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75 lines
2.2 KiB
Plaintext
75 lines
2.2 KiB
Plaintext
NAME
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det - determinant
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SYNOPSIS
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det(m)
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TYPES
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m square matrix with elements of suitable type
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return zero or value of type determined by types of elements
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DESCRIPTION
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The matrix m has to be square, i.e. of dimension 2 with:
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matmax(m,1) - matmin(m,1) == matmax(m,2) - matmin(m,2).
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If the elements of m are numbers (real or complex), det(m)
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returns the value of the determinant of m.
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If some or all of the elements of m are not numbers, the algorithm
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used to evaluate det(m) assumes the definitions of *, unary -, binary -,
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being zero or nonzero, are consistent with commutative ring structure,
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and if the m is larger than 2 x 2, division by nonzero elements is
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consistent with integral-domain structure.
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If m is a 2 x 2 matrix with elements a, b, c, d, where a tests as
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nonzero, det(m) is evaluated by
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det(m) = (a * d) - (c * b).
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If a tests as zero, det(m) = - ((c * b) - (a * d)) is used.
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If m is 3 * 3 with elements a, b, c, d, e, f, g, h, i, where a and
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a * e - d * b test as nonzero, det(m) is evaluated by
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det(m) = ((a * e - d * b) * (a * i - g * c)
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- (a * h - g * b) * (a * f - d * c))/a.
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EXAMPLE
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> mat A[3,3] = {2, 3, 5, 7, 11, 13, 17, 19, 23}
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> c = config("mode", "frac")
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> print det(A), det(A^2), det(A^3), det(A^-1)
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-78 6084 -474552 -1/78
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> obj res {r}
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> global md
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> define res_test(a) = !ismult(a.r, md)
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> define res_sub(a,b) {local obj res v = {(a.r - b.r) % md}; return v;}
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> define res_mul(a,b) {local obj res v = {(a.r * b.r) % md}; return v;}
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> define res_neg(a) {local obj res v = {(-a.r) % md}; return v;}
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> define res(x) {local obj res v = {x % md}; return v;}
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> md = 0
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> mat A[2,2] = {res(2), res(3), res(5), res(7)}
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> md = 5
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> print det(A)
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obj res {4}
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> md = 6
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> print det(A)
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obj res {5}
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Note that if A had been a 3 x 3 or larger matrix, res_div(a,b) for
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non-zero b would have had to be defined (assuming at least one
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division is necessary); for consistent results when md is composite,
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res_div(a,b) should be defined only when b and md are relatively
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prime; there is no problem when md is prime.
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LIMITS
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none
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LIBRARY
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VALUE matdet(MATRIX *m)
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SEE ALSO
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matdim, matmax, matmin, inverse
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