6.6 Arrays

6.6.1 newvect, vector, vect

newvect(len[,list])

vector(len[,list])

:: Creates a new vector object with its length len.

vect([elements])

:: Creates a new vector object by elements.

return

vector

len

non-negative integer

list

list

elements

elements of the vector

• vect creates a new vector object by its elements.
• vector is an alias of newvect.
• newvect creates a new vector object with its length len and its elements all cleared to value 0. If the second argument, a list, is given, the vector is initialized by the list elements. Elements are used from the first through the last. If the list is short for initializing the full vector, 0’s are filled in the remaining vector elements.
• Elements are indexed from 0 through len-1. Note that the first element has not index 1.
• List and vector are different types in Asir. Lists are conveniently used for representing many data objects whose size varies dynamically as computation proceeds. By its flexible expressive power, it is also conveniently used to describe initial values for other structured objects as you see for vectors. Access for an element of a list is performed by following pointers to next elements. By this, access costs for list elements differ for each element. In contrast to lists, vector elements can be accessed in a same time, because they are accessed by computing displacements from the top memory location of the vector object.

  Note also, in Asir, modification of an element of a vector causes modification of the whole vector itself, while modification of a list element does not cause the modification of the whole list object.

  By this, in Asir language, a vector element designator can be a left value of assignment statement, but a list element designator can NOT be a left value of assignment statement.

• No distinction of column vectors and row vectors in Asir. If a matrix is applied to a vector from left, the vector shall be taken as a column vector, and if from right it shall be taken as a row vector.
• The length (or size or dimension) of a vector is given by function size().
• When a vector is passed to a function as its argument (actual parameter), the vector element can be modified in that function.
• A vector is displayed in a similar format as for a list. Note, however, there is a distinction: Elements of a vector are separated simply by a ‘blank space’, while those of a list by a ‘comma.’

[0] A=newvect(5);
[ 0 0 0 0 0 ]
[1] A=newvect(5,[1,2,3,4,[5,6]]);
[ 1 2 3 4 [5,6] ]
[2] A[0];
1
[3] A[4];
[5,6]
[4] size(A);
[5]
[5] length(A);
5
[6] vect(1,2,3,4,[5,6]);
[ 1 2 3 4 [5,6] ]
[7] def afo(V) { V[0] = x; }
[8] afo(A)$
[9] A;
[ x 2 3 4 [5,6] ]

References

newmat, matrix, size, ltov, vtol.

6.6.2 ltov

ltov(list)

:: Converts a list into a vector.

return

vector

list

list

• Converts a list list into a vector of same length. See also newvect().

[3] A=[1,2,3];
[4] ltov(A);
[ 1 2 3 ]

References

newvect, vector, vect, vtol.

6.6.3 vtol

vtol(vect)

:: Converts a vector into a list.

return

list

vect

vector

• Converts a vector vect of length n into a list [vect[0],...,vect[n-1]].
• A conversion from a list to a vector is done by newvect().

[3] A=newvect(3,[1,2,3]);
[ 1 2 3 ]
[4] vtol(A);
[1,2,3]

References

newvect, vector, vect, ltov.

6.6.4 newbytearray

newbytearray(len,[listorstring])

:: Creates a new byte array.

return

byte array

len

non-negative integer

listorstring

list or string

• This function generates a byte array. The specification is similar to that of newvect.
• The initial value can be specified by a character string.
• One can access elements of a byte array just as an array.

[182] A=newbytearray(3);
|00 00 00|
[183] A=newbytearray(3,[1,2,3]);
|01 02 03|
[184] A=newbytearray(3,"abc");  
|61 62 63|
[185] A[0];
97
[186] A[1]=123;
123
[187] A;
|61 7b 63|

References

newvect, vector, vect.

6.6.5 newmat, matrix

newmat(row,col [,[[a,b,...],[c,d,...],...]])

matrix(row,col [,[[a,b,...],[c,d,...],...]])

:: Creates a new matrix with row rows and col columns.

return

matrix

row col

non-negative integer

a b c d

arbitrary

• matrix is an alias of newmat.
• If the third argument, a list, is given, the newly created matrix is initialized so that each element of the list (again a list) initializes each of the rows of the matrix. Elements are used from the first through the last. If the list is short, 0’s are filled in the remaining matrix elements. If no third argument is given all the elements are cleared to 0.
• The size of a matrix is given by function size().
• Let M be a program variable assigned to a matrix. Then, M[I] denotes a (row) vector which corresponds with the I-th row of the matrix. Note that the vector shares its element with the original matrix. Subsequently, if an element of the vector is modified, then the corresponding matrix element is also modified.
• When a matrix is passed to a function as its argument (actual parameter), the matrix element can be modified within that function.

[0] A = newmat(3,3,[[1,1,1],[x,y],[x^2]]);
[ 1 1 1 ]
[ x y 0 ]
[ x^2 0 0 ]
[1] det(A);
-y*x^2
[2] size(A);
[3,3]
[3] A[1];
[ x y 0 ]
[4] A[1][3];
getarray : Out of range
return to toplevel

References

newvect, vector, vect, size, det, nd_det, invmat.

6.6.6 mat, matr, matc

mat(vector[,...])

matr(vector[,...])

:: Creates a new matrix by list of row vectors.

matc(vector[,...])

:: Creates a new matrix by list of column vectors.

return

matrix

vector

array or list

• mat is an alias of matr.
• Each vector has same length. Elements are used from the first through the last. If the list is short, 0’s are filled in the remaining matrix elements.

[0] matr([1,2,3],[4,5,6],[7,8]);
[ 1 2 3 ]
[ 4 5 6 ]
[ 7 8 0 ]
[1] matc([1,2,3],[4,5,6],[7,8]);
[ 1 4 7 ]
[ 2 5 8 ]
[ 3 6 0 ]

References

newmat, matrix

6.6.7 size

size(vect|mat)

:: A list containing the number of elements of the given vector, [size of vect], or a list containing row size and column size of the given matrix, [row size of mat, column size of mat].

return

list

vect

vector

mat

matrix

• Return a list consisting of the dimension of the vector vect, or a list consisting of the row size and column size of the matrix matrix.
• Use length() for the size of list, and nmono() for the number of monomials with non-zero coefficients in a rational expression.

[0] A = newvect(4);
[ 0 0 0 0 ]
[1] size(A);
[4]
[2] length(A);
4
[3] B = newmat(2,3,[[1,2,3],[4,5,6]]);
[ 1 2 3 ]
[ 4 5 6 ]
[4] size(B);
[2,3]

References

car, cdr, cons, append, reverse, length, nmono.

6.6.8 det, nd_det, invmat

det(mat[,mod])

nd_det(mat[,mod])

:: Determinant of mat.

invmat(mat)

:: Inverse matrix of mat.

return

det: expression, invmat: list

mat

matrix

mod

prime

• det and nd_det compute the determinant of matrix mat. invmat computes the inverse matrix of matrix mat. invmat returns a list [num,den], where num is a matrix and num/den represents the inverse matrix.
• The computation is done over GF(mod) if mod is specitied.
• The fraction free Gaussian algorithm is employed. For matrices with multi-variate polynomial entries, minor expansion algorithm sometimes is more efficient than the fraction free Gaussian algorithm.
• nd_det can be used for computing the determinant of a matrix with polynomial entries over the rationals or finite fields. The algorithm is an improved version of the fraction free Gaussian algorithm and it computes the determinant faster than det.

[91] A=newmat(5,5)$                         
[92] V=[x,y,z,u,v];
[x,y,z,u,v]
[93] for(I=0;I<5;I++)for(J=0,B=A[I],W=V[I];J<5;J++)B[J]=W^J;
[94] A;
[ 1 x x^2 x^3 x^4 ]
[ 1 y y^2 y^3 y^4 ]
[ 1 z z^2 z^3 z^4 ]
[ 1 u u^2 u^3 u^4 ]
[ 1 v v^2 v^3 v^4 ]
[95] fctr(det(A));
[[1,1],[u-v,1],[-z+v,1],[-z+u,1],[-y+u,1],[y-v,1],[-y+z,1],[-x+u,1],
[-x+z,1],[-x+v,1],[-x+y,1]]
[96] A = newmat(3,3)$
[97] for(I=0;I<3;I++)for(J=0,B=A[I],W=V[I];J<3;J++)B[J]=W^J;
[98] A;
[ 1 x x^2 ]
[ 1 y y^2 ]
[ 1 z z^2 ]
[99] invmat(A);
[[ -z*y^2+z^2*y z*x^2-z^2*x -y*x^2+y^2*x ]
[ y^2-z^2 -x^2+z^2 x^2-y^2 ]
[ -y+z x-z -x+y ],(-y+z)*x^2+(y^2-z^2)*x-z*y^2+z^2*y]
[100] A*B[0];
[ (-y+z)*x^2+(y^2-z^2)*x-z*y^2+z^2*y 0 0 ]
[ 0 (-y+z)*x^2+(y^2-z^2)*x-z*y^2+z^2*y 0 ]
[ 0 0 (-y+z)*x^2+(y^2-z^2)*x-z*y^2+z^2*y ]
[101] map(red,A*B[0]/B[1]);
[ 1 0 0 ]
[ 0 1 0 ]
[ 0 0 1 ]

References

newmat, matrix.

6.6.9 qsort

qsort(array[,func])

:: Sorts an array array.

return

array (The same as the input; Only the elements are exchanged.)

array

array

func

function for comparison

• This function sorts an array by quick sort.
• If func is not specified, the built-in comparison function is used and the array is sorted in increasing order.
• If a function of two arguments func which returns 0, 1, or -1 is provided, then an ordering is determined so that A<B if func(A,B)=1 holds, and the array is sorted in increasing order with respect to the ordering.
• The returned array is the same as the input. Only the elements are exchanged.

[0] qsort(newvect(10,[1,4,6,7,3,2,9,6,0,-1]));
[ -1 0 1 2 3 4 6 6 7 9 ]
[1] def rev(A,B) { return A>B?-1:(A<B?1:0); }
[2] qsort(newvect(10,[1,4,6,7,3,2,9,6,0,-1]),rev);
[ 9 7 6 6 4 3 2 1 0 -1 ]

References

ord, vars.

6.6.10 rowx, rowm, rowa, colx, colm, cola

rowx(matrix,i,j)

:: Exchanges the i-th and j-th rows.

rowm(matrix,i,c)

:: Multiplies the i-th row by c.

rowa(matrix,i,c)

:: Appends c times the j-th row to the j-th row.

colx(matrix,i,j)

:: Exchanges the i-th and j-th columns.

colm(matrix,i,c)

:: Multiplies the i-th column by c.

cola(matrix,i,c)

:: Appends c times the j-th column to the j-th column.

return

matrix

i, j

integers

c

coefficient

• These operations are destructive for the matrix.

[0] A=newmat(3,3,[[1,2,3],[4,5,6],[7,8,9]]);
[ 1 2 3 ]
[ 4 5 6 ]
[ 7 8 9 ]
[1] rowx(A,1,2)$
[2] A;
[ 1 2 3 ]
[ 7 8 9 ]
[ 4 5 6 ]
[3] rowm(A,2,x);
[ 1 2 3 ]
[ 7 8 9 ]
[ 4*x 5*x 6*x ]
[4] rowa(A,0,1,z);
[ 7*z+1 8*z+2 9*z+3 ]
[ 7 8 9 ]
[ 4*x 5*x 6*x ]

References

newmat, matrix

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