反转一个四乘四的矩阵

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You can use the GNU Scientific Library or look the code up in it.

Edit: You seem to want the Linear Algebra section.

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Here is a small (just one header) C++ vector math library (geared towards 3D programming). If you use it, keep in mind that layout of its matrices in memory is inverted comparing to what OpenGL expects, I had fun time figuring it out...

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最佳答案

here:

bool gluInvertMatrix(const double m[16], double invOut[16])
{
double inv[16], det;
int i;


inv[0] = m[5]  * m[10] * m[15] -
m[5]  * m[11] * m[14] -
m[9]  * m[6]  * m[15] +
m[9]  * m[7]  * m[14] +
m[13] * m[6]  * m[11] -
m[13] * m[7]  * m[10];


inv[4] = -m[4]  * m[10] * m[15] +
m[4]  * m[11] * m[14] +
m[8]  * m[6]  * m[15] -
m[8]  * m[7]  * m[14] -
m[12] * m[6]  * m[11] +
m[12] * m[7]  * m[10];


inv[8] = m[4]  * m[9] * m[15] -
m[4]  * m[11] * m[13] -
m[8]  * m[5] * m[15] +
m[8]  * m[7] * m[13] +
m[12] * m[5] * m[11] -
m[12] * m[7] * m[9];


inv[12] = -m[4]  * m[9] * m[14] +
m[4]  * m[10] * m[13] +
m[8]  * m[5] * m[14] -
m[8]  * m[6] * m[13] -
m[12] * m[5] * m[10] +
m[12] * m[6] * m[9];


inv[1] = -m[1]  * m[10] * m[15] +
m[1]  * m[11] * m[14] +
m[9]  * m[2] * m[15] -
m[9]  * m[3] * m[14] -
m[13] * m[2] * m[11] +
m[13] * m[3] * m[10];


inv[5] = m[0]  * m[10] * m[15] -
m[0]  * m[11] * m[14] -
m[8]  * m[2] * m[15] +
m[8]  * m[3] * m[14] +
m[12] * m[2] * m[11] -
m[12] * m[3] * m[10];


inv[9] = -m[0]  * m[9] * m[15] +
m[0]  * m[11] * m[13] +
m[8]  * m[1] * m[15] -
m[8]  * m[3] * m[13] -
m[12] * m[1] * m[11] +
m[12] * m[3] * m[9];


inv[13] = m[0]  * m[9] * m[14] -
m[0]  * m[10] * m[13] -
m[8]  * m[1] * m[14] +
m[8]  * m[2] * m[13] +
m[12] * m[1] * m[10] -
m[12] * m[2] * m[9];


inv[2] = m[1]  * m[6] * m[15] -
m[1]  * m[7] * m[14] -
m[5]  * m[2] * m[15] +
m[5]  * m[3] * m[14] +
m[13] * m[2] * m[7] -
m[13] * m[3] * m[6];


inv[6] = -m[0]  * m[6] * m[15] +
m[0]  * m[7] * m[14] +
m[4]  * m[2] * m[15] -
m[4]  * m[3] * m[14] -
m[12] * m[2] * m[7] +
m[12] * m[3] * m[6];


inv[10] = m[0]  * m[5] * m[15] -
m[0]  * m[7] * m[13] -
m[4]  * m[1] * m[15] +
m[4]  * m[3] * m[13] +
m[12] * m[1] * m[7] -
m[12] * m[3] * m[5];


inv[14] = -m[0]  * m[5] * m[14] +
m[0]  * m[6] * m[13] +
m[4]  * m[1] * m[14] -
m[4]  * m[2] * m[13] -
m[12] * m[1] * m[6] +
m[12] * m[2] * m[5];


inv[3] = -m[1] * m[6] * m[11] +
m[1] * m[7] * m[10] +
m[5] * m[2] * m[11] -
m[5] * m[3] * m[10] -
m[9] * m[2] * m[7] +
m[9] * m[3] * m[6];


inv[7] = m[0] * m[6] * m[11] -
m[0] * m[7] * m[10] -
m[4] * m[2] * m[11] +
m[4] * m[3] * m[10] +
m[8] * m[2] * m[7] -
m[8] * m[3] * m[6];


inv[11] = -m[0] * m[5] * m[11] +
m[0] * m[7] * m[9] +
m[4] * m[1] * m[11] -
m[4] * m[3] * m[9] -
m[8] * m[1] * m[7] +
m[8] * m[3] * m[5];


inv[15] = m[0] * m[5] * m[10] -
m[0] * m[6] * m[9] -
m[4] * m[1] * m[10] +
m[4] * m[2] * m[9] +
m[8] * m[1] * m[6] -
m[8] * m[2] * m[5];


det = m[0] * inv[0] + m[1] * inv[4] + m[2] * inv[8] + m[3] * inv[12];


if (det == 0)
return false;


det = 1.0 / det;


for (i = 0; i < 16; i++)
invOut[i] = inv[i] * det;


return true;
}

This was lifted from MESA implementation of the GLU library.

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If you need a C++ matrix library with a lot of functions, have a look at Eigen library - http://eigen.tuxfamily.org

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I 'rolled up' the MESA implementation (also wrote a couple of unit tests to ensure it actually works).

Here:

float invf(int i,int j,const float* m){


int o = 2+(j-i);


i += 4+o;
j += 4-o;


#define e(a,b) m[ ((j+b)%4)*4 + ((i+a)%4) ]


float inv =
+ e(+1,-1)*e(+0,+0)*e(-1,+1)
+ e(+1,+1)*e(+0,-1)*e(-1,+0)
+ e(-1,-1)*e(+1,+0)*e(+0,+1)
- e(-1,-1)*e(+0,+0)*e(+1,+1)
- e(-1,+1)*e(+0,-1)*e(+1,+0)
- e(+1,-1)*e(-1,+0)*e(+0,+1);


return (o%2)?inv : -inv;


#undef e


}


bool inverseMatrix4x4(const float *m, float *out)
{


float inv[16];


for(int i=0;i<4;i++)
for(int j=0;j<4;j++)
inv[j*4+i] = invf(i,j,m);


double D = 0;


for(int k=0;k<4;k++) D += m[k] * inv[k*4];


if (D == 0) return false;


D = 1.0 / D;


for (int i = 0; i < 16; i++)
out[i] = inv[i] * D;


return true;


}

I wrote a little about this and display the pattern of positive/negative factors on my blog.

As suggested by @LiraNuna, on many platforms hardware accelerated versions of such routines are available so I'm happy to have a 'backup version' that's readable and concise.

Note: this may run 3.5 times slower or worse than the MESA implementation. You can shift the pattern of factors to remove some additions etc... but it would lose in readability and still won't be very fast.

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You can make it faster according to this blog.

#define SUBP(i,j) input[i][j]
#define SUBQ(i,j) input[i][2+j]
#define SUBR(i,j) input[2+i][j]
#define SUBS(i,j) input[2+i][2+j]


#define OUTP(i,j) output[i][j]
#define OUTQ(i,j) output[i][2+j]
#define OUTR(i,j) output[2+i][j]
#define OUTS(i,j) output[2+i][2+j]


#define INVP(i,j) invP[i][j]
#define INVPQ(i,j) invPQ[i][j]
#define RINVP(i,j) RinvP[i][j]
#define INVPQ(i,j) invPQ[i][j]
#define RINVPQ(i,j) RinvPQ[i][j]
#define INVPQR(i,j) invPQR[i][j]
#define INVS(i,j) invS[i][j]


#define MULTI(MAT1, MAT2, MAT3) \
MAT3(0,0)=MAT1(0,0)*MAT2(0,0) + MAT1(0,1)*MAT2(1,0); \
MAT3(0,1)=MAT1(0,0)*MAT2(0,1) + MAT1(0,1)*MAT2(1,1); \
MAT3(1,0)=MAT1(1,0)*MAT2(0,0) + MAT1(1,1)*MAT2(1,0); \
MAT3(1,1)=MAT1(1,0)*MAT2(0,1) + MAT1(1,1)*MAT2(1,1);


#define INV(MAT1, MAT2) \
_det = 1.0 / (MAT1(0,0) * MAT1(1,1) - MAT1(0,1) * MAT1(1,0)); \
MAT2(0,0) = MAT1(1,1) * _det; \
MAT2(1,1) = MAT1(0,0) * _det; \
MAT2(0,1) = -MAT1(0,1) * _det; \
MAT2(1,0) = -MAT1(1,0) * _det; \


#define SUBTRACT(MAT1, MAT2, MAT3) \
MAT3(0,0)=MAT1(0,0) - MAT2(0,0); \
MAT3(0,1)=MAT1(0,1) - MAT2(0,1); \
MAT3(1,0)=MAT1(1,0) - MAT2(1,0); \
MAT3(1,1)=MAT1(1,1) - MAT2(1,1);


#define NEGATIVE(MAT) \
MAT(0,0)=-MAT(0,0); \
MAT(0,1)=-MAT(0,1); \
MAT(1,0)=-MAT(1,0); \
MAT(1,1)=-MAT(1,1);




void getInvertMatrix(complex<double> input[4][4], complex<double> output[4][4]) {
complex<double> _det;
complex<double> invP[2][2];
complex<double> invPQ[2][2];
complex<double> RinvP[2][2];
complex<double> RinvPQ[2][2];
complex<double> invPQR[2][2];
complex<double> invS[2][2];




INV(SUBP, INVP);
MULTI(SUBR, INVP, RINVP);
MULTI(INVP, SUBQ, INVPQ);
MULTI(RINVP, SUBQ, RINVPQ);
SUBTRACT(SUBS, RINVPQ, INVS);
INV(INVS, OUTS);
NEGATIVE(OUTS);
MULTI(OUTS, RINVP, OUTR);
MULTI(INVPQ, OUTS, OUTQ);
MULTI(INVPQ, OUTR, INVPQR);
SUBTRACT(INVP, INVPQR, OUTP);
}

This is not a complete implementation because P may not be invertible, but you can combine this code with MESA implementation to get a better performance.

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If anyone looking for more costumized code and "easier to read", then I got this

var A2323 = m.m22 * m.m33 - m.m23 * m.m32 ;
var A1323 = m.m21 * m.m33 - m.m23 * m.m31 ;
var A1223 = m.m21 * m.m32 - m.m22 * m.m31 ;
var A0323 = m.m20 * m.m33 - m.m23 * m.m30 ;
var A0223 = m.m20 * m.m32 - m.m22 * m.m30 ;
var A0123 = m.m20 * m.m31 - m.m21 * m.m30 ;
var A2313 = m.m12 * m.m33 - m.m13 * m.m32 ;
var A1313 = m.m11 * m.m33 - m.m13 * m.m31 ;
var A1213 = m.m11 * m.m32 - m.m12 * m.m31 ;
var A2312 = m.m12 * m.m23 - m.m13 * m.m22 ;
var A1312 = m.m11 * m.m23 - m.m13 * m.m21 ;
var A1212 = m.m11 * m.m22 - m.m12 * m.m21 ;
var A0313 = m.m10 * m.m33 - m.m13 * m.m30 ;
var A0213 = m.m10 * m.m32 - m.m12 * m.m30 ;
var A0312 = m.m10 * m.m23 - m.m13 * m.m20 ;
var A0212 = m.m10 * m.m22 - m.m12 * m.m20 ;
var A0113 = m.m10 * m.m31 - m.m11 * m.m30 ;
var A0112 = m.m10 * m.m21 - m.m11 * m.m20 ;


var det = m.m00 * ( m.m11 * A2323 - m.m12 * A1323 + m.m13 * A1223 )
- m.m01 * ( m.m10 * A2323 - m.m12 * A0323 + m.m13 * A0223 )
+ m.m02 * ( m.m10 * A1323 - m.m11 * A0323 + m.m13 * A0123 )
- m.m03 * ( m.m10 * A1223 - m.m11 * A0223 + m.m12 * A0123 ) ;
det = 1 / det;


return new Matrix4x4() {
m00 = det *   ( m.m11 * A2323 - m.m12 * A1323 + m.m13 * A1223 ),
m01 = det * - ( m.m01 * A2323 - m.m02 * A1323 + m.m03 * A1223 ),
m02 = det *   ( m.m01 * A2313 - m.m02 * A1313 + m.m03 * A1213 ),
m03 = det * - ( m.m01 * A2312 - m.m02 * A1312 + m.m03 * A1212 ),
m10 = det * - ( m.m10 * A2323 - m.m12 * A0323 + m.m13 * A0223 ),
m11 = det *   ( m.m00 * A2323 - m.m02 * A0323 + m.m03 * A0223 ),
m12 = det * - ( m.m00 * A2313 - m.m02 * A0313 + m.m03 * A0213 ),
m13 = det *   ( m.m00 * A2312 - m.m02 * A0312 + m.m03 * A0212 ),
m20 = det *   ( m.m10 * A1323 - m.m11 * A0323 + m.m13 * A0123 ),
m21 = det * - ( m.m00 * A1323 - m.m01 * A0323 + m.m03 * A0123 ),
m22 = det *   ( m.m00 * A1313 - m.m01 * A0313 + m.m03 * A0113 ),
m23 = det * - ( m.m00 * A1312 - m.m01 * A0312 + m.m03 * A0112 ),
m30 = det * - ( m.m10 * A1223 - m.m11 * A0223 + m.m12 * A0123 ),
m31 = det *   ( m.m00 * A1223 - m.m01 * A0223 + m.m02 * A0123 ),
m32 = det * - ( m.m00 * A1213 - m.m01 * A0213 + m.m02 * A0113 ),
m33 = det *   ( m.m00 * A1212 - m.m01 * A0212 + m.m02 * A0112 ),
};

I don't write the code, but my program did. I made a small program to make a program that calculate the determinant and inverse of any N-matrix.

I do it because once in the past I need a code that inverses 5x5 matrix, but nobody in the earth have done this so I made one.

Take a look about the program here.

EDIT: The matrix layout is row-by-row (meaning m01 is in the first row and second column). Also the language is C#, but should be easy to convert into C.

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Inspired by @shoosh to check out MESA implementations, I found that matrix inversion looks quite different in more recent mesa releases. I suppose those are good improvements. Here's the matrix inversion code from Mesa-17.3.9:

/* Returns true for success, false for failure (singular matrix) */
bool DirectVolumeRenderer::_mesa_invert_matrix_general( GLfloat out[16], const GLfloat in[16] )
{
/**
* References an element of 4x4 matrix.
* Calculate the linear storage index of the element and references it.
*/
#define MAT(m,r,c) (m)[(c)*4+(r)]
/**
* Swaps the values of two floating point variables.
*/
#define SWAP_ROWS(a, b) { GLfloat *_tmp = a; (a)=(b); (b)=_tmp; }


const GLfloat *m = in;
GLfloat wtmp[4][8];
GLfloat m0, m1, m2, m3, s;
GLfloat *r0, *r1, *r2, *r3;


r0 = wtmp[0], r1 = wtmp[1], r2 = wtmp[2], r3 = wtmp[3];


r0[0] = MAT(m,0,0), r0[1] = MAT(m,0,1),
r0[2] = MAT(m,0,2), r0[3] = MAT(m,0,3),
r0[4] = 1.0, r0[5] = r0[6] = r0[7] = 0.0,


r1[0] = MAT(m,1,0), r1[1] = MAT(m,1,1),
r1[2] = MAT(m,1,2), r1[3] = MAT(m,1,3),
r1[5] = 1.0, r1[4] = r1[6] = r1[7] = 0.0,


r2[0] = MAT(m,2,0), r2[1] = MAT(m,2,1),
r2[2] = MAT(m,2,2), r2[3] = MAT(m,2,3),
r2[6] = 1.0, r2[4] = r2[5] = r2[7] = 0.0,


r3[0] = MAT(m,3,0), r3[1] = MAT(m,3,1),
r3[2] = MAT(m,3,2), r3[3] = MAT(m,3,3),
r3[7] = 1.0, r3[4] = r3[5] = r3[6] = 0.0;


/* choose pivot - or die */
if (fabsf(r3[0])>fabsf(r2[0])) SWAP_ROWS(r3, r2);
if (fabsf(r2[0])>fabsf(r1[0])) SWAP_ROWS(r2, r1);
if (fabsf(r1[0])>fabsf(r0[0])) SWAP_ROWS(r1, r0);
if (0.0F == r0[0])
return false;


/* eliminate first variable     */
m1 = r1[0]/r0[0]; m2 = r2[0]/r0[0]; m3 = r3[0]/r0[0];
s = r0[1]; r1[1] -= m1 * s; r2[1] -= m2 * s; r3[1] -= m3 * s;
s = r0[2]; r1[2] -= m1 * s; r2[2] -= m2 * s; r3[2] -= m3 * s;
s = r0[3]; r1[3] -= m1 * s; r2[3] -= m2 * s; r3[3] -= m3 * s;
s = r0[4];
if (s != 0.0F) { r1[4] -= m1 * s; r2[4] -= m2 * s; r3[4] -= m3 * s; }
s = r0[5];
if (s != 0.0F) { r1[5] -= m1 * s; r2[5] -= m2 * s; r3[5] -= m3 * s; }
s = r0[6];
if (s != 0.0F) { r1[6] -= m1 * s; r2[6] -= m2 * s; r3[6] -= m3 * s; }
s = r0[7];
if (s != 0.0F) { r1[7] -= m1 * s; r2[7] -= m2 * s; r3[7] -= m3 * s; }


/* choose pivot - or die */
if (fabsf(r3[1])>fabsf(r2[1])) SWAP_ROWS(r3, r2);
if (fabsf(r2[1])>fabsf(r1[1])) SWAP_ROWS(r2, r1);
if (0.0F == r1[1])
return false;


/* eliminate second variable */
m2 = r2[1]/r1[1]; m3 = r3[1]/r1[1];
r2[2] -= m2 * r1[2]; r3[2] -= m3 * r1[2];
r2[3] -= m2 * r1[3]; r3[3] -= m3 * r1[3];
s = r1[4]; if (0.0F != s) { r2[4] -= m2 * s; r3[4] -= m3 * s; }
s = r1[5]; if (0.0F != s) { r2[5] -= m2 * s; r3[5] -= m3 * s; }
s = r1[6]; if (0.0F != s) { r2[6] -= m2 * s; r3[6] -= m3 * s; }
s = r1[7]; if (0.0F != s) { r2[7] -= m2 * s; r3[7] -= m3 * s; }


/* choose pivot - or die */
if (fabsf(r3[2])>fabsf(r2[2])) SWAP_ROWS(r3, r2);
if (0.0F == r2[2])
return false;


/* eliminate third variable */
m3 = r3[2]/r2[2];
r3[3] -= m3 * r2[3], r3[4] -= m3 * r2[4],
r3[5] -= m3 * r2[5], r3[6] -= m3 * r2[6],
r3[7] -= m3 * r2[7];


/* last check */
if (0.0F == r3[3])
return false;


s = 1.0F/r3[3];             /* now back substitute row 3 */
r3[4] *= s; r3[5] *= s; r3[6] *= s; r3[7] *= s;


m2 = r2[3];                 /* now back substitute row 2 */
s  = 1.0F/r2[2];
r2[4] = s * (r2[4] - r3[4] * m2), r2[5] = s * (r2[5] - r3[5] * m2),
r2[6] = s * (r2[6] - r3[6] * m2), r2[7] = s * (r2[7] - r3[7] * m2);
m1 = r1[3];
r1[4] -= r3[4] * m1, r1[5] -= r3[5] * m1,
r1[6] -= r3[6] * m1, r1[7] -= r3[7] * m1;
m0 = r0[3];
r0[4] -= r3[4] * m0, r0[5] -= r3[5] * m0,
r0[6] -= r3[6] * m0, r0[7] -= r3[7] * m0;


m1 = r1[2];                 /* now back substitute row 1 */
s  = 1.0F/r1[1];
r1[4] = s * (r1[4] - r2[4] * m1), r1[5] = s * (r1[5] - r2[5] * m1),
r1[6] = s * (r1[6] - r2[6] * m1), r1[7] = s * (r1[7] - r2[7] * m1);
m0 = r0[2];
r0[4] -= r2[4] * m0, r0[5] -= r2[5] * m0,
r0[6] -= r2[6] * m0, r0[7] -= r2[7] * m0;


m0 = r0[1];                 /* now back substitute row 0 */
s  = 1.0F/r0[0];
r0[4] = s * (r0[4] - r1[4] * m0), r0[5] = s * (r0[5] - r1[5] * m0),
r0[6] = s * (r0[6] - r1[6] * m0), r0[7] = s * (r0[7] - r1[7] * m0);


MAT(out,0,0) = r0[4]; MAT(out,0,1) = r0[5],
MAT(out,0,2) = r0[6]; MAT(out,0,3) = r0[7],
MAT(out,1,0) = r1[4]; MAT(out,1,1) = r1[5],
MAT(out,1,2) = r1[6]; MAT(out,1,3) = r1[7],
MAT(out,2,0) = r2[4]; MAT(out,2,1) = r2[5],
MAT(out,2,2) = r2[6]; MAT(out,2,3) = r2[7],
MAT(out,3,0) = r3[4]; MAT(out,3,1) = r3[5],
MAT(out,3,2) = r3[6]; MAT(out,3,3) = r3[7];


#undef SWAP_ROWS
#undef MAT


return true;
}

Note: you can find this piece of code in the mesa code base: mesa-17.3.9/src/mesa/math/m_matrix.c.

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This is the C++ version for @willnode's answer

template<typename Matrix>
static inline void InvertMatrix4(const Matrix& m, Matrix& im, double& det)
{
double A2323 = m(2, 2) * m(3, 3) - m(2, 3) * m(3, 2);
double A1323 = m(2, 1) * m(3, 3) - m(2, 3) * m(3, 1);
double A1223 = m(2, 1) * m(3, 2) - m(2, 2) * m(3, 1);
double A0323 = m(2, 0) * m(3, 3) - m(2, 3) * m(3, 0);
double A0223 = m(2, 0) * m(3, 2) - m(2, 2) * m(3, 0);
double A0123 = m(2, 0) * m(3, 1) - m(2, 1) * m(3, 0);
double A2313 = m(1, 2) * m(3, 3) - m(1, 3) * m(3, 2);
double A1313 = m(1, 1) * m(3, 3) - m(1, 3) * m(3, 1);
double A1213 = m(1, 1) * m(3, 2) - m(1, 2) * m(3, 1);
double A2312 = m(1, 2) * m(2, 3) - m(1, 3) * m(2, 2);
double A1312 = m(1, 1) * m(2, 3) - m(1, 3) * m(2, 1);
double A1212 = m(1, 1) * m(2, 2) - m(1, 2) * m(2, 1);
double A0313 = m(1, 0) * m(3, 3) - m(1, 3) * m(3, 0);
double A0213 = m(1, 0) * m(3, 2) - m(1, 2) * m(3, 0);
double A0312 = m(1, 0) * m(2, 3) - m(1, 3) * m(2, 0);
double A0212 = m(1, 0) * m(2, 2) - m(1, 2) * m(2, 0);
double A0113 = m(1, 0) * m(3, 1) - m(1, 1) * m(3, 0);
double A0112 = m(1, 0) * m(2, 1) - m(1, 1) * m(2, 0);


det = m(0, 0) * ( m(1, 1) * A2323 - m(1, 2) * A1323 + m(1, 3) * A1223 )
- m(0, 1) * ( m(1, 0) * A2323 - m(1, 2) * A0323 + m(1, 3) * A0223 )
+ m(0, 2) * ( m(1, 0) * A1323 - m(1, 1) * A0323 + m(1, 3) * A0123 )
- m(0, 3) * ( m(1, 0) * A1223 - m(1, 1) * A0223 + m(1, 2) * A0123 );
det = 1 / det;


im(0, 0) = det *   ( m(1, 1) * A2323 - m(1, 2) * A1323 + m(1, 3) * A1223 );
im(0, 1) = det * - ( m(0, 1) * A2323 - m(0, 2) * A1323 + m(0, 3) * A1223 );
im(0, 2) = det *   ( m(0, 1) * A2313 - m(0, 2) * A1313 + m(0, 3) * A1213 );
im(0, 3) = det * - ( m(0, 1) * A2312 - m(0, 2) * A1312 + m(0, 3) * A1212 );
im(1, 0) = det * - ( m(1, 0) * A2323 - m(1, 2) * A0323 + m(1, 3) * A0223 );
im(1, 1) = det *   ( m(0, 0) * A2323 - m(0, 2) * A0323 + m(0, 3) * A0223 );
im(1, 2) = det * - ( m(0, 0) * A2313 - m(0, 2) * A0313 + m(0, 3) * A0213 );
im(1, 3) = det *   ( m(0, 0) * A2312 - m(0, 2) * A0312 + m(0, 3) * A0212 );
im(2, 0) = det *   ( m(1, 0) * A1323 - m(1, 1) * A0323 + m(1, 3) * A0123 );
im(2, 1) = det * - ( m(0, 0) * A1323 - m(0, 1) * A0323 + m(0, 3) * A0123 );
im(2, 2) = det *   ( m(0, 0) * A1313 - m(0, 1) * A0313 + m(0, 3) * A0113 );
im(2, 3) = det * - ( m(0, 0) * A1312 - m(0, 1) * A0312 + m(0, 3) * A0112 );
im(3, 0) = det * - ( m(1, 0) * A1223 - m(1, 1) * A0223 + m(1, 2) * A0123 );
im(3, 1) = det *   ( m(0, 0) * A1223 - m(0, 1) * A0223 + m(0, 2) * A0123 );
im(3, 2) = det * - ( m(0, 0) * A1213 - m(0, 1) * A0213 + m(0, 2) * A0113 );
im(3, 3) = det *   ( m(0, 0) * A1212 - m(0, 1) * A0212 + m(0, 2) * A0112 );
}

小开

If you want to compute the inverse matrix of 4x4 matrix, then I recommend to use a library like OpenGL Mathematics (GLM) :

Anyway, you can do it from scratch. The following implementation is similar to the implementation of glm::inverse, but it is not as highly optimized:

bool InverseMat44( const GLfloat m[16], GLfloat invOut[16] )
{
float inv[16], det;
int i;


inv[0]  =  m[5] * m[10] * m[15] - m[5] * m[11] * m[14] - m[9] * m[6] * m[15] + m[9] * m[7] * m[14] + m[13] * m[6] * m[11] - m[13] * m[7] * m[10];
inv[4]  = -m[4] * m[10] * m[15] + m[4] * m[11] * m[14] + m[8] * m[6] * m[15] - m[8] * m[7] * m[14] - m[12] * m[6] * m[11] + m[12] * m[7] * m[10];
inv[8]  =  m[4] * m[9]  * m[15] - m[4] * m[11] * m[13] - m[8] * m[5] * m[15] + m[8] * m[7] * m[13] + m[12] * m[5] * m[11] - m[12] * m[7] * m[9];
inv[12] = -m[4] * m[9]  * m[14] + m[4] * m[10] * m[13] + m[8] * m[5] * m[14] - m[8] * m[6] * m[13] - m[12] * m[5] * m[10] + m[12] * m[6] * m[9];
inv[1]  = -m[1] * m[10] * m[15] + m[1] * m[11] * m[14] + m[9] * m[2] * m[15] - m[9] * m[3] * m[14] - m[13] * m[2] * m[11] + m[13] * m[3] * m[10];
inv[5]  =  m[0] * m[10] * m[15] - m[0] * m[11] * m[14] - m[8] * m[2] * m[15] + m[8] * m[3] * m[14] + m[12] * m[2] * m[11] - m[12] * m[3] * m[10];
inv[9]  = -m[0] * m[9]  * m[15] + m[0] * m[11] * m[13] + m[8] * m[1] * m[15] - m[8] * m[3] * m[13] - m[12] * m[1] * m[11] + m[12] * m[3] * m[9];
inv[13] =  m[0] * m[9]  * m[14] - m[0] * m[10] * m[13] - m[8] * m[1] * m[14] + m[8] * m[2] * m[13] + m[12] * m[1] * m[10] - m[12] * m[2] * m[9];
inv[2]  =  m[1] * m[6]  * m[15] - m[1] * m[7]  * m[14] - m[5] * m[2] * m[15] + m[5] * m[3] * m[14] + m[13] * m[2] * m[7]  - m[13] * m[3] * m[6];
inv[6]  = -m[0] * m[6]  * m[15] + m[0] * m[7]  * m[14] + m[4] * m[2] * m[15] - m[4] * m[3] * m[14] - m[12] * m[2] * m[7]  + m[12] * m[3] * m[6];
inv[10] =  m[0] * m[5]  * m[15] - m[0] * m[7]  * m[13] - m[4] * m[1] * m[15] + m[4] * m[3] * m[13] + m[12] * m[1] * m[7]  - m[12] * m[3] * m[5];
inv[14] = -m[0] * m[5]  * m[14] + m[0] * m[6]  * m[13] + m[4] * m[1] * m[14] - m[4] * m[2] * m[13] - m[12] * m[1] * m[6]  + m[12] * m[2] * m[5];
inv[3]  = -m[1] * m[6]  * m[11] + m[1] * m[7]  * m[10] + m[5] * m[2] * m[11] - m[5] * m[3] * m[10] - m[9]  * m[2] * m[7]  + m[9]  * m[3] * m[6];
inv[7]  =  m[0] * m[6]  * m[11] - m[0] * m[7]  * m[10] - m[4] * m[2] * m[11] + m[4] * m[3] * m[10] + m[8]  * m[2] * m[7]  - m[8]  * m[3] * m[6];
inv[11] = -m[0] * m[5]  * m[11] + m[0] * m[7]  * m[9]  + m[4] * m[1] * m[11] - m[4] * m[3] * m[9]  - m[8]  * m[1] * m[7]  + m[8]  * m[3] * m[5];
inv[15] =  m[0] * m[5]  * m[10] - m[0] * m[6]  * m[9]  - m[4] * m[1] * m[10] + m[4] * m[2] * m[9]  + m[8]  * m[1] * m[6]  - m[8]  * m[2] * m[5];


det = m[0] * inv[0] + m[1] * inv[4] + m[2] * inv[8] + m[3] * inv[12];
if (det == 0) return false;
det = 1.0 / det;


for (i = 0; i < 16; i++)
invOut[i] = inv[i] * det;


return true;
}