クラス
class	MGCurveParameter
	Utility class to compute a curve parameter defined by f(t)=0. [詳解]

関数
template<class func >
double	mgDefint (func &f, double a, double b, double eps, int l=0)
	Integrate f(x)(l=0) or F(x)(l=1, see below) over finite interval (a,b). The DE formula (double exponential formula) is applied. Note Original Fortran program codes are from DEFINT of the book "Fortran77による数値計算ソフトウェア" by M.Sugihara, M.Mori, published by Maruzen K.K. [詳解]

template<class func >
double	mgGausp (func &f, double a, double b, int n=10)
	Legendre-Gauss quadratuer formula over (a,b) . The DE formula (double exponential formula) is applied. Note Original Fortran program codes are from DDGAUSP of the book "Fortran77による数値計算ソフトウェア" by M.Sugihara, M.Mori, published by Maruzen K.K. [詳解]

void	factorizeLU (MGMatrix &W, int *id)
	LU factorization to solve linear equations, general version. This is a general solver using pivotting, and so is inefficient compared with band matrix of diagonally dominant equations. If the equation is known as special equations as band matrix (as of diagonally dominant equations, or as Symetric Tridiagonal), use factorizeBandLU and solveBandLU, factorizeCholeLU and solveCholeLU, or solveSymetricTridiagonal. [詳解]

void	solveLU (const MGMatrix &W, const int *id, const MGBPointSeq &A, MGBPointSeq &X)
	Solve the linear equation W*X=A to get X, inputting factorize W and A. [詳解]

void	solveSymetricTridiagonal (MGBPointSeq &W, const MGBPointSeq &A, MGBPointSeq &X)
	Solve the linear equation M*X=A where M is a symetric tridiagonal matrix of order n. [詳解]

int	factorizeBandLU (MGBPointSeq &W, int nlower)
	factorizeBandLU executes the LU-factorization without pivoting of a banded matrix M. [詳解]

void	solveBandLU (const MGBPointSeq &W, int nlower, const MGBPointSeq &A, MGBPointSeq &X)
	solveBandLU returns the solution of the linear system M*X = A. [詳解]

void	factorizeCholeLU (MGBPointSeq &W)
	Constructs Cholesky factorization. [詳解]

void	solveCholeLU (const MGBPointSeq &W, const MGBPointSeq &A, MGBPointSeq &X)
	Solves the linear system C*X = A , provided W contains the Cholesky factorization. [詳解]

template<class func >
double	mgNlbit (func &fn, double xl, double xr, double eps, int itr, int &ier)
	Compute the solution of fn(x)=0. mgNlbit can be applied when known there exists a solution between (xl, xr). SOLUTION OF A NON-LINEAR EQUATION BY BISECTION METHOD AND REGULA FALSI METHOD. [詳解]

詳解

関数詳解

int factorizeBandLU	(	MGBPointSeq &	W,
		int	nlower
	)

factorizeBandLU executes the LU-factorization without pivoting of a banded matrix M.

M's order is n with (nlower + 1 + nupper) bands of diagonals stored in W. GAUSS Elimination W I T H O U T pivoting is used. The routine is intended for use with matrices M which do not require row interchanges durnig factorization, especially for the TOTALLY POSITIVE matrices which occur in spline calculations. The routine should not be used for arbitrary banded matrices. For arbitary ones, use factorizeLU and solveLU. On function's return, W contains the LU factorization of the matix M. Function's return value is: =0:SUCCESS, =1:FAILURE.

NOTE: This program is translated to C++ from BANFAC of the book "A Practical Guide to Splines" by Carl de Boor, Springer-Verlag.

引数

W

contains interesting part of the matrix M. W.length()=nlower+1+nupper=nband(band width of M), and W.sdim()=n(order of the matrix M). The diagonals(or bands) of M(i+j,j) are stored in W as W(i+nupper,j) for i=-nupper,..., nlower, and j=0,..., n-1. Explicitly, M has nlower bands below the diagonal and nupper bands above the diagonal. Thus the band width nband=nlower+1+nupper(=W.length()). For exapmle, the interesting entries of M of order 9, whose nlower=1 and nupper=2 would appear in the 4 subscripts of W(i, j) as follows:

j=           0  1  2  3  4  5  6  7  8:
i=0:         x  x 02 13 24 35 46 57 68,
i=1:         x 01 12 23 34 45 56 67 78,
i=2:        00 11 22 33 44 55 66 77 88,
i=3:        10 21 32 43 54 65 76 87  x.

All other entries of W not identified in this way with an entry of M are never referenced.

引数

nlower number of bands of the matrix M below the main diagonal.

void factorizeCholeLU ( MGBPointSeq & W )

Constructs Cholesky factorization.

Let C is a symmetric positive semidefinite and banded matrix, having nbands diagonals at and below the main diagonal. Then C can be factorized as C = L*D*L-transpose with L, unit lower triangular and D, the diagonal of C.

****** M E T H O D ****** Gauss elimination , adapted to the symmetry and bandedness of C, is used. Near zero pivots are handled in a special way. The diagonal element C(i,i) = W(0,i) is saved initially in wrok array diag(i), for all i. At the i-th elimination step, the currrent pivot element, viz., W(0,i), is compared with its original value, diag(i). If, as the result of prior elimination steps, this element has been reduced by about a word length, (i.e., if W(0,i)+dang(i) <= diag(i)), then the pivot is declared to be zero. and the entire i-th row is declared to be linearly dependent on the prededing rows. This has the effect of producing X(i)=0 when solving C*X=B for X, regardless of B. Justification for this is as follows. In contemplated applications of this program, the given equations are the normal equations for some least-squares approximation problem, diag(i)=C(i,i) gives the norm-square of the i-th basis function, and, at this point, W(0,i) contains the nrom-square of the error in the least-squares approximation to the i-th basis function by linear combinations of the first i-1. Having W(0,i)+diag(i) <= diag(i) signifies that the n-th function is linearly dependent to machine accuracy on the first i-1 funcitons, therefore can safely be left out from the basis of approximationg functions. The solution of a linear system C*X=B is effected by the succession of the following two calls: factorizeCholeLU to get factorization, and solveCholeLU to solve for X.

NOTE: This program is translated to C++ from BCHFAC of the book "A Practical Guide to Splines" by Carl de Boor, Springer-Verlag.

引数

W

contains nbands diagonals in W(.,j), with the main diagonal in W(.,0). Precisely, W(i,j) contains C(i+j,j), i=0,...,nbands-1, j=0,...,n-1. W.length()=nbands, and W.sdim()=n is the order of the matrix C. For example, the entries of a 7 diagonal symmetric matrix C of order 9 (nbands=4) would be stored in W(i,j) as follows:

        j=          0  1  2  3  4  5  6  7  8:
      i=0:         00 11 22 33 44 55 66 77 88,
      i=1:         10 21 32 43 54 65 76 87  x,
      i=2:         20 31 42 53 64 75 86  x  x,       
      i=3:         30 41 52 63 74 85  x  x  x.

All other entries of W not identified in this way with an entry of C are never referenced. On return, W contains the Cholesky factorization C= L*D*L-transpose with W(0,j) containing 1/D(j,j) for j=0,..., n-1. And W(i,j) containing L(i+j,j) for i=1,..., nbands-1, and j=0, ..., n-i-1.

void factorizeLU	(	MGMatrix &	W,
		int *	id
	)

LU factorization to solve linear equations, general version. This is a general solver using pivotting, and so is inefficient compared with band matrix of diagonally dominant equations. If the equation is known as special equations as band matrix (as of diagonally dominant equations, or as Symetric Tridiagonal), use factorizeBandLU and solveBandLU, factorizeCholeLU and solveCholeLU, or solveSymetricTridiagonal.

factorizeLU factorizes W to LU in the linear equation W*X=A. Using output W, the solution of W*X=A is obtained by solveLU. Here, L is n by n lower triangular matrix and U is n by n upper triangular matrix. U's diagonal is (1). LU factorization using pivot.

引数

W	left-hand side matrix of n*n is input, and factorized LU matrix will be output where n=A.length(), left-bottom triangle including the diagonal will contain L, right-upper triangle excluding the diagonal will contain U.
id	array of int id[n], pivot id will be output in id[i] for i=0,...,n-1

template<class func >

double mgDefint	(	func &	f,
		double	a,
		double	b,
		double	eps,
		int	l = `0`
	)

Integrate f(x)(l=0) or F(x)(l=1, see below) over finite interval (a,b). The DE formula (double exponential formula) is applied. Note Original Fortran program codes are from DEFINT of the book "Fortran77による数値計算ソフトウェア" by M.Sugihara, M.Mori, published by Maruzen K.K.

戻り値

integral of f(x)(when l=0) or F(x)(when l=1).

引数

f	Function object for integrand. f(x) returns the target integration function's value.
a	lower bound of integration
b	upper bound of integration
eps	absolute error tolerance
l	0 or 1 If l=0 then integrate f(x) over (a,b) If l=1 then integrate F(x) over (a,b) In case of l=1 f must be defined as follows: IF (-c .LT. y .LT. 0) THEN f(y) = F(b - y) IF ( 0 .LT. y .LE. c) THEN f(y) = F(a - y) where c = (b - a) / 2.

template<class func >

double mgGausp	(	func &	f,
		double	a,
		double	b,
		int	n = `10`
	)

Legendre-Gauss quadratuer formula over (a,b) . The DE formula (double exponential formula) is applied. Note Original Fortran program codes are from DDGAUSP of the book "Fortran77による数値計算ソフトウェア" by M.Sugihara, M.Mori, published by Maruzen K.K.

戻り値

result of integration.

引数

f	Function object to evaluate the function f(t). f(t) must return double value.
a	lower bound of integration.
b	upper bound of integration.
n	number of points of the formula, must not be larger than 16.

template<class func >

double mgNlbit	(	func &	fn,
		double	xl,
		double	xr,
		double	eps,
		int	itr,
		int &	ier
	)

Compute the solution of fn(x)=0. mgNlbit can be applied when known there exists a solution between (xl, xr). SOLUTION OF A NON-LINEAR EQUATION BY BISECTION METHOD AND REGULA FALSI METHOD.

戻り値

the	solution x obtained.

引数

fn	Function object to evaluate the fucntion, must return double value.
xl	LEFT HAND SIDE VALUE OF THE SOLUTION.
xr	RIGHT HAND SIDE VALUE OF THE SOLUTION .
eps	tolerance allowed for the convergence in world coordnate.
itr	MAXIMUM NUMBER OF REPITITION.
ier	Error code. =0:solution successfully obtained, =1:mgNlbit did not converg, and solution not obtained.

void solveBandLU	(	const MGBPointSeq &	W,
		int	nlower,
		const MGBPointSeq &	A,
		MGBPointSeq &	X
	)

solveBandLU returns the solution of the linear system M*X = A.

solveBandLU is the companion routine of factorizeBandLU, and factorizeBandLU executes the LU-Factorization for M in W.

NOTE: This program is translated to C++ from BANSLV of the book "A Practical Guide to Splines" by Carl de Boor, Springer-Verlag.

引数

W	factorized LU matrix is input which is obtained by factorizeBandLU. Left-bottom triangle including the diagonal contain L, right-upper triangle excluding the diagonal contain U. Refer to factorizeBandLU.
nlower	number of bands of the matrix M below the main diagonal.
A	right hand side vector. A.length() is W.sdim() =n(order of the matrix M).
X	solved X will be output. X.length() will be A.length(), and X.sdim() will be A.sdim().

void solveCholeLU	(	const MGBPointSeq &	W,
		const MGBPointSeq &	A,
		MGBPointSeq &	X
	)

Solves the linear system C*X = A , provided W contains the Cholesky factorization.

The Cholesky factorization is obtained by factorizeCholeLU. See factorizeCholeLU.

NOTE: This program is translated to C++ from BCHSLV of the book "A Practical Guide to Splines" by Carl de Boor, Springer-Verlag.

引数

W	contains the Cholesky factorization for C, can be obtained by factorizeCholeLU. Refer to factorizeCholeLU.
A	right hand side vector. A.length() is W.sdim()=n(order of the matrix C).
X	solved X will be output. X.length() will be A.length(), and X.sdim() will be A.sdim().

void solveLU	(	const MGMatrix &	W,
		const int *	id,
		const MGBPointSeq &	A,
		MGBPointSeq &	X
	)

Solve the linear equation W*X=A to get X, inputting factorize W and A.

W and id are obtained by factorizeLU.

引数

W	factorized LU matrix is input which is obtained by factorizeLU. left-bottom triangle including the diagonal contain L, right-upper triangle excluding the diagonal contain U.
id	array of int id[n] which contain pivot id in id[i] for i=0,...,n-1, this is the output of factorizeLU.
A	right hand side vector. A.length() is W.sdim().
X	solved X will be output. X.length() will be A.length(), and X.sdim() will be A.sdim().

void solveSymetricTridiagonal	(	MGBPointSeq &	W,
		const MGBPointSeq &	A,
		MGBPointSeq &	X
	)

Solve the linear equation M*X=A where M is a symetric tridiagonal matrix of order n.

引数

W	the symetric tridiagonal matrix M is input. three bands of the matrix at and above the diagonal are input, W(i,0)=M(i,i), W(i,1)=M(i,i+1) and M(i+1,i), W(i,2)=M(i,i+2) and M(i+2,i) for i=0,...,n-1, Factorized matrix will be output.
A	right hand side vector. A.length() is W.length().
X	solved X will be output. X.length() will be A.length(), and X.sdim() will be A.sdim().

クラス

関数

詳解

関数詳解