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# inverting a 4×4 matrix

Questions:

i am looking for a sample code implementation on how to invert a 4×4 matrix. i know there is gaussian eleminiation, LU decomposition, etc. but instead of looking at them in detail i am really just looking for the code to do this.

language ideally C++, data is available in array of 16 floats in cloumn-major order.

thank you!

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.

Questions:

If you’re looking for ‘just works’ implementation that is also very optimized without the need to understand the code, I highly reccommend using Intel’s optimized SSE matrix inverse routine described here. There’s also a reference impelementation for both gaussian elimination and Cramer’s rule in C.

I warn though, the SSE code is not pretty if you don’t understand MMX/SSE intrinsics.

Questions:

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

Questions:

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;

}
``````

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.

Questions:

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…

Questions:

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

Edit: You seem to want the Linear Algebra section.

Questions:

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.

Questions:

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 5×5 matrix, but nobody in the earth have done this so I made one.

Take a look about the program here.

Questions: