Posts Tagged ‘programming’

Matrix data types and type inference

Quasar is an array language, this means that array types (vec, mat and cube) are primitive types and have built-in support (for example, this is in contrast with C/C++ where the user has to define it’s own matrix classes).

The reason for the built-in support is of course that this enables easier mapping of Quasar programs to different parallel devices (GPU, …). Moreover, the user is forced to use one representation for its data (rather than using different class libraries, where it is necessary to wrap one matrix class into another matrix class).

On the other hand, by default Quasar abstracts numeric values into one data type scalar. The type scalar just represents a scalar number, and whether this is a floating point number or a fix point number with 16/32/64-bit precision is actually implementation specific (note currently the Quasar runtime system only supports 32-bit and 64-bit floating point numbers).

Type parameters

For efficiency reasons, there is also support for integer data types int, int8, int16, int32, int64, uint8, uint16, uint32, uint64. (Please note that using 64-bit types can suffer from precision errors, because all the calculations are performed in scalar format). To support matrices built of these types, the array types vec, mat and cube are parametric, for example

  • vec[int8] denotes a vector (1D array) of 8-bit signed integers
  • cube[int] denotes a cube (3D array) of signed integers (note: by default, int is 32-bit).

To simplify the types (and to reduce key strokes while programming), there are a number of built-in type aliases:

type vec  : vec[scalar]      % real-valued vector
type cvec : vec[cscalar]    % complex-valued vector

type mat  : mat[scalar]      % real-valued vector
type cmat : mat[cscalar]    % complex-valued vector

type cube  : cube[scalar]    % real-valued vector
type ccube : cube[cscalar]  % complex-valued vector

Please note that these types are just aliases! For example, cube is just cube[scalar] and not cube[something else]:

a = cube[scalar](10)
assert(type(a, "cube"))    % Successful

b = cube[int](10)
assert(type(b, "cube"))    % Unsuccessful - compiler error

However, in case the intention is to check whether a or b is a 3D array regardless of the element type, the special ?? type can be used:

b = cube[int](10)
assert(type(b, "cube[??]"))   % Successful

Type inference

When the type is not specified (for example data that is read dynamically from a file, using the load("data.qd") function), the default data type is ‘??‘. This is a very generic type, every type comparison with ?? results in TRUE. For example:

assert(type(1i+1, '??'))
assert(type([1,2,3], '??'))

However, using variables of type ?? will prevent the compiler to optimize whole operations (for example, applying reductions or automatically generating kernel functions for for-loops). Therefore, it is generally a bad idea to have functions return variables of unspecified type ‘??‘ and correspondingly the compiler gives a warning message when this is the case.

Practically, the type inference starts from the matrix creation functions zeros, ones, imread, … that have a built-in mechanism for deciding the type of the result (based on the parameters of the function).

For example:

  • A = zeros([1,1,4]) creates a vector of length 4 (vec)
  • B = zeros([2,3]) creates a matrix of dimensions 2 x 3 (mat).
  • C = imread("data.tif") creates a cube at all times.

Note that the type inference also works when a variable is passed to the matrix creation functions:

sz = [1,1,4]; A = zeros(sz)

In this case, the compiler knows that sz is a constant vector, it keeps track of the value and uses it for determining the type of zeros.

However: the compiler cannot do this when the variable sz is passed as argument of a function:

function A = create_data(sz)
    A = zeros(sz)

In this case, because the type of sz is unknown, the compiler cannot determine the type of A and will therefore use the default type ??. For convenience, the compiler then also generates a warning message “could not determine the type of output argument A”. The solution is then simply to specify the type of sz:

function A = create_data(sz : ivec2)
    A = zeros(sz)

This way, the compiler knows that sz is a vector of length 2, and can deduce the type of A, which is a matrix (mat).


The type system can be summarized as follows. There are 6 categories of types:

  1. Primitive scalar types scalar, cscalar, int, int8, …
  2. Matrix types vec, mat, cube

    with parametrized versions vec[??], mat[??], cube[??].

  3. Classes: type R : class / type T : mutable class
  4. Function types [?? -> ??], [(??,??)->(??,??)], …

    Device functions: [__device__ ?? -> ??] Kernel functions: [__kernel__ ?? -> ??]

  5. Individual types type
  6. Type classes: T : [scalar|mat|cube]

Finally, different types can be combined to define new types.


  • Figure out what the following type means:
    type X : [vec[ [??->[int|mat|cube[??->??] ] | int -> ?? | __device__ mat->() ] | cscalar ]

    Just kidding;-)

Quasar for Matlab/Octave users

This section gives an overview of various functions and operators in Matlab and Quasar. Please feel free to extend this page if you find something that is missing.

Quick hints (see reference manual for a complete description):

  • Matrices are passed by reference rather than by value. If necessary, use the copy(.) function.
  • Indices start with 0 rather than with 1.
  • end variable, like in A(end,end) is currently not supported.
  • All functions work transparently on the GPU (in MATLAB you need to use special datatypes GPUsingle, GPUdouble, …)

Arithmetic operators

MATLAB/Octave Quasar
Assignment a=1; b=2; a=1; b=2
Multiple assignment [a,b]=[1,2]; [a,b]=[1,2]
Addition a+b a+b
Subtraction a-b a-b
Multiplication a*b a*b
Division a/b a/b
Power a^b a^b
Remainder rem(a,b) periodize(a,b)
Addition (atomic) a+=1 a+=1
Pointwise multiplication a.*b a.*b
Pointwise division a./b a./b
Pointwise power a.^b a.^b

Relational operators

MATLAB/Octave Quasar
Equality a==b a==b
Less than a < b a < b
Greater than a > b a > b
Less than or equal a <= b a <= b
Greater than or equal a >= b a >= b
Inequal a ~= b a != b

Logical operators

MATLAB/Octave Quasar
Logical AND a && b a && b
Logical OR a || b a || b
Logical NOT ~a !a
Logical XOR xor(a,b) xor(a,b)
Bitwise AND a & b and(a,b)
Bitwise OR a | b or(a,b)
Bitwise NOT ~a not(a)
Bitwise XOR xor(a,b) xor(a,b)

Square root, logarithm

MATLAB/Octave Quasar
Square root sqrt(a) sqrt(a)
Logarithm, base e log(a) log(a)
Logarithm, base 2 log2(a) log2(a)
Logarithm, base 10 log10(a) log10(a)
Exponential function exp(a) exp(a)


MATLAB/Octave Quasar
Round round(a) round(a)
Round up ceil(a) ceil(a)
Round down floor(a) floor(a)
Round towards zero fix(a) Not implemented yet
Fractional part frac(a) frac(a)

Mathematical constants

MATLAB/Octave Quasar
pi pi pi
e exp(1) exp(1)

Missing values (IEEE-754 floating point status flags)

MATLAB/Octave Quasar
Not a number NaN 0/0
+Infinity +inf 1/0
-Infinity -inf -1/0
Is not a number isnan(x) isnan(x)
Is infinity isinf(x) isinf(x)
Is finite isfinite(x) isfinite(x)

Complex numbers

MATLAB/Octave Quasar
Imaginary unit i 1i or 1j
Complex number 3+4i z=3+4i z=3+4i or z=complex(3,4)
Modulus abs(z) abs(z)
Argument arg(z) angle(z)
Real part real(z) real(z)
Imaginary part imag(z) imag(z)
Complex conjugate conj(z) conj(z)


MATLAB/Octave Quasar
Sine sin(x) sin(x)
Cosine cos(x) cos(x)
Tangent tan(x) tan(x)
Hyperbolic Sine sinh(x) sinh(x)
Hyperbolic Cosine cosh(x) cosh(x)
Hyperbolic Tangent tanh(x) tanh(x)
Arcus Sine asin(x) asin(x)
Arcus Cosine acos(x) acos(x)
Arcus Tangent atan(x) atan(x)
Arcus Hyperbolic Sine asinh(x) asinh(x)
Arcus Hyperbolic Cosine acosh(x) acosh(x)
Arcus Hyperbolic Tangent atanh(x) atanh(x)

Random numbers

MATLAB/Octave Quasar
Uniform distribution [0,1[ rand(1,10) rand(1,10)
Gaussian distribution randn(1,10) randn(1,10)
Complex Gaussian distribution randn(1,10)+1i*randn(1,10) complex(randn(1,10),randn(1,10))


MATLAB/Octave Quasar Comment
Row vector a=[1 2 3 4]; a=[1,2,3,4]
Column vector a=[1; 2; 3; 4]; a=transpose([1,2,3,4])
Sequence 1:10 1..10
Sequence with steps 1:2:10 1..2..10
Decreasing sequence 10:-1:1 10..-1..1
Linearly spaced vectors linspace(1,10,5) linspace(1,10,5)
Reverse reverse(a) reverse(a) import “system.q”


Quasar notes: please avoid because this programming approach is not very efficient. More efficient is to create the resulting matrix using zeros(x), then use slice accessors to fill in the different parts.

MATLAB/Octave Quasar Comments
Horizontal concatenation [a b] cat(a,b) import “system.q”
Vertical concatenation [a; b] vertcat(a,b) import “system.q”
Vertical concatenation with sequences [0:5; 6:11] vertcat(0..5,6..11) import “system.q”
Repeating repmat(a,[1 2]) repmat(a,[1,2])


MATLAB/Octave Quasar Comment
The first 10 element X+Y a(1:10,1:10); a[0..9,0..9]
Skip the first elements X+Y a(2:end,2:end); a[1..size(a,0)-1,1..size(a,1)-1]
The last elements a(end,end) a[size(a,0..1)-1]
10th row a(10,:) a[9,:]
10th column a(:,10) a[:,9]
Diagonal elements diag(a) diag(a) import “system.q”
Matrix with diagonal 1,2,…,10 diag(1..10) diag(0..9) import “system.q”
Reshaping reshape(a,[1 2 3]) reshape(a,[1,2,3])
Transposing transpose(a) transpose(a)
Hermitian Transpose a’ herm_transpose(a) import “system.q”
Copying matrices a=b a=copy(b) Matrices passed by reference for efficiency
Flip upsize-down flipud(a) flipud(a)
Flip left-right fliplr(a) fliplr(a)

Matrix dimensions

MATLAB/Octave Quasar Comments
Number of elements numel(a) numel(a)
Number of dimensions ndims(a) ndims(a) import “system.q”
Size size(a) size(a)
Size along 2nd dimension size(a,2) size(a,1)
Size along 2nd & 3rd dimension [size(a,2),size(a,3)] size(a,1..2)

Array/matrix creation

MATLAB/Octave Quasar Comment
Identity matrix eye(3) eye(3)
Zero matrix zeros(2,3) zeros(2,3)
Ones matrix ones(2,3) ones(2,3)
Uninitialized matrix uninit(2,3)
Complex identity matrix complex(eye(3)) complex(eye(3))
Complex zero matrix complex(zeros(2,3)) czeros(2,3)
Complex ones matrix complex(ones(2,3)) complex(ones(2,3))
Complex Uninitialized matrix cuninit(2,3)
Vector zeros(1,10) zeros(10) or vec(10) vec enables generic programming
Matrix zeros(10,10) zeros(10,10) or mat(10,10) mat enables generic programming
Cube zeros(10,10,10) zeros(10,10,10) or cube(10,10,10) cube enables generic programming

Maximum and minimum

MATLAB/Octave Quasar
Maximum of two values max(a,b) max(a,b)
Minimum of two values min(a,b) min(a,b)
Maximum of a matrix max(max(a)) max(a)
Minimum of a matrix along dimension X=1,2 min(a,[],X) min(a,X-1)
Maximum of a matrix along dimension X=1,2 max(a,[],X) max(a,X-1)

Matrix assignment

MATLAB/Octave Quasar Comments
Assigning a constant to a matrix slice a(:,1)=44; a[:,0]=44
Assigning a vector to a matrix row slice a(:,1)=0:99 a[:,0]=0..99
Assigning a vector to a matrix column slice a(1,:)=(0:99)’ a[0,:]=transpose(0..99)
Replace all positive elements by one a(a>0)=1 a[a>0]=1 *New*

Boundary handling, clipping (coordinate transforms)

MATLAB/Octave Quasar Comments
Periodic extension rem(x,N) periodize(x,N)
Mirrored extension mirror_ext(x,N)
Clamping clamp(x,N) returns 0 when x=N, otherwise x.
Saturation between 0 and 1 sat(x) returns 0 when x=1, otherwise x.

Fourier transform

MATLAB/Octave Quasar Comments
1D FFT fft(x) fft1(x)
2D FFT fft2(x) fft2(x)
3D FFT fftN(x) fft3(x)
1D inverse FFT ifft(x) ifft1(x)
2D inverse FFT ifft2(x) ifft2(x)
3D inverse FFT ifftN(x) ifft3(x)
1D fftshift fftshift(x) fftshift1(x) import “system.q”
2D fftshift fftshift(x) fftshift2(x) import “system.q”; Works correctly or images with size MxNx3
3D fftshift fftshift(x) fftshift3(x) import “system.q”
1D inverse fftshift ifftshift(x) ifftshift1(x) import “system.q”
2D inverse fftshift ifftshift(x) ifftshift2(x) import “system.q”; Works correctly or images with size MxNx3
3D inverse fftshift ifftshift(x) ifftshift3(x) import “system.q”

Conversion between integer and floating point

MATLAB/Octave Quasar
To single single(x) float(x) 32-bit FP precision mode only
To double double(x) float(x) 64-bit FP precision mode only
To 32-bit signed integer int32(x) int(x) or int32(x) import “inttypes.q”
To 16-bit signed integer int16(x) int16(x) import “inttypes.q”
To 8-bit signed integer int8(x) int8(x) import “inttypes.q”
To 32-bit unsigned integer uint32(x) uint32(x) import “inttypes.q”
To 16-bit unsigned integer uint16(x) uint16(x) import “inttypes.q”
To 8-bit unsigned integer uint8(x) uint8(x) import “inttypes.q”