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M_MATRIX(3) Library Functions Manual M_MATRIX(3) NAME M_Matrix -- Agar-Math matrix-related functions SYNOPSIS #include <agar/core.h> #include <agar/gui.h> #include <agar/math/m.h> DESCRIPTION The M_Vector(3) and M_Matrix interfaces implement linear algebra opera- tions on (real or complex valued) n dimensional vectors, and m by n ma- trices. Optimized interfaces are provided for fixed-dimensional types (which have entries directly accessible as x, y, z and w). Arbitrary- dimensional types may or may not use fixed arrays in memory. For exam- ple, the "sparse" backend uses a sparse matrix representation, and the "db" backend stores vector entries in a database. Backends can be selected at run-time, or Agar-Math can be compiled to provide inline expansions of all operations of a specific backend. Vector extensions (such as SSE and AltiVec) are used by default, if a runtime cpuinfo check determines that they are available (the build re- mains compatible with non-vector platforms, at the cost of extra func- tion calls). For best performance, Agar should be compiled with "--with-sse=inline", or "--with-altivec=inline". M-BY-N MATRICES The following routines operate on dynamically-allocated m by n matri- ces: fpu Native scalar floating point methods. sparse Methods optimized for large, sparse matrices. Based on the excellent Sparse 1.4 package by Kenneth Kundert. M-BY-N MATRICES: INITIALIZATION M_Matrix * M_New(Uint m, Uint n) void M_Free(M_Matrix *M) int M_Resize(M_Matrix *M, Uint m, Uint n) void M_SetIdentity(M_Matrix *M) void M_SetZero(M_Matrix *M) int M_Copy(M_Matrix *D, const M_Matrix *A) M_Matrix * M_Dup(const M_Matrix *M) M_Matrix * M_ReadMatrix(AG_DataSource *ds) void M_WriteMatrix(AG_DataSource *ds, const M_Matrix *A) The M_New() function allocates a new m by n matrix. M_Free() releases all resources allocated for the specified matrix. M_Resize() resizes M to m by n. Existing entries are preserved, but new entries are left uninitialized. If insufficient memory is available, -1 is returned and an error message is set. On success, the function returns 0. M_SetIdentity() initializes M to the identity matrix. M_SetZero() ini- tializes M to all zeros. M_Copy() copies the contents of matrix A into D, which is assumed to have the same dimensions (otherwise, -1 is returned). M_Dup() returns a duplicate of M. The M_ReadMatrix() and M_WriteMatrix() functions are used to (de)seri- alize the contents of matrix A from/to the specified AG_DataSource(3). M-BY-N MATRICES: ACCESSING ELEMENTS M_Real M_Get(M_Matrix *M, Uint i, Uint j) void M_Set(M_Matrix *M, Uint i, Uint j, M_Real val) M_Real * M_GetElement(M_Matrix *M, Uint i, Uint j) void M_ToFloats(float *values, const M_Matrix *A) void M_ToDoubles(double *values, const M_Matrix *A) void M_FromFloats(M_Matrix *A, const float *values) void M_FromDoubles(M_Matrix *A, const double *values) void M_Print(const M_Matrix *A) The M_Get() and M_Set() routines respectively retrieve and set the ele- ment i, j. M_GetElement() returns a pointer to the element i, j. As long as the entry exists, it is safe to read and write the element. The M_ToFloats() and M_ToDoubles() functions return a representation of matrix A as an array of float or double values in row-major order. The M_FromFloats() and M_FromDoubles() functions initialize matrix A from an array of float or double values in row-major order. In both cases, it is assumed that the arrays are of the correct size for the given ma- trix dimensions. M_Print() dumps the individual matrix entries to the standard error output. It is only for debugging purposes. Agar GUI applications can use the provided M_Matview(3) widget to display matrix contents. M-BY-N MATRICES: OPERATIONS M_Matrix * M_Transpose(M_Matrix *M) M_Matrix * M_Add(const M_Matrix *A, const M_Matrix *B) int M_Addv(M_Matrix *A, const M_Matrix *B) void M_AddToDiag(M_Matrix *A, M_Real value) M_Matrix * M_DirectSum(const M_Matrix *A, const M_Matrix *B) M_Matrix * M_Mul(const M_Matrix *A, const M_Matrix *B) int M_Mulv(const M_Matrix *A, const M_Matrix *B, M_Matrix *AB) M_Matrix * M_EntMul(const M_Matrix *A, const M_Matrix *B) int M_EntMulv(const M_Matrix *A, const M_Matrix *B, M_Matrix *AB) void M_Compare(const M_Matrix *A, const M_Matrix *B, M_Real *diff) int M_Trace(M_Real *trace, const M_Matrix *A) void M_IsSquare(M_Matrix *A) M_Matrix * M_GaussJordan(const M_Matrix *A, M_Matrix *b) int M_GaussJordanv(M_Matrix *A, M_Matrix *b) int M_FactorizeLU(M_Matrix *A) void M_BacksubstLU(M_Matrix *LU, M_Vector *b) void M_MNAPreorder(M_Matrix *A) The M_Transpose() function returns the transpose of M (i.e., all i, j elements are swapped against j, i elements). M_Add() returns the sum of the matrices A and B. The M_Addv() variant returns the sum into an existing matrix, returning -1 if the dimensions are incorrect. The M_AddToDiag() routine adds value to each diagonal entry i, i of ma- trix A. M_DirectSum() returns the direct sum of A and B. M_Mul() returns the product of matrices A and B. The M_Mulv() variant returns the product into an existing matrix, returning -1 if the dimen- sions are incorrect. M_EntMul() and M_EntMulv() perform entrywise mul- tiplication as opposed to matrix multiplication. The M_Compare() function compares each entry of A and B, returning the largest difference into diff. M_Trace() returns the trace (the sum of elements on the diagonal) of a square matrix A into trace. The M_IsSquare() function returns 1 if A is a square (n-by-n) matrix. The M_GaussJordan() function solves for x in Ax = b. The solver re- places the contents of A by its inverse, and returns the solution vec- tor into b. The M_FactorizeLU() routine computes the LU factorization of a square matrix A. If successful, the original contents of A are destroyed and replaced by the LU factorization. On error, -1 is returned. Partial pivoting information is recorded in the M_Matrix structure for subse- quent backsubstitution. The M_BacksubstLU() routine solves a system of linear equations repre- sented by a LU factorization LU (previously computed by M_FactorizeLU()) and a right-hand side b. The solution vector is re- turned into b. The M_MNAPreorder() routine attempts to remove zeros from the diagonal, by taking into account the structure of modified node admittance matri- ces (found in applications such as electronic simulators). 4-BY-4 MATRICES The following routines are optimized for 4x4 matrices, as frequently encountered in computer graphics. Entries are directly accessible as structure members. Available backends include: fpu Native scalar floating point methods. sse Accelerate operations using Streaming SIMD Extensions (SSE). 4-BY-4 MATRICES: INITIALIZATION M_Matrix44 M_MatZero44(void) void M_MatZero44v(M_Matrix44 *Z) M_Matrix44 M_MatIdentity44(void) void M_MatIdentity44v(M_Matrix44 *I) void M_MatCopy44(M_Matrix44 *Mdst, const M_Matrix44 *Msrc) The M_MatZero44() and M_MatZero44v() functions initializes the target matrix Z to the zero matrix. M_MatIdentity44() and M_MatIdentity44v() initializes the target matrix I to the identity matrix. The M_MatCopy44() routine copies the contents of matrix Msrc into Mdst. The original contents of Mdst are overwritten. 4-BY-4 MATRICES: ACCESSING ELEMENTS The elements of M_Matrix44 are directly accessible via the m[4][4] mem- ber of the structure. Elements of the matrix are stored in row-major format. The structure is defined as: #ifdef HAVE_SSE typedef union m_matrix44 { struct { __m128 m1, m2, m3, m4; }; float m[4][4]; } M_Matrix44; #else typedef struct m_matrix44 { M_Real m[4][4]; } M_Matrix44; #endif Notice that SIMD extensions force single-precision floats, regardless of the precision for which Agar-Math was built (if a 4x4 matrix of higher precision is required, the general M_Matrix type may be used). The following functions convert between M_Matrix44 and numerical ar- rays: void M_MatToFloats44(float *flts, const M_Matrix44 *A) void M_MatToDoubles44(double *dbls, const M_Matrix44 *A) void M_MatFromFloats44(M_Matrix44 *M, const float *flts) void M_MatFromDoubles44(M_Matrix44 *M, const double *dbls) M_MatToFloats44() converts matrix A to a 4x4 array of floats flts. M_MatToDoubles44() converts matrix A to a 4x4 array of doubles dbls. M_MatFromFloats44() initializes matrix M from the contents of a 4x4 ar- ray of floats flts. M_MatFromDoubles44() initializes matrix M from the contents of a 4x4 array of doubles dbls. 4-BY-4 MATRICES: OPERATIONS M_Matrix44 M_MatTranspose44(M_Matrix44 A) M_Matrix44 M_MatTranspose44p(const M_Matrix44 *A) void M_MatTranspose44v(M_Matrix44 *A) M_Matrix44 M_MatInvert44(M_Matrix44 A) int M_MatInvertElim44(M_Matrix44 A, M_Matrix44 *Ainv) M_Matrix44 M_MatMult44(M_Matrix44 A, M_Matrix44 B) void M_MatMult44v(M_Matrix44 *A, const M_Matrix44 *B) void M_MatMult44pv(M_Matrix44 *AB, const M_Matrix44 *A, const M_Matrix44 *B) M_Vector4 M_MatMultVector44(M_Matrix44 A, M_Vector4 x) M_Vector4 M_MatMultVector44p(const M_Matrix44 *A, const M_Vector4 *x) void M_MatMultVector44v(M_Vector4 *x, const M_Matrix44 *A) void M_MatRotateAxis44(M_Matrix44 *T, M_Real theta, M_Vector3 axis) void M_MatOrbitAxis44(M_Matrix44 *T, M_Vector3 center, M_Vector3 axis, M_Real theta) void M_MatRotateEul44(M_Matrix44 *T, M_Real pitch, M_Real roll, M_Real yaw) void M_MatRotate44I(M_Matrix44 *T, M_Real theta) void M_MatRotate44J(M_Matrix44 *T, M_Real theta) void M_MatRotate44K(M_Matrix44 *T, M_Real theta) void M_MatTranslate44v(M_Matrix44 *T, M_Vector3 v) void M_MatTranslate44(M_Matrix44 *T, M_Real x, M_Real y, M_Real z) void M_MatTranslate44X(M_Matrix44 *T, M_Real c) void M_MatTranslate44Y(M_Matrix44 *T, M_Real c) void M_MatTranslate44Z(M_Matrix44 *T, M_Real c) void M_MatScale44(M_Matrix44 *T, M_Real x, M_Real y, M_Real z, M_Real w) void M_MatUniScale44(M_Matrix44 *T, M_Real c) The M_MatTranspose44(), M_MatTranspose44p() and M_MatTranspose44v() function compute and return the transpose of matrix A (i.e., all ele- ments i,j are swapped for elements j,i). The function M_MatInvert44() computes the inverse of A using Cramer's rule and cofactors. If the matrix is not invertible, the return value is undefined. The M_MatInvertElim44() function computes the inverse of A by system- atic Gaussian elimination. If the matrix is not invertible (singular up to M_MACHEP precision), the function fails. M_MatMult44(), M_MatMult44v() and M_MatMult44pv() compute the product of matrices A and B. The M_MatMultVector44(), M_MatMultVector44p() and M_MatMultVector44v() functions perform matrix-vector multiplication Ax, and returns x. M_MatRotateAxis44() multiplies matrix T against a rotation matrix de- scribing a rotation of theta radians about axis (relative to the ori- gin). The M_MatOrbitAxis44() variant takes axis to be relative to the specified center point as opposed to the origin. M_MatRotateEul44() multiplies T against a matrix describing a rotation about the origin in terms of Euler angles pitch, roll and yaw (given in radians). M_MatRotate44I(), M_MatRotate44J() and M_MatRotate44K() multiply T with a matrix describing a rotation of theta radians about the basis vector i, j or k, respectively. M_MatTranslate44v() multiplies T against a matrix describing a transla- tion by vector v. M_MatTranslate44(), M_MatTranslate44X(), M_MatTranslate44Y() and M_MatTranslate44Z() accept individual coordi- nate arguments. M_MatScale44() multiplies T against a matrix describing uniform/non- uniform scaling by [x,y,z,w]. M_MatUniScale44() performs uniform scal- ing by c. SEE ALSO AG_Intro(3), M_Complex(3), M_Quaternion(3), M_Real(3), M_Vector(3) Sparse 1.4: https://sparse.sourceforge.net/ HISTORY The M_Matrix interface first appeared in Agar 1.3.3. Agar 1.7 December 21, 2022 M_MATRIX(3)
NAME | SYNOPSIS | DESCRIPTION | M-BY-N MATRICES | M-BY-N MATRICES: INITIALIZATION | M-BY-N MATRICES: ACCESSING ELEMENTS | M-BY-N MATRICES: OPERATIONS | 4-BY-4 MATRICES | 4-BY-4 MATRICES: INITIALIZATION | 4-BY-4 MATRICES: ACCESSING ELEMENTS | 4-BY-4 MATRICES: OPERATIONS | SEE ALSO | HISTORY
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