symmetric matrix. The user must input the functions which define the transformation. (Formerly called TRANSFORM.) File: maxima.info, Node: Series, Next: Number Theory, Prev: Ctensor, Up: Top Series ****** * Menu: * Introduction to Series:: * Definitions for Series::