# Matrix Functions, Null Space, and Hessenberg Matrices

by on April 4, 2012

The recently released Multiprecision Computing Toolbox version 3.3.2 introduces multiple new routines and features for advanced numerical computing in arbitrary precision.

• Matrix functions, including logarithm, square root, exponential, trigonometric, and a general matrix function
```funm % Evaluate general matrix function expm % Matrix exponential sqrtm % Matrix square root logm % Matrix logarithm sinm % Matrix sine cosm % Matrix cosine sinhm % Matrix hyperbolic sine coshm % Matrix hyperbolic cosine```
• Special cases of an “economy sized” SVD decomposition
```[U,S,V] = svd(X,0) [U,S,V] = svd(X,'econ')```
• Kernel(null space) matrices, and matrices in Hessenberg form
```null % Null space hess % Hessenberg form of matrix```
• The formatted conversion of multiprecision entities into a string, and to standard data types
```num2str % Convert number to string cast % Cast variable to different data type double % Convert to double precision int16 % Convert to 16-bit signed integer int32 % Convert to 32-bit signed integer int64 % Convert to 64-bit signed integer int8 % Convert to 8-bit signed integer single % Convert to single precision uint16 % Convert to 16-bit unsigned integer uint32 % Convert to 32-bit unsigned integer uint64 % Convert to 64-bit unsigned integer uint8 % Convert to 8-bit unsigned integer```
• Logical Operations
```all % Determine whether all array elements are nonzero or true any % Determine whether any array elements are nonzero not % Find logical NOT of array or scalar input xor % Logical exclusive-OR```

Examples:

```>> mp.Digits(50);   % Matrix Logarithm >> X = mp.rand(5); >> norm(X - expm(logm(X)),1) 1.4833924403864679909085701388241692669212380232102e-54   % Null Space of Matrix >> X = mp.rand(3,5); >> norm(X*null(X),1) 8.6664199965605403870371093638563522224156087802478e-56   % Hessenberg Form of Matrix >> X = mp.rand(10); >> [P,H] = hess(X); >> norm(X - P*H*P',1) 3.3951974810054822928039499037225474000757620280265e-54```

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