• Objective

  Write a CUDA Program to Compute Vector Vector Addition
  

• Description

  The input vectors are generated on host-CPU and transfer the vectors to device-GPU for vector vector vector addition. A simple kernel based on the grid of thread blocks is generated in which thread is given a unique thread ID within its block. Each thread performs partial addition of two vectors and the final resultant value is generated on device-GPU and transfered to host-CPU.

Important Steps :

Steps	Description
1.	Memory allocation on host-CPU and device-GPU : Allocate memory for two input vectors and resultant vector on host-CPU & device-GPU Use cudaMalloc(void** array, int size)
2.	Input data Generation : Fill the input vector with single/double precision real values using randomized data as per input specification
3.	Transfer data from host-CPU to device-GPU: Transfer the host-CPU vector to device-GPU to perform computation Use cudaMemcpy((void*)device_array, (void*)host_array, size , cudaMemcpyHostToDevice )
4.	Launch Kernel : Define the dimensions for Grid and Block on host-CPU and launch the kernel for execution on device-GPU. Computation on device is performed for vector vector addition
5.	Transfer the result from device-GPU to host-CPU : Copy resultant vector to host-CPU from device-GPU. Use cudaMemcpy((void*)host_array, (void*)device_array, size , cudaMemcpyDeviceToHost)
6.	Check correctness of the result on host-CPU Compute vector-vector addition on host-CPU and Compare CPU & GPU results.
7.	Free the memory Free the memory of arrays allocated on host-CPU & device-GPU Use cudaFree(void* array)

• Input

vector size

• Output

execution time in seconds,Gflops achieved

• Objective

  Write a CUDA Program to compute Matrix Matrix Addition.
  

• Description

  The input matices are generated on host-CPU and transfer the matrices to device-GPU to perform matrix matrix multiplication. A simple kernel based on the grid of thread blocks is generated in which thread is given a unique thread ID within its block. Each thread using its threadId performs addition using one element from each matrix and the final resultant value is generated on device-GPU and transfered to host-GPU.

• Input

matrix size

• Output

execution time in seconds,Gflops achieved

• Objective

  Write a CUDA Program to compute Vector Vector Multiplication.
  

• Description

  The input vectors are generated on host-CPU and transfer the vectors to device-GPU for vector vector vector multiplication. A simple kernel based on the grid of thread blocks is generated in which thread is given a unique thread ID within its block. Each thread performs partial multiplication of two vectors and the final resultant value is generated on device-GPU and transfered to host-GPU.

• Input

Vector Size

• Output

execution time in seconds,Gflops achieved

• Objective

  Write a CUDA program to find prefix sum of an given array.
  

• Description

  This is simple program which computes prefix sum of an given array element. Prefix sum of an element of an array is defined by summation of all the element processed this element and the current element. The input array is generated on host-CPU and the Each thread performs summation of current element with previous element of an array. . The output array is generated on the device-GPU and transfer back to host-CPU

• Input

Vector Size

• Output

execution time in seconds,Gflops achieved

• Objective

  Write a program using CUDA to find out transpose of matrix.
  

• Description

  The input matrix is generated on the Host-CPU and transfer to Device-GPU. A simple kernel function is written on GPU which generates the out-put matrix on the Device-GPU and transfer back to Host-CPU . The maxNumThread is used for computations.

• Input

matrix size

• Output

execution time in seconds

• Objective

  Write a CUDA Program to calculate value of PI using numerical integration method.
  

• Description

  This program computes the value of PI over a given interval using Numerical integration. The main thread distributes the given interval uniformly over the number of threads on CUDA device. Each thread calculates its part of the interval and finally thread 0 will add up all the value claculated by indivitual thread. Apart from the conventional threading model, in case of cuda program no lock mechanisim is used. In place of lock , every thread do its part of calculation and put the final result in a global array cell , which is allocated to this perticular thread only. After execution of each thread is over , thread 0 is assigned the job to gather all values and produce final result.
  
  The Grid of Thread Blocks and a unique thread ID is used to ditribute the required number of intervals for computation of PI value.

• Input

Number intervals

• Output

Value of PI

• Objective

  Write a CUDA program to find infinity norm of given matrix.
  

• Description

  Infinity Norm of a Matrix: The Row-Wise infinity norm of a matrix is defined to be the maximum of sums of absolute values of elements in a row, over all rows. After the initial validity checks, each thread is assigned a row using its id. If a thread completes its work, and if still there is some row to be operated, that will be assigned to this thread. The process continue till no row left to be operated. The thread 0 will find out maximum sum among all sum of all element of every row.
  
  The input matrix is generated on the Host-CPU and kernel is executed on Device-GPU in which Grid of thread blocks is used to accumulate the partial sum in an array. The infinity norm is computed using _synchreads()

• Input

matrix size

• Output

execution time in seconds,Gflops achieved

• Objective

  Write a CUDA Program to perform Matrix Vector multiplication.
  

• Description

  Input matrix and the vector are generated on the Host-CPU. A simple kernel based on the grid of thread blocks is generated in which thread is given a unique thread ID within its block. Each thread reads one row of the matrix and performs computation with column of the vector to obtain resultant vector on Device-GPU. The resultant solution vector is transferred back to Host-CPU.

• Input

Matrix Size

• Output

execution time in seconds,Gflops achieved

• Objective

  Write a CUDA Program to perform Matrix Matrix multiplication.
  

• Description

  Two input matrices are generated on the Host-CPU. In simple algorithm, the input matrix is partitoned as per Grid of thread blocks. In Global memory implementation,Each thread reads one row of the matrix and performs computation with one column of the another matrix and compute the correspodning elements of resultant marix on Device-GPU. The resultant matrix is transferred back to Host-CPU.While in local memory implementation, Matrices have been divided into BLOCKS of 16 x 16 sizes. One BLOCK loads one tile of both matrices from global memory to shared memory. Each thread with in BLOCK calculate temporal resultant in the local memory. After all threads within BLOCK completed their part of computation, the BLOCK stores the resultant tile into the global resultant matrix.

• Input

Matrix Size

• Output

execution time in seconds,Gflops achieved

• Objective

  Write a CUBLAS CUDA program, for solving system of linear equations [A]{x} = {b} on CUDA enabled NVIDIA programming environment using Jacobi method
  

• Description

  The Jacobi iterative method is one of the simplest iterative techniques to solve system of linear equations. The i^th equation of a system of linear equations [A]{x}={b} is
  

  

  
  

  If all the diagonal elements of A are nonzero (or are made nonzero by permuting the rows and columns of A), we can rewrite equation (1)

  
  

  

  
  

  The Jacobi method starts with an initial guess x₀ for the solution vector x. This initial vector x₀ is used in the right-hand side of equation (2) to arrive at the next approximation x₁ to the solution vector. The vector x₁ is then used in the right hand side of equation (2), and the process continues until a close enough approximation to the actual solution is found. A typical iteration step in the Jacobi method is

  

  
  We now express the iteration step of equation 3 in terms of residual r_k. Equation (3) can be rewritten as
  

  

  
  

  Each process computes n/p values of the vector x in each iteration. These values are gathered by all the processes and each process tests for convergence. If the values have been computed upto a certain accuracy the iterations are stopped otherwise the processes use these values in the next iterations to compute a new set of values.

• Implementation The input matrix and the right hand-side vector, intial soultion vector is generated on Host-CPU and transferred to Device-GPU. In simple algorithm, the input matrix is partitoned as per Grid of thread blocks. Each thread reads one row of the matrix A and performs computation with vector and update the solution vector. Convergence of the solution is checked and the solution vector is transferred back to Host-CPU.
  
  
• Input

Size of Input Matrix and the Vector

• Output

The solution of matrix system of linear equations Ax = b

• Objective

  CUDA implementaiton for Conjugate Gradient Method to solve the system of linear equations [A]{x}={b}. Assume that A is symmetric positive definite matrix.

• Description

  Description of conjugate gradient method :
  
  The conjugate gradient (CG) method is an example of minimizing method. A real n x n matrix A is positive definite if x^T A x > {0} for any n x 1 real, nonzero vector x. For a symmetric positive definite matrix A, the unique vector x that minimizes the quadratic functional
  
  f(x) = (1/2)x^TAx-x^Tb 
  
  is the solution to the system Ax = b, here x and b are n x 1 vectors. It is not particularly relevant when n is very large, since the conjugating time for that number of iterations is usually prohibitive and the property does not hold in presence of rounding errors. The reason is that the gradient of functional f (x) is Ax - b, which is zero when f (x) is minimum. The gradient of a function is a n x 1 vector. We explain some important steps in the algorithm. An iteration of a minimization method is of the form
  
  

  x_k+1 = x_k  + tau_kd_k     (1) 
   
  where tau_k is a scalar step size and d_kis the direction vector, d_k is a descent direction for f at x.  We now consider the problem of determining tau_k, given x_k and d_k, so that f(x) is minimized on the line x = x_k + tau_k d_k, for tau_k. The function f(x_k+ tau d_k) is quadratic in tau, and its minimization leads to the condition
  
  tau _k =  g_k^Tg_k / d_k^TAd_k ,    (2) 
  where g_k=Ax_k - b is the gradient (residue)  vector after k iterations. The residual need not be computed explicitly in each iteration because it can be computed incrementally by using its value from the previous iteration. In the (k+1)^th iteration, the residual g_k+1 can be expressed as follows: 
  
       g_k+1  = Ax_k+1 - b        = A(x_k+ tau_k d_k) - b 
       = Ax_k- b + tau_k Ad_k  
  
       = g_k + tau_k Ad_k     (3)
  
  

  Thus, the only matrix-vector product computed in each iteration is Ad_k, which is already required to compute tau_k  in the equation (2). If A is a symmetric positive definite matrix and d₁, d_2,..., d_n are direction vectors that are conjugate with respect to A (that is, d_i^T Ad_j=0 for all 0<n, j<=n, i!=j), then x_k+1 in the Equation (1) converges to the solution of Ax = bin at most n iterations, assuming no rounding errors.
  
  In practice, however, the number of iterations that yields an acceptable approximation to the solution is much smaller than n.  It  also  makes the gradient at x_k+1 orthogonal to search direction, i.e  d_k^T g_k+1  = 0.  Now we suppose that the search directions are determined by an iteration of the form 
  
  

  d_k+1 = -g_k+1+ beta_k d_k      (4)
  
  

  where d₀ = -g₀  and beta₀, beta₁ , ...... remain to be determined. We find the new search direction in the  plane spanned by the gradient at the most recent point and previous search direction. The parameter beta_k+1is determined by following equation
  
  

  Beta_k+1 = g^T_k+1Ad_k/ d^T_kAd_k     (5)
  And, one can derive orthogonality relations
    g^T_kg _l= 0 (l != k);                 d^T_kAd_l = 0 (l !=k)
  
  The derivation of the above equation (5)  and orthogonality relations is beyond the scope of this document. For details please refer [ ]. Using equation (3) and orthogonality relations, the equation (5)  can be further reduced to
  
  Beta_k+1 = g^T_k+1g_k+1/ g^T_kg_k<     (6)
  
  

  The above equations (1) to (6) lead to CG algorithm. The algorithm terminates when the square of the Euclidean vector norm of gradient (residual) falls below a predetermined tolerance value.  Although all of the versions of the conjugate gradient method obtained by combining the formulas for g_k, Beta_k, and tau_k in various ways are mathematically equivalent, their computer implementation is not. The following version is compared with respect to computational labor, storage requirements, and accuracy. The following sequence of steps are widely accepted.
  

  1. tau _k         =  g_k^Tg_k / d_k^TAd_k 
  2. x_k+1        = x_k  + tau_k d_k 
  3. g_k+1        =  g_k + tau_k Ad_k  
  4. Beta_k+1 = g^T_k+1g_k+1/ g^T_kg_k 
  5. d_k+1        = -g_k+1 + Beta_k d_k 
  
  where k = 0, 1, 2, .......... Initially we choose x₀, calculate g₀ = Ax₀ - b , and put d₀= -g₀ 
  
  The computer implementation of this algorithm is explained as follows :
  
  void CongugateGradient(float x₀ [ ], float b [ ], float d) 
      { 
         float    g, Delta0, Delta1, beta; 
         float    temp, tau; 
         int        iteration;
         iteration = 0;
         x   = x_0;          g = b; 
         g = A x - g; 
         Delta0 = g^T * g;   
         if ( Delta0 <= EPSILON) 
             return;  
         d = -g; 
         do { 
            iteration = iteration + 1; 
            temp = A * d;  
            tau = Delta0 / d^T  * temp;  
            x = x + tau * d;  
            g = g + tau * temp;  
            Delta1 = g^T  * g;
            if ( Delta1 <= EPSILON )  
                   break; 
            beta = Delta1 / Delta0;  
            Delta0 = Delta1;  
            d = -g + beta * d;  
        } while(Delta0 > EPSILON && Iteration < MAX_ITERATIONS);
         return;
   

  Regarding one-dimensional arrays of size n x 1 are required for temp, g, x, d. The storage requirement for matrix. A is depends upon the structure ( dense, band, sparse ) of the matrix.The two dimensional n x n array is the simplest structure to store matrix A. For large sparse matrix A this structure wastes a large amount of storage space, for such matrix A suitable storage scheme should be used.
  
  

• The preconditioned conjugate gradient algorithm  




Let C be a positive definite matrix factored in the form C = E E^T, and let the quadratic functional 


f(x) = (1/2)x^TAx - x^Tb + C , 
We define second quadratic functional g(y) by the transformation y = E^Tx,

g(x) = g(E^-Ty) = (1/2)y^TA^*y - y^Tb ^* + C^*         where A ^* = E^-1AE^-T, b^* = E^-1b, C^* = C. 
Here, A^* is symmetric and positive definite. The similarity transformation

E^-TA^*E^T = E^-TE^-1A = C^-1A 

reveals that A^* and A have same eigen values. If C can be found such that the condition number of the matrix A^* is less than the condition number of the matrix A, then the rate of convergence of the preconditioned method is better than that of conjugate gradient method. We call C the preconditioning matrix, A^* the preconditioned matrix, We assume that the matrix C = EE^T is positive definite, since E is nonsingular by assumption. If the coefficient matrix A has l distinct eigen values, the CG algorithm converges to the solution of the system Ax = b in at most l iterations (assuming no rounding errors). Therefore, if A has many distinct eigen values that vary widely in magnitude, the CG algorithm may require a large number of iterations to converge to an acceptable approximation to the solution.

The speed of convergence of the CG algorithm can be increased by preconditioning A with the congruence transformation A* = E^-1AE^-T where E is a nonsingular matrix. E is chosen such that A* has fewer distinct eigen values than A. The CG algorithm is then used to solve A* y =b*, where x =(E^T)^-1y . The resulting algorithm is called the preconditioned conjugate gradient (PCG) algorithm. The step performed in each iteration of the preconditioned conjugate gradient algorithm are as follows



1.   tau _k         =  g_k^Th_k / d_k^TAd_k 
2.   x_k+1       = x_k + tau_k d_k 
3.  g_k+1       =  g_k + tau_k Ad_k  
4.   h_k+1       =  C^-1g_k+1 
5. &nsbp; beta_k+1 = g^T_k+1h_k+1/ g^T_kh_k 
6.   d_k+1       = -h_k+1 + beta_k+1d_k

where k = 0, 1, 2, .......... Initially we choose x₀, calculate g₀ = Ax₀ - b, h₀= C^-1g₀ and d₀ = -h₀. The multiplication by C^-1 in step (4) is to be interpreted as solving a system of equations with coefficient matrix C. A source of preconditioning matrices is the class of stationary iterative methods for solving the system Ax^* = b. 
• Parallel implementations of the PCG algorithm

The parallel conjugate gradient algorithm involves the following  type of computations and communications

Partitioning of a matrix : The matrix A is obtained by discretization of partial differential equations by finite element, or finite difference method. In such cases, the matrix is either sparse or banded. Consequently, the partition  of the matrix onto p processes play a vital role for performance. For, simplicity , we assume that A is symmetric positive definite and is rowwise block-striped partitioned. 


Scalar Multiplication of a vector and addition of vectors :
Each of these computations can be performed sequentially regardless of the preconditioner and the type of coefficient matrix. If all vectors are distributed identically among the processes, these steps require no communication in a parallel implementation.

Vector inner products :
In some situations, partial vectors are available on each  processes.  MPI Collective library calls are necessary to perform vector inner products  If the parallel computer supports fast reduction operations, such as optimized MPI, then the communication time for the inner-product calculations can be made minimum.

Matrix-vector multiplication :
The computation and the communication cost of the matrix-vector multiplication; depends on the structure of the matrix A. The parallel implementation of the PCG algorithm for three cases one in which A is a block-tridiagonal matrix of the type, two in which it is banded unstructured sparse matrix, and three in which the matrix is sparse give different performance on parallel computers. Various parts of the algorithm in each of the three cases dominate in terms of communication overheads.

Solving the preconditioned system :
The PCG algorithm solves system of linear equations in each iteration The preconditioner C is chosen so that solving the system modified system is in expensive compared to solving the original system of equations Ax = b. Nevertheless, preconditioning increases the amount of computation in each iteration. For good preconditioners, however, the increase is compensated by a reduction in the number of iterations required to achieve acceptable convergence. The computation and the communication requirements of this step depends on the type of preconditioner used. preconditioning method such as diagonal preconditioning, in which the preconditioning matrix C has nonzero elements only along the principle diagonal does not involve any communication Also, Incomplete Cholesky (IC) preconditioning, in which C is based on incomplete Cholesky factorization of A and it may involve different computations and communications in parallel implementation.

The convergence of CG method iterations performed by checking the error criteria i.e. eulicidean norm of the residual vector should be less than prescribed tolerance. This convergence check involves gathering of real value from all processes, which may be very costly operation.

We consider parallel implementations of the PCG algorithm using diagonal preconditioner for dense coefficient matrix type. As we will see, if C is a diagonal preconditioner, then solving the modified system does not require any interprocessor communication. Hence, the communication time in a CG iteration with diagonal preconditioning is the same as that in an iteration of the unpreconditioned algorithm.

Thus the operations that involve any communication overheads are computation of inner products, matrix-vector multiplication and, in case of IC preconditioner solving the system.

• Input

  Input Matrix and Right Hand side Vector

• Output

  Solution x of linear system of matrix equations Ax = b

The input to the problem is given as arguments in the command line. It should be given in the following format ; Suppose that the number of rows of the sparse matrix is n (only square matrices are considered) and the sparsity i.e. the percentage of number of zero's (given in the range 0 to 1) is m, then the program must be run as,

./program_name n m

CPU generates the sparse matrix, the vector to be multiplied using random values and the row_ptr and col_idx vectors based on the sparse matrix.

The CPU prints the time taken for the computation.

• Objective

  Write a CUDA Program to compute Vector Vector Multiplication using multi-GPU
  

• Description

  The input vectors are generated on host-CPU and transfer partial vector elements to each device-GPU for vector vector vector multiplication. A simple kernel based on the grid of thread blocks is generated in which thread is given a unique thread ID within its block. Each thread performs partial multiplication of two vectors on device-GPU and the accumulate the partial result on device-GPU and transfered to Host-CPU-GPU.

• Input

Vector Size

• Output

execution time in seconds,Gflops achieved

• Objective

  Write a CUDA Program to perform Matrix Matrix multiplication on multi-GPU
  

• Description

  Two input matrices are generated on the Host-CPU and divided among multiple GPU-devices for computations. We create a two-dimensional grid that is overlaid on matrices. Matrices have been divided into thread blocks of 16 x 16 sizes. One BLOCK loads one tile of both matrices from global memory to shared memory. Each thread with in BLOCK calculate temporal resultant in the register. After all threads within block completed their part of computation, the BLOCK stores the resultant tile into the global resultant matrix. The resultant matrix is transferred back to Host-CPU. Timing has been recorded for the kernel launch using Cuda Events. Matrix Multiplication has been also computed on Host-CPU and after that both Host-CPU result and device-GPU result has been compared to check correctness of result.
  


Important Steps :

Steps	Description
1.	Memory allocation on host-CPU and device-GPU : Allocate memory for two input matrices and resultant matrix on host-CPU & device-GPU Use cudaMalloc(void** array, int size)
2.	Input data Generation : Fill the input matrices with single/double precision real values using randomized data as per input specification
3.	Transfer data from host-CPU to device-GPU: Transfer the host-CPU matrices to device-GPU to perform computation Use cudaMemcpy((void*)device_array, (void*)host_array, size , cudaMemcpyHostToDevice )
4.	Launch Kernel : Define the dimensions for Grid and Block on host-CPU and launch the kernel for execution on device-GPU.
5.	Transfer the result from device-GPU to host-CPU : Copy resultant matrix to host-CPU from device-GPU. Use cudaMemcpy((void*)host_array, (void*)device_array, size , cudaMemcpyDeviceToHost)
6.	Check correctness of the result on host-CPU Compute matrix-matrix multiplication on host-CPU; Compare CPU and GPU results.
7.	Free the memory Free the memory of arrays allocated on host-CPU & device-GPU Use cudaFree(void* array)

• Input

Matrix Size

• Output

execution time in seconds,Gflops achieved.

• Objective

  Write a CUDA Program to perform Vector Vector multiplication on multi-GPU
  

• Description

  The input vectors are generated on host-CPU divided among multiple GPU-devices for computations. Transfer the vectors to available device_GPUs for vector multiplication using CUBLAS1 library call. The final output value is transferred back to Host-CPU.
  
  Computation on device is performed by CULBAS routines

Important Steps :

Steps	Description
1.	Memory allocation on host-CPU and device-GPU : Allocate memory for two input vectors on host-CPU & device-GPU Use cublasAlloc()
2.	Input data Generation : Fill the input vectors with single/double precision real values using randomized data as per input specification
3.	> Transfer data from host-CPU to device-GPU: Use cublasSetVector()
4.	Perform computation using cublasDdot() CUBLAS library call
5.	Transfer the result from device-GPU to host-CPU :
6.	Check correctness of the result on host-CPU Compute vector-vector multiplication on host-CPU; Compare CPU and GPU results.
7.	Free the memory Free the memory of arrays allocated on host-CPU & device-GPU

• Input

Vector Size

• Output

Execution time in seconds, Gflops acheived

• Objective

  Write a CUDA Program to perform Matrix Matrix multiplication on multi-GPU using CUBLAS Libraries
  

• Description

  Two input matrices are generated on the Host-CPU In simple algorithm, the input matrix is partitoned as per Grid of thread blocks. Each thread reads one row of the matrix and performs computation with one column of the another matrix and compute the correspodning elements of resultant marix on device-GPU. The resultant matrix is transferred back to Host-CPU.
  
  We create a two-dimensional grid that is overlaid on matrices. Matrices have been divided into thread blocks of 16 x 16 sizes. One BLOCK loads one tile of both matrices from global memory to shared memory. Each thread with in BLOCK calculate temporal resultant in the register. After all threads within block completed their part of computation, the BLOCK stores the resultant tile into the global resultant matrix. The resultant matrix is transferred back to Host-CPU. Timing has been recorded for the kernel launch using Cuda Events. Matrix Multiplication has been also computed on Host-CPU and after that both Host-CPU result and device-GPU result has been compared to check correctness of result. Computation on device is performed for CULBAS routines

Important Steps :

Steps	Description
1.	Memory allocation on host-CPU and device-GPU : Allocate memory for two input matrices on host-CPU & device-GPU Use cublasAlloc()
2.	Input data Generation : Fill the input vectors with single/double precision real values using randomized data as per input specification
3.	Transfer data from host-CPU to device-GPU: Use cublasSetVector()
4.	Perform computation using cublasDgemm() CUBLAS library call
5.	Transfer the result from device-GPU to host-CPU :
6.	Check correctness of the result on host-CPU Compute matrix-matrix multiplication on host-CPU; Compare CPU and GPU results.
7.	Free the memory Free the memory of arrays allocated on host-CPU & device-GPU

• Input

Matrix Size

• Output

Execution time in seconds , Gflops acheived

char name[256];
size_t totalGlobalMem;
size_t sharedMemBlock;
int regPerBlock;
int warpSize;
size_t memPitch;
int maxThreadsPerBlock;
int maxThreadsDim[1];
int maxGridSize[3];
size_t totalConstMem;
int major;
int minor;
int clockRate;
size_t texturealignment;
int deviceOverlap;
int multiProcessorcount;
int KerneExecutionTimeoutEnabled;
int integrated;
int canMapHostMemory;
int computeMode;
int maxTexture1D;
int maxTexture2d[2];
int maxTexture3d[3];
int maxTexture2dArray[3];
int concurrentKernels;