Files
file	AlgMatrix.c
	Matrix allocation and maintenance functions.

file	AlgMatrixCG.c
	Conjugate Gradient iterative method with preconditioning for the solution of linear systems with the form \(\mathbf{A} \mathbf{x} = \mathbf{b}\). A must be a symmetric postive definite matrix, i.e. \({\mathbf{x}}^T \mathbf{A} \mathbf{x} < 0\), \(\forall \mathbf{x} \not= \mathbf{0}\), \(\mathbf{x} \in R^n\).

file	AlgMatrixGauss.c
	Provides a function for solving matrix equations of the form: A.x = b for x using Gaussian elimination with partial pivoting.

file	AlgMatrixLSQR.c
	Provides functions for solving matrix equations using LSQR. This software is based on lsqr.c, a C version of LSQR derived by James W. Howse jhows.nosp@m.e@la.nosp@m.nl.go.nosp@m.v from the Fortran 77 implementation of C. C. Paige and M. A. Saunders. In most cases the extensive comments from Howse's lsqr.c have been preserved with little change.

file	AlgMatrixLU.c
	Provides functions for solving matrix equations of the form: A.x = b for x, inverting a matrix, calculating the determinant of a matrix and performing LU decomposition.

file	AlgMatrixMath.c
	Functions for basic arithmatic with matricies.

file	AlgMatrixRSEigen.c
	Functions to find the eigenvalues and eigenvectors of a real symmetric matrix.

file	AlgMatrixRSTDiag.c
	Reduces a real symmetric matrix to symmetric tridiagonal form by orthogonal similarity transformation and construction of the right operator of the reduction.

file	AlgMatrixSV.c
	Provides functions for singular value decomposition.

file	AlgMatrixTDiagQLI.c
	Determines the eigenvalues and eigenvectors of a real symmetric tridiagonal matrix using implicit shifts.

Functions
AlgMatrix	AlgMatrixNew (AlgMatrixType aType, size_t nR, size_t nC, size_t nE, double tol, AlgError *dstErr)
	Allocates a new matrix or the requested type. More...

AlgMatrixRect *	AlgMatrixRectNew (size_t nR, size_t nC, AlgError *dstErr)
	Allocates a new rectangular matrix. More...

AlgMatrixSym *	AlgMatrixSymNew (size_t nN, AlgError *dstErr)
	Allocates a new symmetric matrix. More...

AlgMatrixLLR *	AlgMatrixLLRNew (size_t nR, size_t nC, size_t nE, double tol, AlgError *dstErr)
	Allocates a new linked list row matrix. More...

AlgMatrixLLRE *	AlgMatrixLLRENew (AlgMatrixLLR *mat)
	Gets a linked list row matrix entry from the matrix. The return value may be NULL if none are available and the entries need to be expanded sing AlgMatrixLLRExpand(). More...

void	AlgMatrixLLREFree (AlgMatrixLLR mat, AlgMatrixLLRE p)
	Returns a linked list row matrix entry to the free stack of the matrix. More...

AlgError	AlgMatrixWriteAscii (AlgMatrix mat, FILE *fP)
	Writes a matrix in numeric ASCI format to the given file file. The rows are on separate lines and the columns of each row are white space seperated. More...

AlgMatrix	AlgMatrixReadAscii (AlgMatrixType mType, double tol, FILE fP, const char fSep, size_t recMax, AlgError *dstErr)
	Reads a matrix of the requested type from an ASCII file. If the type is ALG_MATRIX_SYM then only the first half of each row will be used, although all must be given. More...

AlgError	AlgMatrixSet (AlgMatrix mat, size_t row, size_t col, double val)

AlgError	AlgMatrixLLRCopyInPlace (AlgMatrixLLR aM, AlgMatrixLLR bM)
	Copy the second LLR matrix to the first. More...

AlgError	AlgMatrixLLRSet (AlgMatrixLLR *mat, size_t row, size_t col, double val)
	This can have one of three actions: More...

double	AlgMatrixValue (AlgMatrix mat, size_t row, size_t col)
	Returns the value in the matrix at the given coordinates. More...

double	AlgMatrixLLRValue (AlgMatrixLLR *mat, size_t row, size_t col)
	Returns the value in the matrix at the given coordinates. More...

AlgError	AlgMatrixLLRExpand (AlgMatrixLLR *mat, size_t nE)
	Ensures that there are at least the requested number of free linked list row matrix entries available. More...

void	AlgMatrixZero (AlgMatrix mat)
	Sets all elements of the matrix to zero. More...

void	AlgMatrixRectZero (AlgMatrixRect *mat)
	Sets all elements of the matrix to zero. More...

void	AlgMatrixSymZero (AlgMatrixSym *mat)
	Sets all elements of the matrix to zero. More...

void	AlgMatrixLLRZero (AlgMatrixLLR *mat)
	Sets all elements of the matrix to zero. More...

AlgError	AlgMatrixSetAll (AlgMatrix mat, double val)
	Sets all elements of the given matrix to the given value. More...

void	AlgMatrixRectSetAll (AlgMatrixRect *mat, double val)
	Sets all elements of the given rectangular matrix to the given value. More...

void	AlgMatrixSymSetAll (AlgMatrixSym *mat, double val)
	Sets all elements of the given symmetric matrix to the given value. More...

AlgError	AlgMatrixLLRSetAll (AlgMatrixLLR *mat, double val)
	Sets all elements of the given linked list row matrix to the given value. This is not an efficient use of a linked list row matrix since all values will be allocate and set. More...

void	AlgMatrixLLRERemove (AlgMatrixLLR *mat, size_t row, size_t col)
	Removes the entry at the given coordinates. More...

AlgError	AlgMatrixCGSolve (AlgMatrix aM, double xV, double bV, AlgMatrix wM, void(pFn)(void , AlgMatrix, double , double ), void pDat, double tol, int maxItr, double dstTol, int *dstItr)
	Conjugate Gradient iterative method with preconditioning for the solution of linear systems with the form \(\mathbf{A} \mathbf{x} = \mathbf{b}\). \(\mathbf{A}\) must be a symmetric postive definite matrix, i.e. \({\mathbf{x}}^T \mathbf{A} \mathbf{x} < 0\), \(\forall \mathbf{x} \not= \mathbf{0}\), \(\mathbf{x} \in R^n\). Convergence is tested using: \( \frac{\\| \mathbf{b} - mathbf{A} \mathbf{x} \\|} {\\|\mathbf{b}\\|} < \delta\). If the preconditioning function pFn is non NULL then it is called, passing the preconditioning data pDat, as: (pFn)(void pDat, double *aM, double r, double *z, int n) to solve \(\mathbf{A} \mathbf{z} = \mathbf{r}\) for \(\mathbf{z}\) with the solution overwriting the initial contents of z. More...

AlgError	AlgMatrixGaussSolve (AlgMatrix aMat, double *xMat)
	Solves the matrix equation A.x = b for x by Gaussian elimination with partial pivoting. Matrix A is the matrix of coefficients. More...

AlgError	AlgMatrixSolveLSQR (AlgMatrix aM, double bV, double xV, double damping, double relErrA, double relErrB, long maxItr, long condLim, int dstTerm, long dstItr, double dstFNorm, double dstCondN, double dstResNorm, double dstResNormA, double *dstNormX)
	This function finds a solution x to the following problems: More...

AlgError	AlgMatrixLUSolveRaw3 (double *aM, double bV, int bSz)
	Solves the matrix equation A.x = b for x, where A is a 3x3 libAlc double array matrix. On return the matrix A is overwritten with its LU decomposition and the column vector b is overwritten with the solution column matrix x. More...

AlgError	AlgMatrixLUSolveRaw4 (double *aM, double bV, int bSz)
	Solves the matrix equation A.x = b for x, where A is a 4x4 libAlc double array matrix. On return the matrix A is overwritten with its LU decomposition and the column vector b is overwritten with the solution column matrix x. More...

AlgError	AlgMatrixLUSolve (AlgMatrix aM, double *bV, int bSz)
	Solves the matrix equation A.x = b for x, where A is a square matrix. On return the matrix A is overwritten with its LU decomposition and the column vector b is overwritten with the solution column matrix x. More...

AlgError	AlgMatrixLUSolveRaw (double *aM, int aSz, double bV, int bSz)
	Solves the matrix equation A.x = b for x, where A is a square libAlc double array matrix. On return the matrix A is overwritten with its LU decomposition and the column vector b is overwritten with the solution column matrix x. More...

AlgError	AlgMatrixLUInvertRaw3 (double **aM)
	Calculates the inverse of a 3x3 libAlc double array matrix. More...

AlgError	AlgMatrixLUInvertRaw4 (double **aM)
	Calculates the inverse of a 4x4 libAlc double array matrix. More...

AlgError	AlgMatrixLUInvert (AlgMatrix aM)
	Calculates the inverse of a square matrix. More...

AlgError	AlgMatrixLUInvertRaw (double **aM, int aSz)
	Calculates the inverse of a square matrix. More...

AlgError	AlgMatrixLUDetermRaw3 (double *aM, double det)
	Calculates the determinant of a 3x3 double libAlc array matrix. The matrix is overwitten with its LU decomposition on return. More...

AlgError	AlgMatrixLUDetermRaw4 (double *aM, double det)
	Calculates the determinant of a 4x4 double libAlc array matrix. The matrix is overwitten with its LU decomposition on return. More...

AlgError	AlgMatrixLUDeterm (AlgMatrix aM, double *det)
	Calculates the determinant of a square matrix. The matrix is overwitten with its LU decomposition on return. More...

AlgError	AlgMatrixLUDetermRaw (double *aM, int aSz, double det)
	Calculates the determinant of a square double libAlc array matrix. The matrix is overwitten with its LU decomposition on return. More...

AlgError	AlgMatrixLUDecomp (AlgMatrix aM, int iV, double evenOdd)
	Replaces the given matrix with the LU decomposition of a row-wise permutation of itself. The given index vector is used to record the permutation effected by the partial pivoting. More...

AlgError	AlgMatrixLUDecompRaw (double *aM, int aSz, int iV, double *evenOdd)
	Replaces the given matrix with the LU decomposition of a row-wise permutation of itself. The given index vector is used to record the permutation effected by the partial pivoting. More...

AlgError	AlgMatrixLUBackSub (AlgMatrix aM, int iV, double bV)
	Solves the set of of linear equations A.x = b where A is input as its LU decomposition determined with AlgMatrixLUDecomp() of a row-wise permutation of itself. The given index vector is used to record the permutation effected by the partial pivoting. More...

AlgError	AlgMatrixLUBackSubRaw (double *aM, int aSz, int iV, double *bV)
	Solves the set of of linear equations A.x = b where A is input as its LU decomposition determined with AlgMatrixLUDecompRaw() of a row-wise permutation of itself. The given index vector is used to record the permutation effected by the partial pivoting. More...

void	AlgMatrixAdd (AlgMatrix aM, AlgMatrix bM, AlgMatrix cM)
	Computes the sum of two matricies and returns the result in a third supplied matrix: \[ \mathbf{A} = \mathbf{B} + \mathbf{C} \] The dimensions of the all three matricies must be nR, nC. It is safe to supply the same matrix as any combination of aM, bM and cM. More...

void	AlgMatrixSub (AlgMatrix aM, AlgMatrix bM, AlgMatrix cM)
	Subtracts on matrix from another and returns the result in a third supplied matrix: \[ \mathbf{A} = \mathbf{B} - \mathbf{C} \] The dimensions of the all three matricies must be nR, nC. It is safe to supply the same matrix as any combination of aM, bM and cM. More...

void	AlgMatrixMul (AlgMatrix aM, AlgMatrix bM, AlgMatrix cM)
	Computes the product of two matricies and returns the result in a third supplied matrix: \[ \mathbf{A} = \mathbf{B} \mathbf{C} \] The dimensions of the result matrix (aM) must be cR, bC. All the matrices must be valid and of the same type except in the case of multiplying tow symmetric matrices when the result matrix must be rectangular (because in general the product of two symmetric matrices is not symmetric). More...

double	AlgMatrixTrace (AlgMatrix aM)
	Computes the trace of the given matrix. More...

void	AlgMatrixTranspose (AlgMatrix aM, AlgMatrix bM)
	Computes the transpose of the given matrix: \[ \mathbf{A} = \mathbf{B^T} \] aM = transpose(bM). The dimensions of the result matrix (aM) must be aR == bC, aC == bR. More...

void	AlgMatrixCopy (AlgMatrix aM, AlgMatrix bM)
	Copies the values of the matrix bM to the result matric aM: \[ \mathbf{A} = \mathbf{B} \] . More...

void	AlgMatrixScale (AlgMatrix aM, AlgMatrix bM, double sv)
	Multiplies the given matrix by the given scalar: \[ \mathbf{A} = s \mathbf{B} \] . More...

void	AlgMatrixScaleAdd (AlgMatrix aM, AlgMatrix bM, AlgMatrix cM, double sv)
	Multiplies the a matrix by a scalar and then adds another matrix: \[ \mathbf{A} = \mathbf{B} + s \mathbf{C}. \] All the matrices must have the same dimensions. More...

void	AlgMatrixScalar (AlgMatrix aM, double sv)
	Sets the elements of the given square matrix so that it is a scalar matrix: \[ \mathbf{A} = s \mathbf{I} \] . More...

void	AlgMatrixVectorMul (double aV, AlgMatrix bM, double cV)
	Multiplies the matrix \(\mathbf{B}\) by the vector \(\mathbf{c}\): \[ \mathbf{a} = \mathbf{B} \mathbf{c} \] . More...

void	AlgMatrixVectorMulAdd (double aV, AlgMatrix bM, double cV, double *dV)
	Multiplies the matrix \(\mathbf{B}\) by the vector \(\mathbf{c}\) and adds the vector \(\mathbf{d}\): \[ \mathbf{a} = \mathbf{B} \mathbf{c} + \mathbf{d} \] . More...

void	AlgMatrixVectorMulWAdd (double aV, AlgMatrix bM, double cV, double *dV, double s, double t)
	Multiplies the matrix \(\mathbf{B}\) by the vector \(\mathbf{c}\) and adds the vector \(\mathbf{d}\) using the given weights \(s\) and \(t\): \[ \mathbf{a} = s \mathbf{B} \mathbf{c} + t \mathbf{d} \] . More...

void	AlgMatrixTVectorMul (double aV, AlgMatrix bM, double cV)
	Multiplies the transpose of matrix \(\mathbf{B}\) by the vector \(\mathbf{c}\): \[ \mathbf{a} = \mathbf{B}^T \mathbf{c} \] . More...

void	AlgMatrixTVectorMulAdd (double aV, AlgMatrix bM, double cV, double *dV)
	Multiplies the transpose of matrix \(\mathbf{B}\) by the vector \(\mathbf{c}\): \[ \mathbf{a} = \mathbf{B}^T \mathbf{c} + \mathbf{d} \] . More...

AlgError	AlgMatrixRawInv2x2 (double a00, double a01, double a10, double a11)
	Inverts a raw 2 x 2 matrix. All matrix values are passed using pointers and their values are set to those of the inverse matrix on return. If the given matrix is singular the ALG_ERR_MATRIX_SINGULAR error code is returned and no values are changed. More...

AlgError	AlgMatrixRawInv3x3 (double a00, double a01, double a02, double a10, double a11, double a12, double a20, double a21, double *a22)
	Inverts a raw 2 x 2 matrix. All matrix values are passed using pointers and their values are set to those of the inverse matrix on return. If the given matrix is singular the ALG_ERR_MATRIX_SINGULAR error code is returned and no values are changed. More...

AlgError	AlgMatrixRSEigen (AlgMatrix aM, double *vM, int reqEV)
	Determines the eigenvalues and eigenvectors of a real symmetric matrix by calling AlgMatrixRSTDiag() to create a tridiagonal symmetric matrix and then AlgMatrixTDiagQLI() to compute its eigenvalues and eigenvectors. The eigenvectors and eigenvalues are returned in descending eigenvalue order. For efficiency, the eigenvectors should only be computed if required. More...

AlgError	AlgMatrixRSTDiag (AlgMatrix aMat, double dVec, double oVec)
	An implementation of Householder's alorithm which reduces a real aSz x aSz symmetric matrix to symmetric tridiagonal form by aSz - 2 orthogonal similarity transformations. This is a translation of the functions/subroutines 'tred2' in the fortran EISPACK library and the Numerical Recipies book, both of these are descended from the algol procedure tred2 in Wilkinson J.H and Reinsch C. Linear Algebra, Volume II Handbook for Automatic Computation. Springer-Verlag, 1971. See Numerical Recipies in C: The Art of Scientific Computing. Cambridge University Press, 1992. More...

AlgError	AlgMatrixSVSolve (AlgMatrix aMat, double bVec, double tol, int dstIC)
	Solves the matrix equation A.x = b for x, where A is a matrix with at least as many rows as columns. On return the matrix A is overwritten by the matrix U in the singular value decomposition: A = U.W.V'. More...

AlgError	AlgMatrixSVDecomp (AlgMatrix aMat, double *wVec, AlgMatrix vMat)
	Performs singular value decomposition of the given matrix ( \(A\) ) and computes two additional matricies ( \(U\) ) and ( \(V\) ) such that: \[ A = U W V^T \] where \(V^T\) is the transpose of \(V\). The code for AlgMatrixSVDecomp was derived from: Numerical Recipies function svdcmp(), EISPACK subroutine SVD(), CERN subroutine SVD() and ACM algorithm 358. The singular values ( \(W\) ) and singular vectors ( \(V\) ) are returned unsorted. See AlgMatrixSVSolve() for a usage example. More...

AlgError	AlgMatrixSVBackSub (AlgMatrix uMat, double wVec, AlgMatrix vMat, double bVec)
	Solves the set of of linear equations A.x = b where A is input as its singular value decomposition in the three matricies U, W and V, as returned by AlgMatrixSVDecomp(). The code for AlgMatrixSVBackSub was derived from: Numerical Recipies function svbksb(). More...

AlgError	AlgMatrixTDiagQLI (double dM, double oM, int aSz, AlgMatrix zM)
	Determines the eigenvalues and eigenvectors of a real symmetric tridiagonal matrix using the QL algorithm with implicit shifts. This is a based on the function 'tqli' in Numerical Recipies in C: The Art of Scientific Computiung. Cambridge University Press, 1992. More...

Detailed Description

Function Documentation

AlgMatrix AlgMatrixNew	(	AlgMatrixType	aType,
		size_t	nR,
		size_t	nC,
		size_t	nE,
		double	tol,
		AlgError *	dstErr
	)

Allocates a new matrix or the requested type.

Returns: New matrix or matrix with core NULL on error.

Parameters

aType	Matrix type.
nR	Number of rows.
nC	Number of columns.
nE	Number of list entries to allocate, only used for linked list row matrices, may be zero.
tol	Matrix tollerance value, only used for linked list row matrices.
dstErr	Destination error pointer, may be NULL.

References ALG_ERR_MATRIX_TYPE, ALG_ERR_NONE, ALG_MATRIX_LLR, ALG_MATRIX_RECT, ALG_MATRIX_SYM, AlgMatrixLLRNew(), AlgMatrixRectNew(), AlgMatrixSymNew(), _AlgMatrix::core, _AlgMatrix::llr, _AlgMatrix::rect, and _AlgMatrix::sym.

Referenced by WlzKrigReallocBuffers2D(), and WlzKrigReallocBuffers3D().

AlgMatrixRect* AlgMatrixRectNew	(	size_t	nR,
		size_t	nC,
		AlgError *	dstErr
	)

Allocates a new rectangular matrix.

Returns: New rectangular matrix or NULL on error.

Parameters

nR	Number of rows.
nC	Number of columns.
dstErr	Destination error pointer, may be NULL.

References AlcCalloc(), AlcDouble2Calloc(), AlcFree(), ALG_ERR_MALLOC, ALG_ERR_NONE, ALG_MATRIX_RECT, _AlgMatrixRect::array, _AlgMatrixRect::maxC, _AlgMatrixRect::maxR, _AlgMatrixRect::nC, _AlgMatrixRect::nR, and _AlgMatrixRect::type.

Referenced by AlgMatrixNew(), AlgMatrixSVSolve(), AlgPolynomialLSq(), WlzAffineTransformLSqDQ2D(), WlzAffineTransformLSqDQ3D(), WlzAffineTransformLSqGen2D(), WlzAffineTransformLSqGen3D(), WlzAffineTransformLSqReg2D(), WlzAffineTransformLSqReg3D(), WlzBasisFnConf2DFromCPts(), WlzBasisFnGauss2DFromCPts(), WlzBasisFnIMQ2DFromCPts(), WlzBasisFnIMQ3DFromCPts(), WlzBasisFnMQ2DFromCPts(), WlzBasisFnMQ3DFromCPts(), WlzBasisFnPoly2DFromCPts(), WlzBasisFnScalarMOS3DFromCPts(), WlzBasisFnTPS2DFromCPts(), WlzCMeshCompSurfMap(), WlzCMeshExpansion(), WlzCMeshExpValues(), WlzDGTensorPDFeature(), WlzDGTensorSDFeature(), WlzFitPlaneSVD(), WlzGeomCurvature(), and WlzGeometryLSqOPlane().

AlgMatrixSym* AlgMatrixSymNew	(	size_t	nN,
		AlgError *	dstErr
	)

Allocates a new symmetric matrix.

Returns: New symmetric matrix or NULL on error.

Parameters

nN	Number of rows and columns.
dstErr	Destination error pointer, may be NULL.

References AlcCalloc(), AlcFree(), AlcMalloc(), ALG_ERR_MALLOC, ALG_ERR_NONE, ALG_MATRIX_SYM, _AlgMatrixSym::array, _AlgMatrixSym::maxN, _AlgMatrixSym::nC, _AlgMatrixSym::nR, and _AlgMatrixSym::type.

Referenced by AlgMatrixNew(), and AlgMatrixReadAscii().

AlgMatrixLLR* AlgMatrixLLRNew	(	size_t	nR,
		size_t	nC,
		size_t	nE,
		double	tol,
		AlgError *	dstErr
	)

Allocates a new linked list row matrix.

Returns: New linked list row matrix or NULL on error.

Parameters

nR	Number of rows.
nC	Number of columns.
nE	Number of list entries to allocate, may be zero.
tol	Matrix tollerance value.
dstErr	Destination error pointer, may be NULL.

References AlcCalloc(), ALG_ERR_MALLOC, ALG_ERR_NONE, ALG_MATRIX_LLR, AlgMatrixLLRExpand(), AlgMatrixLLRFree(), _AlgMatrixLLR::nC, _AlgMatrixLLR::nR, _AlgMatrixLLR::tbl, and _AlgMatrixLLR::type.

Referenced by AlgMatrixNew(), AlgMatrixReadAscii(), and WlzCMeshCompSurfMap().

AlgMatrixLLRE* AlgMatrixLLRENew ( AlgMatrixLLR * mat )

Gets a linked list row matrix entry from the matrix. The return value may be NULL if none are available and the entries need to be expanded sing AlgMatrixLLRExpand().

Returns: Linked list row matrix entry or NULL if none available.

Note: Call this function rather than manipulating the free entries directly.

Parameters

mat	Linked list row matrix.

References _AlgMatrixLLR::freeStk, _AlgMatrixLLR::numEnt, and _AlgMatrixLLRE::nxt.

Referenced by AlgMatrixLLRCopyInPlace(), AlgMatrixLLRSet(), AlgMatrixReadAscii(), and AlgMatrixScalar().

void AlgMatrixLLREFree	(	AlgMatrixLLR *	mat,
		AlgMatrixLLRE *	p
	)

Returns a linked list row matrix entry to the free stack of the matrix.

Note: Call this function rather than manipulating the free entries directly.

Parameters

mat	Linked list row matrix.
p	Linked list row matrix entry.

References _AlgMatrixLLR::freeStk, _AlgMatrixLLR::numEnt, and _AlgMatrixLLRE::nxt.

Referenced by AlgMatrixLLRERemove(), and AlgMatrixLLRExpand().

AlgError AlgMatrixWriteAscii	(	AlgMatrix	mat,
		FILE *	fP
	)

Writes a matrix in numeric ASCI format to the given file file. The rows are on separate lines and the columns of each row are white space seperated.

Returns: Alg error code.

Parameters

mat	Given matrix.
fP	Output file pointer.

References ALG_ERR_MATRIX_TYPE, ALG_ERR_NONE, ALG_ERR_WRITE, ALG_MATRIX_LLR, ALG_MATRIX_RECT, ALG_MATRIX_SYM, _AlgMatrixRect::array, _AlgMatrixSym::array, _AlgMatrixLLRE::col, _AlgMatrix::core, _AlgMatrix::llr, _AlgMatrixRect::nC, _AlgMatrixSym::nC, _AlgMatrixLLR::nC, _AlgMatrixRect::nR, _AlgMatrixSym::nR, _AlgMatrixLLR::nR, _AlgMatrixLLRE::nxt, _AlgMatrix::rect, _AlgMatrix::sym, _AlgMatrixLLR::tbl, _AlgMatrixCore::type, and _AlgMatrixLLRE::val.

Referenced by WlzBasisFnScalarMOS3DFromCPts(), and WlzCMeshCompSurfMap().

AlgMatrix AlgMatrixReadAscii	(	AlgMatrixType	mType,
		double	tol,
		FILE *	fP,
		const char *	fSep,
		size_t	recMax,
		AlgError *	dstErr
	)

Reads a matrix of the requested type from an ASCII file. If the type is ALG_MATRIX_SYM then only the first half of each row will be used, although all must be given.

Returns: New matrix read from the file. Matrix core member will be NULL on error.

Parameters

mType	Required type of matrix.
tol	Tolerance value for ALG_MATRIX_LLR matrices (unused for other types).
fP	Input file pointer.
fSep	Field separator string containing possible field separator characters.
recMax	Maximum number of characters per input record.
dstErr	Destination error pointer, may be NULL.

References ALC_ER_ALLOC, ALC_ER_NONE, ALC_ER_READ, AlcCalloc(), AlcVecReadDouble2Asci(), AlcVectorFree(), AlcVectorItemGet(), AlcVectorSetArray1D(), AlcVectorToArray2D(), ALG_ERR_FUNC, ALG_ERR_MALLOC, ALG_ERR_MATRIX_TYPE, ALG_ERR_NONE, ALG_ERR_READ, ALG_MATRIX_LLR, ALG_MATRIX_RECT, ALG_MATRIX_SYM, AlgMatrixFree(), AlgMatrixLLRENew(), AlgMatrixLLRExpand(), AlgMatrixLLRNew(), AlgMatrixSymNew(), _AlgMatrixRect::array, _AlgMatrixSym::array, _AlgMatrixLLRE::col, _AlgMatrix::core, _AlgMatrix::llr, _AlgMatrixRect::nC, _AlgMatrixRect::nR, _AlgMatrixLLRE::nxt, _AlgMatrix::rect, _AlgMatrix::sym, _AlgMatrixLLR::tbl, _AlgMatrixRect::type, and _AlgMatrixLLRE::val.

AlgError AlgMatrixSet	(	AlgMatrix	mat,
		size_t	row,
		size_t	col,
		double	val
	)

Returns

Alg error code.

            Errors can occur because of a memory allocation
            failure in ALG_MATRIX_LLR matrices or an invalid
            matrix type.

Parameters

mat	Given matrix.
row	Row coordinate.
col	Column coordinate.
val	Matrix value.

References ALG_ERR_NONE, ALG_MATRIX_LLR, ALG_MATRIX_RECT, ALG_MATRIX_SYM, AlgMatrixLLRSet(), _AlgMatrixRect::array, _AlgMatrixSym::array, _AlgMatrix::core, _AlgMatrix::llr, _AlgMatrix::rect, _AlgMatrix::sym, and _AlgMatrixCore::type.

Referenced by WlzCMeshCompSurfMap().

AlgError AlgMatrixLLRCopyInPlace	(	AlgMatrixLLR *	aM,
		AlgMatrixLLR *	bM
	)

Copy the second LLR matrix to the first.

Returns: Alg error code.

Parameters

aM	First given matrix.
bM	Second given matrix.

References ALG_ERR_NONE, AlgMatrixLLRENew(), AlgMatrixLLRExpand(), AlgMatrixLLRZero(), _AlgMatrixLLRE::col, _AlgMatrixLLR::nR, _AlgMatrixLLR::numEnt, _AlgMatrixLLRE::nxt, _AlgMatrixLLR::tbl, and _AlgMatrixLLRE::val.

Referenced by AlgMatrixCopy().

AlgError AlgMatrixLLRSet	(	AlgMatrixLLR *	mat,
		size_t	row,
		size_t	col,
		double	val
	)

This can have one of three actions:

Returns

Alg error code.

            1. If no value exists at the given coordinates then
            the new value is added with a new entry.
            2. If a value exists at the given coordinate and the
            given value is non-zero then the entry value is
            replaced with the given value.
            3. If a value exists at the given coordinate and the
            given value is zero then the corresponding entry is
            removed.
            Errors can only occur because of a memory allocation
            failure. These can be avoided by preallocating
            sufficient entries using AlgMatrixLLRExpand().

Parameters

mat	Linked list row matrix.
row	Row coordinate.
col	Column coordinate.
val	Matrix value.

References ALG_ERR_NONE, AlgMatrixLLRENew(), AlgMatrixLLRERemove(), AlgMatrixLLRExpand(), _AlgMatrixLLRE::col, _AlgMatrixLLR::freeStk, _AlgMatrixLLR::tol, and _AlgMatrixLLRE::val.

Referenced by AlgMatrixAdd(), AlgMatrixLLRSetAll(), AlgMatrixMul(), AlgMatrixScale(), AlgMatrixScaleAdd(), AlgMatrixSet(), AlgMatrixSub(), and AlgMatrixTranspose().

double AlgMatrixValue	(	AlgMatrix	mat,
		size_t	row,
		size_t	col
	)

Returns the value in the matrix at the given coordinates.

Returns: Value in matrix at given coordinates.

Parameters

mat	Given matrix.
row	Given row.
col	Given column.

References ALG_MATRIX_LLR, ALG_MATRIX_RECT, ALG_MATRIX_SYM, AlgMatrixLLRValue(), _AlgMatrixSym::array, _AlgMatrix::core, _AlgMatrix::llr, _AlgMatrix::sym, and _AlgMatrixCore::type.

double AlgMatrixLLRValue	(	AlgMatrixLLR *	mat,
		size_t	row,
		size_t	col
	)

Returns the value in the matrix at the given coordinates.

Returns: Value in matrix at given coordinates.

Parameters

mat	Linked list row matrix.
row	Given row.
col	Given column.

References _AlgMatrixLLRE::col, _AlgMatrixLLRE::nxt, _AlgMatrixLLR::tbl, and _AlgMatrixLLRE::val.

Referenced by AlgMatrixMul(), AlgMatrixTrace(), and AlgMatrixValue().

AlgError AlgMatrixLLRExpand	(	AlgMatrixLLR *	mat,
		size_t	nE
	)

Ensures that there are at least the requested number of free linked list row matrix entries available.

Returns: Alg error code.

Parameters

mat	Linked list row matrix.
nE	Requested minimum number of free entries or if zero a default number of entries are added.

References AlcCalloc(), AlcFree(), AlcFreeStackPush(), ALG_ERR_MALLOC, ALG_ERR_NONE, AlgMatrixLLREFree(), _AlgMatrixLLR::blk, _AlgMatrixLLRE::col, _AlgMatrixLLR::freeStk, _AlgMatrixLLR::maxEnt, _AlgMatrixLLR::numEnt, _AlgMatrixLLRE::nxt, _AlgMatrixLLR::tbl, and _AlgMatrixLLRE::val.

Referenced by AlgMatrixLLRCopyInPlace(), AlgMatrixLLRNew(), AlgMatrixLLRSet(), AlgMatrixLLRSetAll(), AlgMatrixReadAscii(), and AlgMatrixScalar().

void AlgMatrixZero ( AlgMatrix mat )

Sets all elements of the matrix to zero.

Parameters

mat	Given matrix.

References ALG_MATRIX_LLR, ALG_MATRIX_RECT, ALG_MATRIX_SYM, AlgMatrixLLRZero(), AlgMatrixRectZero(), AlgMatrixSymZero(), _AlgMatrix::core, _AlgMatrix::llr, _AlgMatrix::rect, _AlgMatrix::sym, and _AlgMatrixCore::type.

Referenced by AlgMatrixCGSolve(), and WlzGeometryLSqOPlane().

void AlgMatrixRectZero ( AlgMatrixRect * mat )

Sets all elements of the matrix to zero.

Parameters

mat	Given rectangular matrix.

References _AlgMatrixRect::array, _AlgMatrixRect::nC, and _AlgMatrixRect::nR.

Referenced by AlgMatrixScalar(), and AlgMatrixZero().

void AlgMatrixSymZero ( AlgMatrixSym * mat )

Sets all elements of the matrix to zero.

Parameters

mat	Given symmetric matrix.

References _AlgMatrixSym::array, and _AlgMatrixSym::nR.

Referenced by AlgMatrixScalar(), and AlgMatrixZero().

void AlgMatrixLLRZero ( AlgMatrixLLR * mat )

Sets all elements of the matrix to zero.

Parameters

mat	Given linked list row matrix.

References _AlgMatrixLLR::freeStk, _AlgMatrixLLR::nR, _AlgMatrixLLR::numEnt, _AlgMatrixLLRE::nxt, and _AlgMatrixLLR::tbl.

Referenced by AlgMatrixAdd(), AlgMatrixLLRCopyInPlace(), AlgMatrixLLRSetAll(), AlgMatrixScalar(), AlgMatrixScaleAdd(), AlgMatrixSub(), AlgMatrixTranspose(), and AlgMatrixZero().

AlgError AlgMatrixSetAll	(	AlgMatrix	mat,
		double	val
	)

Sets all elements of the given matrix to the given value.

Returns: Alg error code. An error may occur if the matrix type is not appropriate (ie not one of ALG_MATRIX_RECT, ALG_MATRIX_SYM or ALG_MATRIX_LLR) or if the elements of a ALG_MATRIX_LLR matrix can not be allocated.

Parameters

mat	Given matrix.
val	Given value.

References ALG_ERR_MATRIX_TYPE, ALG_ERR_NONE, ALG_MATRIX_LLR, ALG_MATRIX_RECT, ALG_MATRIX_SYM, AlgMatrixLLRSetAll(), AlgMatrixRectSetAll(), AlgMatrixSymSetAll(), _AlgMatrix::core, _AlgMatrix::llr, _AlgMatrix::rect, _AlgMatrix::sym, and _AlgMatrixCore::type.

Referenced by WlzCMeshCompSurfMap().

void AlgMatrixRectSetAll	(	AlgMatrixRect *	mat,
		double	val
	)

Sets all elements of the given rectangular matrix to the given value.

Parameters

mat	Given rectangular matrix.
val	Given value.

References _AlgMatrixRect::array, _AlgMatrixRect::nC, and _AlgMatrixRect::nR.

Referenced by AlgMatrixSetAll().

void AlgMatrixSymSetAll	(	AlgMatrixSym *	mat,
		double	val
	)

Sets all elements of the given symmetric matrix to the given value.

Parameters

mat	Given symmetric matrix.
val	Given value.

References _AlgMatrixSym::array, and _AlgMatrixSym::nR.

Referenced by AlgMatrixSetAll().

AlgError AlgMatrixLLRSetAll	(	AlgMatrixLLR *	mat,
		double	val
	)

Sets all elements of the given linked list row matrix to the given value. This is not an efficient use of a linked list row matrix since all values will be allocate and set.

Returns: Error code.

Parameters

mat	Given linked list row matrix.
val	Given value.

References ALG_ERR_NONE, AlgMatrixLLRExpand(), AlgMatrixLLRSet(), AlgMatrixLLRZero(), _AlgMatrixLLR::nC, _AlgMatrixLLR::nR, and _AlgMatrixLLR::tol.

Referenced by AlgMatrixSetAll().

void AlgMatrixLLRERemove	(	AlgMatrixLLR *	mat,
		size_t	row,
		size_t	col
	)

Removes the entry at the given coordinates.

Parameters

mat	Linked list row matrix.
row	Row coordinate.
col	Column coordinate.

References AlgMatrixLLREFree(), _AlgMatrixLLRE::col, _AlgMatrixLLR::nR, _AlgMatrixLLRE::nxt, and _AlgMatrixLLR::tbl.

Referenced by AlgMatrixLLRSet(), and AlgMatrixScale().

AlgError AlgMatrixCGSolve	(	AlgMatrix	aM,
		double *	xV,
		double *	bV,
		AlgMatrix	wM,
		void()(void , AlgMatrix, double , double )	pFn,
		void *	pDat,
		double	tol,
		int	maxItr,
		double *	dstTol,
		int *	dstItr
	)

Conjugate Gradient iterative method with preconditioning for the solution of linear systems with the form \(\mathbf{A} \mathbf{x} = \mathbf{b}\). \(\mathbf{A}\) must be a symmetric postive definite matrix, i.e. \({\mathbf{x}}^T \mathbf{A} \mathbf{x} < 0\), \(\forall \mathbf{x} \not= \mathbf{0}\), \(\mathbf{x} \in R^n\). Convergence is tested using: \( \frac{\| \mathbf{b} - mathbf{A} \mathbf{x} \|} {\|\mathbf{b}\|} < \delta\). If the preconditioning function pFn is non NULL then it is called, passing the preconditioning data pDat, as: (*pFn)(void *pDat, double **aM, double *r, double *z, int n) to solve \(\mathbf{A} \mathbf{z} = \mathbf{r}\) for \(\mathbf{z}\) with the solution overwriting the initial contents of z.

Returns: Error code.

Parameters

aM	Matrix \(\mathbf{A}\).
xV	Matrix \(\mathbf{x}\) which should contain an initial estimate although this may be \(\mathbf{0}\).
bV	Matrix \(\mathbf{b}\).
wM	Matrix with dimensions [4, n], this must be a rectangular matrix.
pFn	Preconditioning function.
pDat	Data to be passed to preconditioning function.
tol	Tolerance required, \(\delta\).
maxItr	The maximum number of itterations.
dstTol	Destination pointer for the residual after the final iteration, may be NULL.
dstItr	Destination pointer for the actual number of itterations performed, may be NULL.

References ALG_ERR_CONVERGENCE, ALG_ERR_FUNC, ALG_ERR_NONE, ALG_MATRIX_LLR, ALG_MATRIX_RECT, ALG_MATRIX_SYM, AlgMatrixVectorMul(), AlgMatrixZero(), AlgVectorCopy(), AlgVectorDot(), AlgVectorNorm(), AlgVectorScaleAdd(), AlgVectorSub(), _AlgMatrixRect::array, _AlgMatrix::core, _AlgMatrixCore::nC, _AlgMatrixCore::nR, _AlgMatrix::rect, and _AlgMatrixCore::type.

Referenced by WlzAffineTransformLSqDQ3D(), WlzBasisFnScalarMOS3DFromCPts(), and WlzCMeshCompSurfMap().

AlgError AlgMatrixGaussSolve	(	AlgMatrix	aMat,
		double *	xMat
	)

Solves the matrix equation A.x = b for x by Gaussian elimination with partial pivoting. Matrix A is the matrix of coefficients.

Returns: Error code.

Parameters

aMat	The augmented matrix of size aSz x (aSz + 1), whose columns 0 - (aSz - 1) are the corresponding columns of A, and aSz'th column is b. Overwritten on exit.
xMat	On exit contains the solution matrix x.

References ALG_DBG, ALG_DBG_LVL_1, ALG_DBG_LVL_FN, ALG_ERR_FUNC, ALG_ERR_MATRIX_HOMOGENEOUS, ALG_ERR_MATRIX_SINGULAR, ALG_ERR_NONE, ALG_MATRIX_RECT, _AlgMatrixRect::array, _AlgMatrix::core, _AlgMatrixCore::nC, _AlgMatrixCore::nR, _AlgMatrixRect::nR, _AlgMatrix::rect, and _AlgMatrixCore::type.

Referenced by AlgPolynomialLSq().

AlgError AlgMatrixSolveLSQR	(	AlgMatrix	aM,
		double *	bV,
		double *	xV,
		double	damping,
		double	relErrA,
		double	relErrB,
		long	maxItr,
		long	condLim,
		int *	dstTerm,
		long *	dstItr,
		double *	dstFNorm,
		double *	dstCondN,
		double *	dstResNorm,
		double *	dstResNormA,
		double *	dstNormX
	)

This function finds a solution x to the following problems:

Returns

    1. Unsymmetric equations --    solve  A*x = b

    2. Linear least squares  --    solve  A*x = b
                                   in the least-squares sense

    3. Damped least squares  --    solve  (   A    )*x = ( b )
                                          ( damp*I )     ( 0 )
                                   in the least-squares sense

    where 'A' is a matrix with 'm' rows and 'n' columns, 'b' is an
    'm'-vector, and 'damp' is a scalar.  (All quantities are real.)
    The matrix 'A' is intended to be large and sparse.  

    Notation
    --------
    The following quantities are used in discussing the subroutine
    parameters:

    'Abar'   =  (   A    ),          'bbar'  =  ( b )
                ( damp*I )                      ( 0 )

    'r'      =  b  -  A*x,           'rbar'  =  bbar  -  Abar*x

    'rnorm'  =  sqrt( norm(r)**2  +  damp**2 * norm(x)**2 )
             =  norm( rbar )

    'rel_prec'  =  the relative precision of floating-point arithmetic
                   on the machine being used.  Typically 2.22e-16
                   with 64-bit arithmetic.

    LSQR  minimizes the function 'rnorm' with respect to 'x'.

    References
    ----------
    C.C. Paige and M.A. Saunders,  LSQR: An algorithm for sparse
         linear equations and sparse least squares,
         ACM Transactions on Mathematical Software 8, 1 (March 1982),
         pp. 43-71.
    C.C. Paige and M.A. Saunders,  Algorithm 583, LSQR: Sparse
         linear equations and least-squares problems,
         ACM Transactions on Mathematical Software 8, 2 (June 1982),
         pp. 195-209.
    C.L. Lawson, R.J. Hanson, D.R. Kincaid and F.T. Krogh,
         Basic linear algebra subprograms for Fortran usage,
         ACM Transactions on Mathematical Software 5, 3 (Sept 1979),
         pp. 308-323 and 324-325.

Parameters

aM	Matrix A with nR rows and nC columns. The values of A are not modified by this function.
bV	Vector b with nR entries which are modified by this function.
xV	Vector x for with initial guess at solution and for the return of the solution with nC entries.
damping	Damping parameter, set to 0.0 for no damping.
relErrA	An estimate of the relative error in matrix A. If A is accurate to about 6 digits, set to 1.0e-06.
relErrB	An estimate of the relative error in vector b. Set as for relErrA.
maxItr	An upper limit on the number of iterations. If zero set to nC.
condLim	Upper limit on the condition number of matrix 'Abar'. Iterations will be terminated if a computed estimate of exceeds this condition number.
dstTerm	Destination pointer for termination code, which may be NULL. The values used are: 0 x = x0 is the exact solution. No iterations were performed. 1 The equations Ax = b are probably compatible. Norm(Ax - b) is sufficiently small, given the values of relErrA and relErrB. 2 The system Ax = b is probably not compatible. A least-squares solution has been obtained that is sufficiently accurate, given the value of relErrA. 3 An estimate of cond('Abar') has exceeded condLim. The system Ax = b appears to be ill-conditioned. 4 The equations Ax = b are probably compatible. Norm(Ax - b) is as small as seems reasonable on this machine. 5 The system A*x = b is probably not compatible. A least-squares solution has been obtained that is as accurate as seems reasonable on this machine. 6 Cond('Abar') seems to be so large that there is no point in doing further iterations, given the precision of this machine. 7 The iteration limit maxItr was reached.
dstItr	Destination pointer for the number of iterations performed, which may be NULL.
dstFNorm	Destination pointer for an estimate of the Frobenius norm of 'Abar', may be NULL. This is the square-root of the sum of squares of the elements of 'Abar'. If 'damping' is small and if the columns of 'A' have all been scaled to have length 1.0, then the Frobenius norm should increase to roughly sqrt('nC').
dstCondN	Destination pointer for an estimate of the condition number of 'Abar', may be NULL.
dstResNorm	Destination pointer for an estimate of the final value of norm('rbar'), the function being minimized. This will be small if A*x = b has a solution. May be NULL.
dstResNormA	Destination pointer for an estimate of the final value of the residual for the usual normal equations. This should be small in all cases. May be NULL.
dstNormX	Destination pointer for an estimate of the final solution vector 'x'.

References AlcCalloc(), AlcFree(), ALG_ERR_CONVERGENCE, ALG_ERR_MALLOC, ALG_ERR_MATRIX_CONDITION, ALG_ERR_NONE, ALG_SQR, AlgMatrixTVectorMulAdd(), AlgMatrixVectorMulAdd(), AlgVectorCopy(), AlgVectorNorm(), AlgVectorScale(), _AlgMatrix::core, _AlgMatrixCore::nC, and _AlgMatrixCore::nR.

Referenced by WlzCMeshCompSurfMap().

AlgError AlgMatrixLUSolveRaw3	(	double **	aM,
		double *	bV,
		int	bSz
	)

Solves the matrix equation A.x = b for x, where A is a 3x3 libAlc double array matrix. On return the matrix A is overwritten with its LU decomposition and the column vector b is overwritten with the solution column matrix x.

Returns: Error code.

Parameters

aM	Raw 3x3 array matrix A.
bV	Column vector b.
bSz	Size (number of columns) of vector b (and x).

References ALG_ERR_NONE, AlgMatrixLUBackSubRaw(), and AlgMatrixLUDecompRaw().

Referenced by AlgMatrixLUInvertRaw3(), and Wlz3DViewGetIntersectionPoint().

AlgError AlgMatrixLUSolveRaw4	(	double **	aM,
		double *	bV,
		int	bSz
	)

Solves the matrix equation A.x = b for x, where A is a 4x4 libAlc double array matrix. On return the matrix A is overwritten with its LU decomposition and the column vector b is overwritten with the solution column matrix x.

Returns: Error code.

Parameters

aM	Raw 4x4 array matrix A.
bV	Column vector b.
bSz	Size (number of columns) of vector b (and x).

References ALG_ERR_NONE, AlgMatrixLUBackSubRaw(), and AlgMatrixLUDecompRaw().

Referenced by AlgMatrixLUInvertRaw4(), and WlzAffineTransformLSqRegWlz2D().

AlgError AlgMatrixLUSolve	(	AlgMatrix	aM,
		double *	bV,
		int	bSz
	)

Solves the matrix equation A.x = b for x, where A is a square matrix. On return the matrix A is overwritten with its LU decomposition and the column vector b is overwritten with the solution column matrix x.

Returns: Error code.

Parameters

aM	Square matrix.
bV	Column vector b.
bSz	Size (number of columns) of vector b (and x).

References ALG_ERR_FUNC, ALG_ERR_MATRIX_TYPE, ALG_ERR_NONE, ALG_MATRIX_RECT, AlgMatrixLUSolveRaw(), _AlgMatrixRect::array, _AlgMatrix::core, _AlgMatrixRect::nR, _AlgMatrix::rect, and _AlgMatrixCore::type.

AlgError AlgMatrixLUSolveRaw	(	double **	aM,
		int	aSz,
		double *	bV,
		int	bSz
	)

Solves the matrix equation A.x = b for x, where A is a square libAlc double array matrix. On return the matrix A is overwritten with its LU decomposition and the column vector b is overwritten with the solution column matrix x.

Returns: Error code.

Parameters

aM	Square libalc array matrix.
aSz	Size of matrix A
bV	Column vector b.
bSz	Size (number of columns) of vector b (and x).

References AlcFree(), AlcMalloc(), ALG_ERR_FUNC, ALG_ERR_MALLOC, ALG_ERR_NONE, AlgMatrixLUBackSubRaw(), and AlgMatrixLUDecompRaw().

Referenced by AlgMatrixLUInvertRaw(), and AlgMatrixLUSolve().

AlgError AlgMatrixLUInvertRaw3 ( double ** aM )

Calculates the inverse of a 3x3 libAlc double array matrix.

Returns: Error code.

Parameters

aM	Raw double array of size 3x3.

References ALG_ERR_NONE, and AlgMatrixLUSolveRaw3().

AlgError AlgMatrixLUInvertRaw4 ( double ** aM )

Calculates the inverse of a 4x4 libAlc double array matrix.

Returns: Error code.

Parameters

aM	Raw double array of size 4x4.

References ALG_ERR_NONE, and AlgMatrixLUSolveRaw4().

Referenced by WlzGeomTetraAffineSolveLU().

AlgError AlgMatrixLUInvert ( AlgMatrix aM )

Calculates the inverse of a square matrix.

Returns: Error code.

Parameters

aM	Square libAlc double array matrix A.

References ALG_ERR_FUNC, ALG_ERR_MATRIX_TYPE, ALG_ERR_NONE, ALG_MATRIX_RECT, AlgMatrixLUInvertRaw(), _AlgMatrixRect::array, _AlgMatrix::core, _AlgMatrixRect::nR, _AlgMatrix::rect, and _AlgMatrixCore::type.

AlgError AlgMatrixLUInvertRaw	(	double **	aM,
		int	aSz
	)

Calculates the inverse of a square matrix.

Returns: Error code.

Parameters

aM	Square libAlc double array matrix A.
aSz	Size of matrix A.

References AlcCalloc(), AlcFree(), ALG_ERR_FUNC, ALG_ERR_MALLOC, ALG_ERR_NONE, and AlgMatrixLUSolveRaw().

Referenced by AlgMatrixLUInvert(), and WlzAffineTransformInverse().

AlgError AlgMatrixLUDetermRaw3	(	double **	aM,
		double *	det
	)

Calculates the determinant of a 3x3 double libAlc array matrix. The matrix is overwitten with its LU decomposition on return.

Returns: Error code.

Parameters

aM	Raw 3x3 array matrix.
det	Destination ptr for the determinant (may be NULL).

References ALG_ERR_NONE, and AlgMatrixLUDecompRaw().

AlgError AlgMatrixLUDetermRaw4	(	double **	aM,
		double *	det
	)

Calculates the determinant of a 4x4 double libAlc array matrix. The matrix is overwitten with its LU decomposition on return.

Returns: Error code.

Parameters

aM	Raw 4x4 array matrix.
det	Destination ptr for the determinant (may be NULL).

References ALG_ERR_NONE, and AlgMatrixLUDecompRaw().

AlgError AlgMatrixLUDeterm	(	AlgMatrix	aM,
		double *	det
	)

Calculates the determinant of a square matrix. The matrix is overwitten with its LU decomposition on return.

Returns: Error code.

Parameters

aM	Matrix A.
det	Destination ptr for the determinant (may be NULL).

References ALG_ERR_FUNC, ALG_ERR_MATRIX_TYPE, ALG_ERR_NONE, ALG_MATRIX_RECT, AlgMatrixLUDetermRaw(), _AlgMatrixRect::array, _AlgMatrix::core, _AlgMatrixRect::nR, _AlgMatrix::rect, and _AlgMatrixCore::type.

AlgError AlgMatrixLUDetermRaw	(	double **	aM,
		int	aSz,
		double *	det
	)

Calculates the determinant of a square double libAlc array matrix. The matrix is overwitten with its LU decomposition on return.

Returns: Error code.

Parameters

aM	Raw matrix A.
aSz	Size of matrix A.
det	Destination ptr for the determinant (may be NULL).

References AlcFree(), AlcMalloc(), ALG_ERR_FUNC, ALG_ERR_MALLOC, ALG_ERR_NONE, and AlgMatrixLUDecompRaw().

Referenced by AlgMatrixLUDeterm(), and WlzCMeshStrainTensorAtPts().

AlgError AlgMatrixLUDecomp	(	AlgMatrix	aM,
		int *	iV,
		double *	evenOdd
	)

Replaces the given matrix with the LU decomposition of a row-wise permutation of itself. The given index vector is used to record the permutation effected by the partial pivoting.

Returns: Error code.

Parameters

aM	Matrix A.
iV	Index vector.
evenOdd	Set to +/-1.0 depending on whether the permutation was even or odd (may be NULL).

References ALG_ERR_FUNC, ALG_ERR_MATRIX_TYPE, ALG_ERR_NONE, ALG_MATRIX_RECT, AlgMatrixLUDecompRaw(), _AlgMatrixRect::array, _AlgMatrix::core, _AlgMatrixRect::nR, _AlgMatrix::rect, and _AlgMatrixCore::type.

AlgError AlgMatrixLUDecompRaw	(	double **	aM,
		int	aSz,
		int *	iV,
		double *	evenOdd
	)

Replaces the given matrix with the LU decomposition of a row-wise permutation of itself. The given index vector is used to record the permutation effected by the partial pivoting.

Returns: Error code.

Parameters

aM	Given raw matrix A.
aSz	Size of matrix A.
iV	Index vector.
evenOdd	Set to +/-1.0 depending on whether the permutation was even or odd (may be NULL).

References AlcFree(), AlcMalloc(), ALG_ERR_FUNC, ALG_ERR_MALLOC, ALG_ERR_MATRIX_SINGULAR, and ALG_ERR_NONE.

Referenced by AlgMatrixLUDecomp(), AlgMatrixLUDetermRaw(), AlgMatrixLUDetermRaw3(), AlgMatrixLUDetermRaw4(), AlgMatrixLUSolveRaw(), AlgMatrixLUSolveRaw3(), AlgMatrixLUSolveRaw4(), WlzKrigOSetModelSV2D(), and WlzKrigOSetModelSV3D().

AlgError AlgMatrixLUBackSub	(	AlgMatrix	aM,
		int *	iV,
		double *	bV
	)

Solves the set of of linear equations A.x = b where A is input as its LU decomposition determined with AlgMatrixLUDecomp() of a row-wise permutation of itself. The given index vector is used to record the permutation effected by the partial pivoting.

Returns: Error code.

Parameters

aM	Matrix A.
iV	Index vector.
bV	Column vector b.

References ALG_ERR_FUNC, ALG_ERR_MATRIX_TYPE, ALG_ERR_NONE, ALG_MATRIX_RECT, AlgMatrixLUBackSubRaw(), _AlgMatrixRect::array, _AlgMatrix::core, _AlgMatrixRect::nR, _AlgMatrix::rect, and _AlgMatrixCore::type.

Referenced by WlzKrigOWeightsSolve().

AlgError AlgMatrixLUBackSubRaw	(	double **	aM,
		int	aSz,
		int *	iV,
		double *	bV
	)

Solves the set of of linear equations A.x = b where A is input as its LU decomposition determined with AlgMatrixLUDecompRaw() of a row-wise permutation of itself. The given index vector is used to record the permutation effected by the partial pivoting.

Returns: Error code.

Parameters

aM	Given raw libAlc array matrix A.
aSz	Size of matrix A.
iV	Index vector.
bV	Column vector b.

References ALG_ERR_FUNC, and ALG_ERR_NONE.

Referenced by AlgMatrixLUBackSub(), AlgMatrixLUSolveRaw(), AlgMatrixLUSolveRaw3(), and AlgMatrixLUSolveRaw4().

void AlgMatrixAdd	(	AlgMatrix	aM,
		AlgMatrix	bM,
		AlgMatrix	cM
	)

Computes the sum of two matricies and returns the result in a third supplied matrix:

\[ \mathbf{A} = \mathbf{B} + \mathbf{C} \]

The dimensions of the all three matricies must be nR, nC. It is safe to supply the same matrix as any combination of aM, bM and cM.

Returns: void

Note: For efficiency the given parameters are not checked.; Matrix size is limited only by address space.; All matrices must be of the same type (not checked for).

Parameters

aM	Supplied matrix for result, \(\mathbf{A}\).
bM	First matrix in the sum, \(\mathbf{B}\).
cM	Second matrix in the sum, \(\mathbf{C}\).

References ALG_MATRIX_LLR, ALG_MATRIX_RECT, ALG_MATRIX_SYM, AlgMatrixLLRSet(), AlgMatrixLLRZero(), _AlgMatrixRect::array, _AlgMatrixSym::array, _AlgMatrixLLRE::col, _AlgMatrix::core, _AlgMatrix::llr, _AlgMatrixCore::nC, _AlgMatrixCore::nR, _AlgMatrixLLRE::nxt, _AlgMatrix::rect, _AlgMatrix::sym, _AlgMatrixLLR::tbl, _AlgMatrixCore::type, and _AlgMatrixLLRE::val.

void AlgMatrixSub	(	AlgMatrix	aM,
		AlgMatrix	bM,
		AlgMatrix	cM
	)

Subtracts on matrix from another and returns the result in a third supplied matrix:

\[ \mathbf{A} = \mathbf{B} - \mathbf{C} \]

The dimensions of the all three matricies must be nR, nC. It is safe to supply the same matrix as any combination of aM, bM and cM.

Returns: void

Note: For efficiency the given parameters are not checked.; Matrix size is limited only by address space.

Parameters

aM	Supplied matrix for result, \(\mathbf{A}\).
bM	First matrix in the subtraction, \(\mathbf{B}\).
cM	Second matrix in the subtraction, \(\mathbf{C}\).

References ALG_MATRIX_LLR, ALG_MATRIX_RECT, ALG_MATRIX_SYM, AlgMatrixLLRSet(), AlgMatrixLLRZero(), _AlgMatrixRect::array, _AlgMatrixSym::array, _AlgMatrixLLRE::col, _AlgMatrix::core, _AlgMatrix::llr, _AlgMatrixCore::nC, _AlgMatrixCore::nR, _AlgMatrixLLRE::nxt, _AlgMatrix::rect, _AlgMatrix::sym, _AlgMatrixLLR::tbl, _AlgMatrixCore::type, and _AlgMatrixLLRE::val.

Referenced by WlzAffineTransformLSqDQ2D(), and WlzAffineTransformLSqDQ3D().

void AlgMatrixMul	(	AlgMatrix	aM,
		AlgMatrix	bM,
		AlgMatrix	cM
	)

Computes the product of two matricies and returns the result in a third supplied matrix:

\[ \mathbf{A} = \mathbf{B} \mathbf{C} \]

The dimensions of the result matrix (aM) must be cR, bC. All the matrices must be valid and of the same type except in the case of multiplying tow symmetric matrices when the result matrix must be rectangular (because in general the product of two symmetric matrices is not symmetric).

Returns: void

Note: For efficiency the given parameters are not fully checked.; Matrix size is limited only by address space.

Parameters

aM	Supplied matrix for result, \(\mathbf{A}\)
bM	First matrix in the product, \(\mathbf{B}\)
cM	Second matrix in the product, \(\mathbf{C}\)

References ALG_ERR_MATRIX_TYPE, ALG_ERR_NONE, ALG_MATRIX_LLR, ALG_MATRIX_RECT, ALG_MATRIX_SYM, AlgMatrixLLRSet(), AlgMatrixLLRValue(), _AlgMatrixRect::array, _AlgMatrixLLRE::col, _AlgMatrix::core, _AlgMatrix::llr, _AlgMatrixCore::nC, _AlgMatrixCore::nR, _AlgMatrixLLRE::nxt, _AlgMatrix::rect, _AlgMatrixLLR::tbl, _AlgMatrixCore::type, and _AlgMatrixLLRE::val.

Referenced by WlzAffineTransformLSqDQ2D(), and WlzAffineTransformLSqDQ3D().

double AlgMatrixTrace ( AlgMatrix aM )

Computes the trace of the given matrix.

Returns: The trace of the matrix.

Note: For efficiency the given parameters are not checked.; Matrix size is limited only by address space.

Parameters

aM	Supplied matrix.

References ALG_MATRIX_LLR, ALG_MATRIX_RECT, ALG_MATRIX_SYM, ALG_MIN, AlgMatrixLLRValue(), _AlgMatrixRect::array, _AlgMatrixSym::array, _AlgMatrix::core, _AlgMatrix::llr, _AlgMatrixCore::nC, _AlgMatrixCore::nR, _AlgMatrix::rect, _AlgMatrix::sym, and _AlgMatrixCore::type.

void AlgMatrixTranspose	(	AlgMatrix	aM,
		AlgMatrix	bM
	)

Computes the transpose of the given matrix:

\[ \mathbf{A} = \mathbf{B^T} \]

aM = transpose(bM). The dimensions of the result matrix (aM) must be aR == bC, aC == bR.

Returns: void

Note: For efficiency the given parameters are not checked.; Matrix size is limited only by address space.

Parameters

aM	Supplied matrix for result, \(\mathbf{A}\).
bM	Matrix to transpose, \(\mathbf{B}\).

References ALG_MATRIX_LLR, ALG_MATRIX_RECT, ALG_MATRIX_SYM, AlgMatrixLLRSet(), AlgMatrixLLRZero(), _AlgMatrixRect::array, _AlgMatrixSym::array, _AlgMatrixLLRE::col, _AlgMatrix::core, _AlgMatrix::llr, _AlgMatrixCore::nC, _AlgMatrixCore::nR, _AlgMatrixLLRE::nxt, _AlgMatrix::rect, _AlgMatrix::sym, _AlgMatrixLLR::tbl, _AlgMatrixCore::type, and _AlgMatrixLLRE::val.

Referenced by WlzAffineTransformLSqDQ2D(), and WlzAffineTransformLSqDQ3D().

void AlgMatrixCopy	(	AlgMatrix	aM,
		AlgMatrix	bM
	)

Copies the values of the matrix bM to the result matric aM:

\[ \mathbf{A} = \mathbf{B} \]

.

Returns: void

Note: For efficiency the given parameters are not checked.; Matrix size is limited only by address space.

Parameters

aM	Supplied matrix for result, \(\mathbf{A}\).
bM	Matrix to copy, \(\mathbf{B}\).

References ALG_MATRIX_LLR, ALG_MATRIX_RECT, ALG_MATRIX_SYM, AlgMatrixLLRCopyInPlace(), AlgVectorCopy(), _AlgMatrixRect::array, _AlgMatrixSym::array, _AlgMatrix::core, _AlgMatrix::llr, _AlgMatrixCore::nC, _AlgMatrixCore::nR, _AlgMatrix::rect, _AlgMatrix::sym, and _AlgMatrixCore::type.

Referenced by WlzAffineTransformLSqDQ3D().

void AlgMatrixScale	(	AlgMatrix	aM,
		AlgMatrix	bM,
		double	sv
	)

Multiplies the given matrix by the given scalar:

\[ \mathbf{A} = s \mathbf{B} \]

.

Note: For efficiency the given parameters are not checked.; Matrix size is limited only by address space.

Parameters

aM	Supplied matrix for result, \(\mathbf{A}\).
bM	Given matrix to scale, \(\mathbf{B}\).
sv	Scalar value, \(s\).

References ALG_MATRIX_LLR, ALG_MATRIX_RECT, ALG_MATRIX_SYM, AlgMatrixLLRERemove(), AlgMatrixLLRSet(), AlgVectorScale(), _AlgMatrixRect::array, _AlgMatrixSym::array, _AlgMatrixLLRE::col, _AlgMatrix::core, _AlgMatrix::llr, _AlgMatrixCore::nC, _AlgMatrixCore::nR, _AlgMatrixLLR::nR, _AlgMatrixLLRE::nxt, _AlgMatrix::rect, _AlgMatrix::sym, _AlgMatrixLLR::tbl, _AlgMatrixLLR::tol, _AlgMatrixCore::type, and _AlgMatrixLLRE::val.

Referenced by WlzAffineTransformLSqDQ2D(), and WlzAffineTransformLSqDQ3D().

void AlgMatrixScaleAdd	(	AlgMatrix	aM,
		AlgMatrix	bM,
		AlgMatrix	cM,
		double	sv
	)

Multiplies the a matrix by a scalar and then adds another matrix:

\[ \mathbf{A} = \mathbf{B} + s \mathbf{C}. \]

All the matrices must have the same dimensions.

Returns: void

Note: For efficiency the given parameters are not checked.; Matrix size is limited only by address space.

Parameters

aM	Supplied matrix for result, \(\mathbf{A}\).
bM	Given matrix to scale, \(\mathbf{B}\).
cM	Matrix too add, \(\mathbf{C}\).
sv	Scalar value, \(s\).

References ALG_MATRIX_LLR, ALG_MATRIX_RECT, ALG_MATRIX_SYM, AlgMatrixLLRSet(), AlgMatrixLLRZero(), _AlgMatrixRect::array, _AlgMatrixSym::array, _AlgMatrixLLRE::col, _AlgMatrix::core, _AlgMatrix::llr, _AlgMatrixCore::nC, _AlgMatrixCore::nR, _AlgMatrixLLRE::nxt, _AlgMatrix::rect, _AlgMatrix::sym, _AlgMatrixLLR::tbl, _AlgMatrixCore::type, and _AlgMatrixLLRE::val.

void AlgMatrixScalar	(	AlgMatrix	aM,
		double	sv
	)

Sets the elements of the given square matrix so that it is a scalar matrix:

\[ \mathbf{A} = s \mathbf{I} \]

.

Note: For efficiency the given parameters are not checked.; This function assumes that the matrix has been allocated by AlcDouble2Malloc().

Parameters

aM	Supplied matrix for result, \(\mathbf{A}\).
sv	Scalar value, \(s\).

References ALG_ERR_NONE, ALG_MATRIX_LLR, ALG_MATRIX_RECT, ALG_MATRIX_SYM, ALG_MIN, AlgMatrixLLRENew(), AlgMatrixLLRExpand(), AlgMatrixLLRZero(), AlgMatrixRectZero(), AlgMatrixSymZero(), _AlgMatrixRect::array, _AlgMatrixLLRE::col, _AlgMatrix::core, _AlgMatrix::llr, _AlgMatrixCore::nC, _AlgMatrixCore::nR, _AlgMatrixLLRE::nxt, _AlgMatrix::rect, _AlgMatrix::sym, _AlgMatrixLLR::tbl, _AlgMatrixCore::type, and _AlgMatrixLLRE::val.

void AlgMatrixVectorMul	(	double *	aV,
		AlgMatrix	bM,
		double *	cV
	)

Multiplies the matrix \(\mathbf{B}\) by the vector \(\mathbf{c}\):

\[ \mathbf{a} = \mathbf{B} \mathbf{c} \]

.

Note: This function assumes that the matrix has been allocated by either AlcDouble2Malloc() or AlcSymDouble2Malloc().; Matrix size is limited only by address space.

Parameters

aV	Supplied vector for result.
bM	Matrix \(mathbf{B}\).
cV	Vector \(\mathbf{c}\). \(mathbf{B}\).

References ALG_MATRIX_LLR, ALG_MATRIX_RECT, ALG_MATRIX_SYM, _AlgMatrixRect::array, _AlgMatrixSym::array, _AlgMatrixLLRE::col, _AlgMatrix::core, _AlgMatrix::llr, _AlgMatrixRect::nC, _AlgMatrixRect::nR, _AlgMatrixSym::nR, _AlgMatrixLLR::nR, _AlgMatrixLLRE::nxt, _AlgMatrix::rect, _AlgMatrix::sym, _AlgMatrixLLR::tbl, _AlgMatrixCore::type, and _AlgMatrixLLRE::val.

Referenced by AlgMatrixCGSolve(), and WlzCMeshCompSurfMap().

void AlgMatrixVectorMulAdd	(	double *	aV,
		AlgMatrix	bM,
		double *	cV,
		double *	dV
	)

Multiplies the matrix \(\mathbf{B}\) by the vector \(\mathbf{c}\) and adds the vector \(\mathbf{d}\):

\[ \mathbf{a} = \mathbf{B} \mathbf{c} + \mathbf{d} \]

.

Note: This function assumes that the matrix has been allocated by either AlcDouble2Malloc() or AlcSymDouble2Malloc().; Matrix size is limited only by address space.

Parameters

aV	Supplied vector for result.
bM	Matrix \(mathbf{B}\).
cV	Vector \(\mathbf{c}\).
dV	Vector \(\mathbf{d}\). \(mathbf{B}\).

References ALG_MATRIX_LLR, ALG_MATRIX_RECT, ALG_MATRIX_SYM, _AlgMatrixRect::array, _AlgMatrixSym::array, _AlgMatrixLLRE::col, _AlgMatrix::core, _AlgMatrix::llr, _AlgMatrixRect::nC, _AlgMatrixRect::nR, _AlgMatrixSym::nR, _AlgMatrixLLR::nR, _AlgMatrixLLRE::nxt, _AlgMatrix::rect, _AlgMatrix::sym, _AlgMatrixLLR::tbl, _AlgMatrixCore::type, and _AlgMatrixLLRE::val.

Referenced by AlgMatrixSolveLSQR().

void AlgMatrixVectorMulWAdd	(	double *	aV,
		AlgMatrix	bM,
		double *	cV,
		double *	dV,
		double	s,
		double	t
	)

Multiplies the matrix \(\mathbf{B}\) by the vector \(\mathbf{c}\) and adds the vector \(\mathbf{d}\) using the given weights \(s\) and \(t\):

\[ \mathbf{a} = s \mathbf{B} \mathbf{c} + t \mathbf{d} \]

.

Note: This function assumes that the matrix has been allocated by either AlcDouble2Malloc() or AlcSymDouble2Malloc().; Matrix size is limited only by address space.

Parameters

aV	Supplied vector for result.
bM	Matrix \(mathbf{B}\).
cV	Vector \(\mathbf{c}\).
dV	Vector \(\mathbf{d}\).
s	First weighting scalar \(s\).
t	Second weighting scalar \(t\).

References ALG_MATRIX_LLR, ALG_MATRIX_RECT, ALG_MATRIX_SYM, _AlgMatrixRect::array, _AlgMatrixSym::array, _AlgMatrixLLRE::col, _AlgMatrix::core, _AlgMatrix::llr, _AlgMatrixRect::nC, _AlgMatrixRect::nR, _AlgMatrixSym::nR, _AlgMatrixLLR::nR, _AlgMatrixLLRE::nxt, _AlgMatrix::rect, _AlgMatrix::sym, _AlgMatrixLLR::tbl, _AlgMatrixCore::type, and _AlgMatrixLLRE::val.

void AlgMatrixTVectorMul	(	double *	aV,
		AlgMatrix	bM,
		double *	cV
	)

Multiplies the transpose of matrix \(\mathbf{B}\) by the vector \(\mathbf{c}\):

\[ \mathbf{a} = \mathbf{B}^T \mathbf{c} \]

.

Note: This function assumes that the matrix has been allocated by either AlcDouble2Malloc() or AlcSymDouble2Malloc().; Matrix size is limited only by address space.

Parameters

aV	Supplied vector for result.
bM	Matrix \(mathbf{B}\).
cV	Vector \(\mathbf{c}\). \(mathbf{B}\).

References ALG_MATRIX_LLR, ALG_MATRIX_RECT, ALG_MATRIX_SYM, AlgVectorZero(), _AlgMatrixRect::array, _AlgMatrixSym::array, _AlgMatrixLLRE::col, _AlgMatrix::core, _AlgMatrix::llr, _AlgMatrixRect::nC, _AlgMatrixLLR::nC, _AlgMatrixRect::nR, _AlgMatrixSym::nR, _AlgMatrixLLR::nR, _AlgMatrixLLRE::nxt, _AlgMatrix::rect, _AlgMatrix::sym, _AlgMatrixLLR::tbl, _AlgMatrixCore::type, and _AlgMatrixLLRE::val.

void AlgMatrixTVectorMulAdd	(	double *	aV,
		AlgMatrix	bM,
		double *	cV,
		double *	dV
	)

Multiplies the transpose of matrix \(\mathbf{B}\) by the vector \(\mathbf{c}\):

\[ \mathbf{a} = \mathbf{B}^T \mathbf{c} + \mathbf{d} \]

.

Note: This function assumes that the matrix has been allocated by either AlcDouble2Malloc() or AlcSymDouble2Malloc().; Matrix size is limited only by address space.

Parameters

aV	Supplied vector for result.
bM	Matrix \(mathbf{B}\).
cV	Vector \(\mathbf{c}\).
dV	Vector \(\mathbf{d}\). \(mathbf{B}\).

References ALG_MATRIX_LLR, ALG_MATRIX_RECT, ALG_MATRIX_SYM, AlgVectorCopy(), _AlgMatrixSym::array, _AlgMatrixLLRE::col, _AlgMatrix::core, _AlgMatrix::llr, _AlgMatrixRect::nC, _AlgMatrixLLR::nC, _AlgMatrixRect::nR, _AlgMatrixSym::nR, _AlgMatrixLLR::nR, _AlgMatrixLLRE::nxt, _AlgMatrix::rect, _AlgMatrix::sym, _AlgMatrixLLR::tbl, _AlgMatrixCore::type, and _AlgMatrixLLRE::val.

Referenced by AlgMatrixSolveLSQR().

AlgError AlgMatrixRawInv2x2	(	double *	a00,
		double *	a01,
		double *	a10,
		double *	a11
	)

Inverts a raw 2 x 2 matrix. All matrix values are passed using pointers and their values are set to those of the inverse matrix on return. If the given matrix is singular the ALG_ERR_MATRIX_SINGULAR error code is returned and no values are changed.

Returns: Alg error code, either ALG_ERR_NONE or ALG_ERR_MATRIX_SINGULAR.

Parameters

a00	Matrix element row 0, column 0.
a01	Matrix element row 0, column 1.
a10	Matrix element row 1, column 0.
a11	Matrix element row 1, column 1.

References ALG_ERR_MATRIX_SINGULAR, and ALG_ERR_NONE.

AlgError AlgMatrixRawInv3x3	(	double *	a00,
		double *	a01,
		double *	a02,
		double *	a10,
		double *	a11,
		double *	a12,
		double *	a20,
		double *	a21,
		double *	a22
	)

Inverts a raw 2 x 2 matrix. All matrix values are passed using pointers and their values are set to those of the inverse matrix on return. If the given matrix is singular the ALG_ERR_MATRIX_SINGULAR error code is returned and no values are changed.

Returns: Alg error code, either ALG_ERR_NONE or ALG_ERR_MATRIX_SINGULAR.

Parameters

a00	Matrix element row 0, column 0.
a01	Matrix element row 0, column 1.
a02	Matrix element row 0, column 2.
a10	Matrix element row 1, column 0.
a11	Matrix element row 1, column 1.
a12	Matrix element row 1, column 2.
a20	Matrix element row 2, column 0.
a21	Matrix element row 2, column 1.
a22	Matrix element row 2, column 2.

References ALG_ERR_MATRIX_SINGULAR, and ALG_ERR_NONE.

AlgError AlgMatrixRSEigen	(	AlgMatrix	aM,
		double *	vM,
		int	reqEV
	)

Determines the eigenvalues and eigenvectors of a real symmetric matrix by calling AlgMatrixRSTDiag() to create a tridiagonal symmetric matrix and then AlgMatrixTDiagQLI() to compute its eigenvalues and eigenvectors. The eigenvectors and eigenvalues are returned in descending eigenvalue order. For efficiency, the eigenvectors should only be computed if required.

Returns: Error code.

Parameters

aM	Given real symmetric matrix which contains the eigenvectors in it's columns on return if required.
vM	Given vector for the return of the eigenvalues.
reqEV	Non zero if the eigenvectors are required.

References AlcFree(), AlcMalloc(), ALG_ERR_FUNC, ALG_ERR_MALLOC, ALG_ERR_NONE, ALG_MATRIX_RECT, AlgMatrixRSTDiag(), AlgMatrixTDiagQLI(), _AlgMatrixRect::array, _AlgMatrix::core, _AlgMatrixCore::nC, _AlgMatrixCore::nR, _AlgMatrix::rect, and _AlgMatrixCore::type.

Referenced by WlzAffineTransformLSqDQ2D(), WlzAffineTransformLSqDQ3D(), WlzCMeshExpansion(), and WlzCMeshStrainTensorAtPts().

AlgError AlgMatrixRSTDiag	(	AlgMatrix	aMat,
		double *	dVec,
		double *	oVec
	)

An implementation of Householder's alorithm which reduces a real aSz x aSz symmetric matrix to symmetric tridiagonal form by aSz - 2 orthogonal similarity transformations. This is a translation of the functions/subroutines 'tred2' in the fortran EISPACK library and the Numerical Recipies book, both of these are descended from the algol procedure tred2 in Wilkinson J.H and Reinsch C. Linear Algebra, Volume II Handbook for Automatic Computation. Springer-Verlag, 1971. See Numerical Recipies in C: The Art of Scientific Computing. Cambridge University Press, 1992.

Returns: Error code.

Parameters

aMat	Given real symmetric matrix.
dVec	Given array for the return of the diagonal elements.
oVec	Given array for the return of the off diagonal elements with the first element set to 0.0.

References ALG_ERR_FUNC, ALG_ERR_NONE, ALG_MATRIX_RECT, _AlgMatrixRect::array, _AlgMatrix::core, _AlgMatrixCore::nC, _AlgMatrixCore::nR, _AlgMatrixRect::nR, _AlgMatrix::rect, and _AlgMatrixCore::type.

Referenced by AlgMatrixRSEigen().

AlgError AlgMatrixSVSolve	(	AlgMatrix	aMat,
		double *	bVec,
		double	tol,
		int *	dstIC
	)

Solves the matrix equation A.x = b for x, where A is a matrix with at least as many rows as columns. On return the matrix A is overwritten by the matrix U in the singular value decomposition: A = U.W.V'.

Returns: Error code.

Parameters

aMat	Matrix A.
bVec	Column matrix b, overwritten by matrix x on return.
tol	Tolerance for singular values, 1.0e-06 should be suitable as a default value.
dstIC	Destination pointer for ill-conditioned flag, which may be NULL, but if not NULL is set to the number of singular values which are smaller than the maximum singular value times the given threshold.

References AlcCalloc(), AlcFree(), ALG_DBG, ALG_DBG_LVL_1, ALG_DBG_LVL_FN, ALG_ERR_FUNC, ALG_ERR_MALLOC, ALG_ERR_NONE, ALG_MATRIX_RECT, AlgMatrixRectFree(), AlgMatrixRectNew(), AlgMatrixSVBackSub(), AlgMatrixSVDecomp(), _AlgMatrix::core, _AlgMatrixCore::nC, _AlgMatrixRect::nC, _AlgMatrixCore::nR, _AlgMatrix::rect, and _AlgMatrixCore::type.

Referenced by WlzAffineTransformLSqDQ3D(), and WlzGeomCurvature().

AlgError AlgMatrixSVDecomp	(	AlgMatrix	aMat,
		double *	wVec,
		AlgMatrix	vMat
	)

Performs singular value decomposition of the given matrix ( \(A\) ) and computes two additional matricies ( \(U\) ) and ( \(V\) ) such that:

\[ A = U W V^T \]

where \(V^T\) is the transpose of \(V\). The code for AlgMatrixSVDecomp was derived from: Numerical Recipies function svdcmp(), EISPACK subroutine SVD(), CERN subroutine SVD() and ACM algorithm 358. The singular values ( \(W\) ) and singular vectors ( \(V\) ) are returned unsorted. See AlgMatrixSVSolve() for a usage example.

Returns: Error code.

Parameters

aMat	The given matrix A, and U on return.
wVec	The diagonal matrix of singular values, returned as a vector.
vMat	The matrix V (not it's transpose).

References AlcCalloc(), AlcFree(), ALG_DBG, ALG_DBG_LVL_1, ALG_DBG_LVL_FN, ALG_ERR_FUNC, ALG_ERR_MALLOC, ALG_ERR_MATRIX_SINGULAR, ALG_ERR_NONE, ALG_MATRIX_RECT, _AlgMatrixRect::array, _AlgMatrix::core, _AlgMatrixCore::nC, _AlgMatrixRect::nC, _AlgMatrixCore::nR, _AlgMatrixRect::nR, _AlgMatrix::rect, and _AlgMatrixCore::type.

Referenced by AlgMatrixSVSolve(), WlzAffineTransformLSqReg2D(), WlzAffineTransformLSqReg3D(), WlzBasisFnConf2DFromCPts(), WlzBasisFnGauss2DFromCPts(), WlzBasisFnIMQ2DFromCPts(), WlzBasisFnIMQ3DFromCPts(), WlzBasisFnMQ2DFromCPts(), WlzBasisFnMQ3DFromCPts(), WlzBasisFnPoly2DFromCPts(), WlzBasisFnScalarMOS3DFromCPts(), WlzBasisFnTPS2DFromCPts(), WlzCMeshExpValues(), WlzFitPlaneSVD(), and WlzGeometryLSqOPlane().

AlgError AlgMatrixSVBackSub	(	AlgMatrix	uMat,
		double *	wVec,
		AlgMatrix	vMat,
		double *	bVec
	)

Solves the set of of linear equations A.x = b where A is input as its singular value decomposition in the three matricies U, W and V, as returned by AlgMatrixSVDecomp(). The code for AlgMatrixSVBackSub was derived from: Numerical Recipies function svbksb().

Returns: Error code.

Parameters

uMat	Given matrix U with nM rows and nN columns..
wVec	The diagonal matrix of singular values, returned as a vector with nN elements.
vMat	The matrix V (not it's transpose) with nN columns.
bVec	Column matrix b with nM elements, overwritten by column matrix x on return.

References AlcCalloc(), AlcFree(), ALG_DBG, ALG_DBG_LVL_1, ALG_DBG_LVL_FN, ALG_ERR_FUNC, ALG_ERR_MALLOC, ALG_ERR_NONE, ALG_MATRIX_RECT, _AlgMatrixRect::array, _AlgMatrix::core, _AlgMatrixCore::nC, _AlgMatrixRect::nC, _AlgMatrixCore::nR, _AlgMatrixRect::nR, _AlgMatrix::rect, and _AlgMatrixCore::type.

Referenced by AlgMatrixSVSolve(), WlzBasisFnConf2DFromCPts(), WlzBasisFnGauss2DFromCPts(), WlzBasisFnIMQ2DFromCPts(), WlzBasisFnIMQ3DFromCPts(), WlzBasisFnMQ2DFromCPts(), WlzBasisFnMQ3DFromCPts(), WlzBasisFnPoly2DFromCPts(), WlzBasisFnScalarMOS3DFromCPts(), and WlzBasisFnTPS2DFromCPts().

AlgError AlgMatrixTDiagQLI	(	double *	dM,
		double *	oM,
		int	aSz,
		AlgMatrix	zM
	)

Determines the eigenvalues and eigenvectors of a real symmetric tridiagonal matrix using the QL algorithm with implicit shifts. This is a based on the function 'tqli' in Numerical Recipies in C: The Art of Scientific Computiung. Cambridge University Press, 1992.

Returns: Error code.

Parameters

dM	Given diagonal elements of the tridiagonal matrix. On return it contains the eigenvalues.
oM	Given off diagonal elements of the tridiagonal matrix, with an arbitrary first element. On return it's contents are destroyed.
aSz	Size of the array.
zM	Given array for the return of the eigenvectors, may be NULL if the eigenvectors are not required. If required the i'th eigenvector is returned in the i'th column of zM.

References ALG_ERR_CONVERGENCE, ALG_ERR_FUNC, ALG_ERR_NONE, ALG_MATRIX_RECT, _AlgMatrixRect::array, _AlgMatrix::core, _AlgMatrix::rect, and _AlgMatrixCore::type.

Referenced by AlgMatrixRSEigen().

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