text:
f11 Chapter Contents
NAG Library Manual

# NAG Library Chapter Introductionf11 – Large Scale Linear Systems

## 1  Scope of the Chapter

This chapter provides functions for the solution of large sparse systems of simultaneous linear equations. These include iterative methods for real nonsymmetric and symmetric, complex non-Hermitian and Hermitian linear systems and direct methods for general real linear systems. Further direct methods are currently available in Chapters f01 and f04.

## 2  Background to the Problems

This section is only a brief introduction to the solution of sparse linear systems. For a more detailed discussion see for example Duff et al. (1986) and Demmel et al. (1999) for direct methods, or Barrett et al. (1994) for iterative methods.

### 2.1  Sparse Matrices and Their Storage

A matrix $A$ may be described as sparse if the number of zero elements is sufficiently large that it is worthwhile using algorithms which avoid computations involving zero elements.
If $A$ is sparse, and the chosen algorithm requires the matrix coefficients to be stored, a significant saving in storage can often be made by storing only the nonzero elements. A number of different formats may be used to represent sparse matrices economically. These differ according to the amount of storage required, the amount of indirect addressing required for fundamental operations such as matrix-vector products, and their suitability for vector and/or parallel architectures. For a survey of some of these storage formats see Barrett et al. (1994).
Some of the functions in this chapter have been designed to be independent of the matrix storage format. This allows you to choose your own preferred format, or to avoid storing the matrix altogether. Other functions are the so-called Black Boxes, which are easier to use, but are based on fixed storage formats. Three fixed storage formats for sparse matrices are currently used. These are known as coordinate storage (CS) format, symmetric coordinate storage (SCS) format and compressed column storage (CCS) format.

#### 2.1.1  Coordinate storage (CS) format

This storage format represents a sparse matrix $A$, with nnz nonzero elements, in terms of three one-dimensional arrays – a double or Complex array a and two Integer arrays irow and icol. These arrays are all of dimension at least nnz. a contains the nonzero elements themselves, while irow and icol store the corresponding row and column indices respectively.
For example, the matrix
 $A= 1 -2 -1 -1 -3 0 -1 0 0 -4 3 0 0 0 2 2 0 4 1 1 -2 0 0 0 1$
might be represented in the arrays airow and icol as
• ${\mathbf{a}}=\left(1,2,-1,-1,-3,-1,-4,3,2,2,4,1,1,-2,1\right)$
• ${\mathbf{irow}}=\left(1,1,1,1,1,2,2,3,3,4,4,4,4,5,5\right)$
• ${\mathbf{icol}}=\left(1,2,3,4,5,2,5,1,5,1,3,4,5,1,5\right)$.
Notes
 (i) The general format specifies no ordering of the array elements, but some functions may impose a specific ordering. For example, the nonzero elements may be required to be ordered by increasing row index and by increasing column index within each row, as in the example above. Utility functions are provided to order the elements appropriately (see Section 2.2). (ii) With this storage format it is possible to enter duplicate elements. These may be interpreted in various ways (e.g., raising an error, ignoring all but the first entry, all but the last, or summing).

#### 2.1.2  Symmetric coordinate storage (SCS) format

This storage format is suitable for symmetric and Hermitian matrices, and is identical to the CS format described in Section 2.1.1, except that only the lower triangular nonzero elements are stored. Thus, for example, the matrix
 $A= 4 -1 -0 -0 -1 2 1 5 0 2 0 0 0 0 2 1 0 -1 0 2 1 3 1 0 -1 0 0 1 4 0 2 0 -1 0 0 3$
might be represented in the arrays airow and icol as
• ${\mathbf{a}}=\left(4,1,5,2,2,1,3,-1,1,4,2,-1,3\right)$.
• ${\mathbf{irow}}=\left(1,2,2,3,4,4,4,5,5,5,6,6,6\right)$,
• ${\mathbf{icol}}=\left(1,1,2,3,2,3,4,1,4,5,1,3,6\right)$.

#### 2.1.3  Compressed column storage (CCS) format

This storage format also uses three one-dimensional arrays – a double or Complex array a and two Integer arrays irowix and icolzp. The array a and irowix are of dimension at least $\mathit{nnz}$, while icolzp is of dimension at least ${\mathbf{n}}+1$. a contains the nonzero elements, going down the first column, then the second and so on. For example, the matrix in Section 2.1.1 above will be represented by
• ${\mathbf{a}}=\left(1,3,2,-2,2,-1,-1,4,-1,1,-3,-4,2,1,1\right)$.
irowix records the row index for each entry in a, so the same matrix will have
• ${\mathbf{irowix}}=\left(1,3,4,5,1,2,1,4,1,4,1,2,3,4,5\right)$.
icolzp records the index into a which starts each new column. The last entry of icolzp is equal to $\mathit{nnz}+1$. An empty column (one filled with zeros, that is) is signalled by an index that is the same as the next non-empty column, or $\mathit{nnz}+1$ if all subsequent columns are empty. The above example corresponds to
• ${\mathbf{icolzp}}=\left(1,5,7,9,11,16\right)$
The example in Section 2.1.2 above will be represented by
• ${\mathbf{a}}=\left(4,1,-1,2,1,5,2,2,1,-1,2,1,3,1,-1,1,4,2,-1,3\right)$
• ${\mathbf{irowix}}=\left(1,2,5,6,1,2,4,3,4,6,2,3,4,5,1,4,5,1,3,6\right)$
• ${\mathbf{icolzp}}=\left(1,5,8,11,15,18,21\right)$

### 2.2  Direct Methods

Direct methods for the solution of the linear algebraic system
 $Ax=b$ (1)
aim to determine the solution vector $x$ in a fixed number of arithmetic operations, which is determined a priori by the number of unknowns. For example, an $LU$ factorization of $A$ followed by forward and backward substitution is a direct method for (1).
If the matrix $A$ is sparse it is possible to design direct methods which exploit the sparsity pattern and are therefore much more computationally efficient than the algorithms in Chapter f07, which in general take no account of sparsity. However, if the matrix is very large and sparse, then iterative methods, with an appropriate preconditioner, (see Section 2.3) may be more efficient still.
This chapter provides a direct $LU$ factorization method for sparse real systems. This method is based on special coding for supernodes, broadly defined as groups of consecutive columns with the same nonzero structure, which enables use of dense BLAS kernels. The algorithms contained here come from the SuperLU software suite (see Demmel et al. (1999)). An important requirement of sparse $LU$ factorization is keeping the factors as sparse as possible. It is well known that certain column orderings can produce much sparser factorizations than the normal left-to-right ordering. It is well worth the effort, then, to find such column orderings since they reduce both storage requirements of the factors, the time taken to compute them and the time taken to solve the linear system. The row reorderings, demanded by partial pivoting in order to keep the factorization stable, can further complicate the choice of the column ordering, but quite good and fast algorithms have been developed to make possible a fairly reliable computation of an appropriate column ordering for any sparsity pattern. We provide one such algorithm (known in the literature as COLAMD) through one function in the suite. Similar to the case for dense matrices, functions are provided to compute the $LU$ factorization with partial row pivoting for numerical stability, solve (1) by performing the forward and backward substitutions for multiple right hand side vectors, refine the solution, minimize the backward error and estimate the forward error of the solutions, compute norms, estimate condition numbers and perform diagnostics of the factorization. It is also possible to explicitly construct, column by column, the dense inverse of the matrix by solving equation (1) for right hand sides corresponding to columns of the identity matrix. Blocks of dense columns can be handled at one time and then stored in some chosen sparse format, as system memory allows. For more details see Section 3.4.
It is also possible to use iterative method functions in this chapter to compute a direct factorization. Such methods are available for sparse real nonsymmetric, complex non-Hermitian, real symmetric positive definite and complex Hermitian positive definite systems. Further direct methods may be found in Chapters f01f04 and f07.

### 2.3  Iterative Methods

In contrast to the direct methods discussed in Section 2.2, iterative methods for (1) approach the solution through a sequence of approximations until some user-specified termination criterion is met or until some predefined maximum number of iterations has been reached. The number of iterations required for convergence is not generally known in advance, as it depends on the accuracy required, and on the matrix $A$ – its sparsity pattern, conditioning and eigenvalue spectrum.
Faster convergence can often be achieved using a preconditioner (see Golub and Van Loan (1996) and Barrett et al. (1994)). A preconditioner maps the original system of equations onto a different system
 $A-x-=b-,$ (2)
which hopefully exhibits better convergence characteristics. For example, the condition number of the matrix $\stackrel{-}{A}$ may be better than that of $A$, or it may have eigenvalues of greater multiplicity.
An unsuitable preconditioner or no preconditioning at all may result in a very slow rate or lack of convergence. However, preconditioning involves a trade-off between the reduction in the number of iterations required for convergence and the additional computational costs per iteration. Setting up a preconditioner may also involve non-negligible overheads. The application of preconditioners to real nonsymmetric, complex non-Hermitian, real symmetric and complex Hermitian and real symmetric systems of equations is further considered in Sections 2.4 and 2.5.

### 2.4  Iterative Methods for Real Nonsymmetric and Complex Non-Hermitian Linear Systems

Many of the most effective iterative methods for the solution of (1) lie in the class of non-stationary Krylov subspace methods (see Barrett et al. (1994)). For real nonsymmetric and complex non-Hermitian matrices this class includes:
Here we just give a brief overview of these algorithms as implemented in this chapter. For full details see the function documents for nag_sparse_nsym_basic_setup (f11bdc) and nag_sparse_nherm_basic_setup (f11brc).
RGMRES is based on the Arnoldi method, which explicitly generates an orthogonal basis for the Krylov subspace $\mathrm{span}\left\{{A}^{k}{r}_{0}\right\}$, $k=0,1,2,\dots \text{}$, where ${r}_{0}$ is the initial residual. The solution is then expanded onto the orthogonal basis so as to minimize the residual norm. For real nonsymmetric and complex non-Hermitian matrices the generation of the basis requires a ‘long’ recurrence relation, resulting in prohibitive computational and storage costs. RGMRES limits these costs by restarting the Arnoldi process from the latest available residual every $m$ iterations. The value of $m$ is chosen in advance and is fixed throughout the computation. Unfortunately, an optimum value of $m$ cannot easily be predicted.
CGS is a development of the bi-conjugate gradient method where the nonsymmetric Lanczos method is applied to reduce the coefficient matrix to tridiagonal form: two bi-orthogonal sequences of vectors are generated starting from the initial residual ${r}_{0}$ and from the shadow residual ${\stackrel{^}{r}}_{0}$ corresponding to the arbitrary problem ${A}^{\mathrm{H}}\stackrel{^}{x}=\stackrel{^}{b}$, where $\stackrel{^}{b}$ is chosen so that ${r}_{0}={\stackrel{^}{r}}_{0}$. In the course of the iteration, the residual and shadow residual ${r}_{i}={P}_{i}\left(A\right){r}_{0}$ and ${\stackrel{^}{r}}_{i}={P}_{i}\left({A}^{\mathrm{H}}\right){\stackrel{^}{r}}_{0}$ are generated, where ${P}_{i}$ is a polynomial of order $i$, and bi-orthogonality is exploited by computing the vector product ${\rho }_{i}=\left({\stackrel{^}{r}}_{i},{r}_{i}\right)=\left({P}_{i}\left({A}^{\mathrm{H}}\right){\stackrel{^}{r}}_{0}{P}_{i}\left(A\right){r}_{0}\right)=\left({\stackrel{^}{r}}_{0},{P}_{i}^{2}\left(A\right){r}_{0}\right)$. Applying the ‘contraction’ operator ${P}_{i}\left(A\right)$ twice, the iteration coefficients can still be recovered without advancing the solution of the shadow problem, which is of no interest. The CGS method often provides fast convergence; however, there is no reason why the contraction operator should also reduce the once reduced vector ${P}_{i}\left(A\right){r}_{0}$: this can lead to a highly irregular convergence.
Bi-CGSTAB$\left(\ell \right)$ is similar to the CGS method. However, instead of generating the sequence $\left\{{P}_{i}^{2}\left(A\right){r}_{0}\right\}$, it generates the sequence $\left\{{Q}_{i}\left(A\right){P}_{i}\left(A\right){r}_{0}\right\}$ where the ${Q}_{i}\left(A\right)$ are polynomials chosen to minimize the residual after the application of the contraction operator ${P}_{i}\left(A\right)$. Two main steps can be identified for each iteration: an OR (Orthogonal Residuals) step where a basis of order $\ell$ is generated by a Bi-CG iteration and an MR (Minimum Residuals) step where the residual is minimized over the basis generated, by a method similar to GMRES. For $\ell =1$, the method corresponds to the Bi-CGSTAB method of Van der Vorst (1989). For $\ell >1$, more information about complex eigenvalues of the iteration matrix can be taken into account, and this may lead to improved convergence and robustness. However, as $\ell$ increases, numerical instabilities may arise.
The transpose-free quasi-minimal residual method (TFQMR) (see Freund and Nachtigal (1991) and Freund (1993)) is conceptually derived from the CGS method. The residual is minimized over the space of the residual vectors generated by the CGS iterations under the simplifying assumption that residuals are almost orthogonal. In practice, this is not the case but theoretical analysis has proved the validity of the method. This has the effect of remedying the rather irregular convergence behaviour with wild oscillations in the residual norm that can degrade the numerical performance and robustness of the CGS method. In general, the TFQMR method can be expected to converge at least as fast as the CGS method, in terms of number of iterations, although each iteration involves a higher operation count. When the CGS method exhibits irregular convergence, the TFQMR method can produce much smoother, almost monotonic convergence curves. However, the close relationship between the CGS and TFQMR method implies that the overall speed of convergence is similar for both methods. In some cases, the TFQMR method may converge faster than the CGS method.
Faster convergence can usually be achieved by using a preconditioner. A left preconditioner ${M}^{-1}$ can be used by the RGMRES, CGS and TFQMR methods, such that $\stackrel{-}{A}={M}^{-1}A\sim {I}_{n}$ in (2), where ${I}_{n}$ is the identity matrix of order $n$; a right preconditioner ${M}^{-1}$ can be used by the Bi-CGSTAB$\left(\ell \right)$ method, such that $\stackrel{-}{A}=A{M}^{-1}\sim {I}_{n}$. These are formal definitions, used only in the design of the algorithms; in practice, only the means to compute the matrix-vector products $v=Au$ and $v={A}^{\mathrm{H}}u$ (the latter only being required when an estimate of ${‖A‖}_{1}$ or ${‖A‖}_{\infty }$ is computed internally), and to solve the preconditioning equations $Mv=u$ are required, that is, explicit information about $M$, or its inverse is not required at any stage.
Preconditioning matrices $M$ are typically based on incomplete factorizations (see Meijerink and Van der Vorst (1981)), or on the approximate inverses occurring in stationary iterative methods (see Young (1971)). A common example is the incomplete $LU$ factorization
 $M=PLDUQ=A-R$
where $L$ is lower triangular with unit diagonal elements, $D$ is diagonal, $U$ is upper triangular with unit diagonals, $P$ and $Q$ are permutation matrices, and $R$ is a remainder matrix. A zero-fill incomplete $LU$ factorization is one for which the matrix
 $S=PL+D+UQ$
has the same pattern of nonzero entries as $A$. This is obtained by discarding any fill elements (nonzero elements of $S$ arising during the factorization in locations where $A$ has zero elements). Allowing some of these fill elements to be kept rather than discarded generally increases the accuracy of the factorization at the expense of some loss of sparsity. For further details see Barrett et al. (1994).

### 2.5  Iterative Methods for Real Symmetric and Complex Hermitian Linear Systems

Three of the best known iterative methods applicable to real symmetric and complex Hermitian linear systems are the conjugate gradient (CG) method (see Hestenes and Stiefel (1952) and Golub and Van Loan (1996)) and Lanczos type methods based on SYMMLQ and MINRES (see Paige and Saunders (1975)). The description of these methods given below is for the real symmetric cases. The generalization to complex Hermitian matrices is straightforward.
For the CG method the matrix $A$ should ideally be positive definite. The application of CG to indefinite matrices may lead to failure, or to lack of convergence. The SYMMLQ and MINRES methods are suitable for both positive definite and indefinite symmetric matrices. They are more robust than CG, but less efficient when $A$ is positive definite.
The methods start from the residual ${r}_{0}=b-A{x}_{0}$, where ${x}_{0}$ is an initial estimate for the solution (often ${x}_{0}=0$), and generate an orthogonal basis for the Krylov subspace $\mathrm{span}\left\{{A}^{k}{r}_{0}\right\}$, for $\mathit{k}=0,1,\dots$, by means of three-term recurrence relations (see Golub and Van Loan (1996)). A sequence of symmetric tridiagonal matrices $\left\{{T}_{k}\right\}$ is also generated. Here and in the following, the index $k$ denotes the iteration count. The resulting symmetric tridiagonal systems of equations are usually more easily solved than the original problem. A sequence of solution iterates $\left\{{x}_{k}\right\}$ is thus generated such that the sequence of the norms of the residuals $\left\{‖{r}_{k}‖\right\}$ converges to a required tolerance. Note that, in general, the convergence is not monotonic.
In exact arithmetic, after $n$ iterations, this process is equivalent to an orthogonal reduction of $A$ to symmetric tridiagonal form, ${T}_{n}={Q}^{\mathrm{T}}AQ$; the solution ${x}_{n}$ would thus achieve exact convergence. In finite-precision arithmetic, cancellation and round-off errors accumulate causing loss of orthogonality. These methods must therefore be viewed as genuinely iterative methods, able to converge to a solution within a prescribed tolerance.
The orthogonal basis is not formed explicitly in either method. The basic difference between the methods lies in the method of solution of the resulting symmetric tridiagonal systems of equations: the CG method is equivalent to carrying out an $LD{L}^{\mathrm{T}}$ (Cholesky) factorization whereas the Lanczos method (SYMMLQ) uses an $LQ$ factorization. The MINRES method on the other hand minimizes the residual into 2-norm.
A preconditioner for these methods must be symmetric and positive definite, i.e., representable by $M=E{E}^{\mathrm{T}}$, where $M$ is nonsingular, and such that $\stackrel{-}{A}={E}^{-1}A{E}^{-\mathrm{T}}\sim {I}_{n}$ in (2), where ${I}_{n}$ is the identity matrix of order $n$. These are formal definitions, used only in the design of the algorithms; in practice, only the means to compute the matrix-vector products $v=Au$ and to solve the preconditioning equations $Mv=u$ are required.
Preconditioning matrices $M$ are typically based on incomplete factorizations (see Meijerink and Van der Vorst (1977)), or on the approximate inverses occurring in stationary iterative methods (see Young (1971)). A common example is the incomplete Cholesky factorization
 $M=PLDLTPT=A-R$
where $P$ is a permutation matrix, $L$ is lower triangular with unit diagonal elements, $D$ is diagonal and $R$ is a remainder matrix. A zero-fill incomplete Cholesky factorization is one for which the matrix
 $S=PL+D+LTPT$
has the same pattern of nonzero entries as $A$. This is obtained by discarding any fill elements (nonzero elements of $S$ arising during the factorization in locations where $A$ has zero elements). Allowing some of these fill elements to be kept rather than discarded generally increases the accuracy of the factorization at the expense of some loss of sparsity. For further details see Barrett et al. (1994).

## 3  Recommendations on Choice and Use of Available Functions

### 3.1  Types of Function Available

The direct method functions available in this chapter largely follow the LAPACK scheme in that four different functions separately handle the tasks of factorizing, solving, refining and condition number estimating. See Section 3.4.
The iterative method functions available in this chapter divide essentially into three types: basic functions, utility functions and Black Box functions.
Basic functions are grouped in suites of three, and implement the underlying iterative method. Each suite comprises a setup function, a solver, and a function to return additional information. The solver function is independent of the matrix storage format (indeed the matrix need not be stored at all) and the type of preconditioner. It uses reverse communication, i.e., it returns repeatedly to the calling program with the argument irevcm set to specified values which require the calling program to carry out a specific task (either to compute a matrix-vector product or to solve the preconditioning equation), to signal the completion of the computation or to allow the calling program to monitor the solution. Reverse communication has the following advantages.
 (i) Maximum flexibility in the representation and storage of sparse matrices. All matrix operations are performed outside the solver function, thereby avoiding the need for a complicated interface with enough flexibility to cope with all types of storage schemes and sparsity patterns. This also applies to preconditioners. (ii) Enhanced user interaction: you can closely monitor the solution and tidy or immediate termination can be requested. This is useful, for example, when alternative termination criteria are to be employed or in case of failure of the external functions used to perform matrix operations.
At present there are suites of basic functions for real symmetric and nonsymmetric systems, and for complex Hermitian and non-Hermitian systems.
Utility functions perform such tasks as initializing the preconditioning matrix $M$, solving linear systems involving $M$, or computing matrix-vector products, for particular preconditioners and matrix storage formats. Used in combination, basic functions and utility functions therefore provide iterative methods with a considerable degree of flexibility, allowing you to select from different termination criteria, monitor the approximate solution, and compute various diagnostic arguments. The tasks of computing the matrix-vector products and dealing with the preconditioner are removed from you, but at the expense of sacrificing some flexibility in the choice of preconditioner and matrix storage format.
Black Box functions call basic and utility functions in order to provide easy-to-use functions for particular preconditioners and sparse matrix storage formats. They are much less flexible than the basic functions, but do not use reverse communication, and may be suitable in many simple cases.
The structure of this chapter has been designed to cater for as many types of application as possible. If a Black Box function exists which is suitable for a given application you are recommended to use it. If you then decide you need some additional flexibility it is easy to achieve this by using basic and utility functions which reproduce the algorithm used in the Black Box, but allow more access to algorithmic control arguments and monitoring. If you wish to use a preconditioner or storage format for which no utility functions are provided, you must call basic functions, and provide your own utility functions.

### 3.2  Iterative Methods for Real Nonsymmetric and Complex Non-Hermitian Linear Systems

The suite of basic functions nag_sparse_nsym_basic_setup (f11bdc)nag_sparse_nsym_basic_solver (f11bec) and nag_sparse_nsym_basic_diagnostic (f11bfc) implements either RGMRES, CGS, Bi-CGSTAB$\left(\ell \right)$, or TFQMR, for the iterative solution of the real sparse nonsymmetric linear system $Ax=b$. These functions allow a choice of termination criteria and the norms used in them, allow monitoring of the approximate solution, and can return estimates of the norm of $A$ and the largest singular value of the preconditioned matrix $\stackrel{-}{A}$.
In general, it is not possible to recommend one of these methods (RGMRES, CGS, Bi-CGSTAB$\left(\ell \right)$, or TFQMR) in preference to another. RGMRES is popular, but requires the most storage, and can easily stagnate when the size $m$ of the orthogonal basis is too small, or the preconditioner is not good enough. CGS can be the fastest method, but the computed residuals can exhibit instability which may greatly affect the convergence and quality of the solution. Bi-CGSTAB$\left(\ell \right)$ seems robust and reliable, but it can be slower than the other methods. TFQMR can be viewed as a more robust variant of the CGS method: it shares the CGS method speed but avoids the CGS fluctuations in the residual, which may give, rise to instability. Some further discussion of the relative merits of these methods can be found in Barrett et al. (1994).
The utility functions provided for real nonsymmetric matrices use the coordinate storage (CS) format described in Section 2.1.1. nag_sparse_nsym_fac (f11dac) computes a preconditioning matrix based on incomplete $LU$ factorization, and nag_sparse_nsym_precon_ilu_solve (f11dbc) solves linear systems involving the preconditioner generated by nag_sparse_nsym_fac (f11dac). The amount of fill-in occurring in the incomplete factorization can be controlled by specifying either the level of fill, or the drop tolerance. Partial or complete pivoting may optionally be employed, and the factorization can be modified to preserve row-sums.
nag_sparse_nsym_precon_bdilu (f11dfc) is a generalization of nag_sparse_nsym_fac (f11dac). It computes incomplete $LU$ factorizations on a set of (possibly overlapping) block diagonal matrices, using a prescribed block structure, to provide a block Jacobi or additive Schwartz preconditioner. To solve the linear system defined by the preconditioner generated by nag_sparse_nsym_precon_bdilu (f11dfc), a sequence of calls to nag_sparse_nsym_precon_ilu_solve (f11dbc) (one for each block) would be required.
nag_sparse_nsym_precon_ssor_solve (f11ddc) is similar to nag_sparse_nsym_precon_ilu_solve (f11dbc), but solves linear systems involving the preconditioner corresponding to symmetric successive-over-relaxation (SSOR). The value of the relaxation argument $\omega$ must currently be supplied by you. Automatic procedures for choosing $\omega$ will be included in the chapter at a future mark.
nag_sparse_nsym_jacobi (f11dkc) applies the iterated Jacobi method to a symmetric or nonsymmetric system of linear equations and can be used as a preconditioner. However, the domain of validity of the Jacobi method is rather restricted; you should read the function document for nag_sparse_nsym_jacobi (f11dkc) before using it.
nag_sparse_nsym_matvec (f11xac) computes matrix-vector products for real nonsymmetric matrices stored in ordered CS format. An additional utility function nag_sparse_nsym_sort (f11zac) orders the nonzero elements of a real sparse nonsymmetric matrix stored in general CS format. The same function can be used to convert a matrix from CS format to CCS format.
The Black Box function nag_sparse_nsym_fac_sol (f11dcc) makes calls to nag_sparse_nsym_basic_setup (f11bdc)nag_sparse_nsym_basic_solver (f11bec)nag_sparse_nsym_basic_diagnostic (f11bfc)nag_sparse_nsym_precon_ilu_solve (f11dbc) and nag_sparse_nsym_matvec (f11xac), to solve a real sparse nonsymmetric linear system, represented in CS format, using RGMRES, CGS, Bi-CGSTAB$\left(\ell \right)$, or TFQMR, with incomplete $LU$ preconditioning. nag_sparse_nsym_sol (f11dec) is similar, but has options for no preconditioning, Jacobi preconditioning or SSOR preconditioning. nag_sparse_nsym_precon_bdilu_solve (f11dgc) is also similar to nag_sparse_nsym_fac_sol (f11dcc), but uses block Jacobi or additive Schwartz preconditioning.
For complex non-Hermitian sparse matrices there is an equivalent suite of functions. nag_sparse_nherm_basic_setup (f11brc)nag_sparse_nherm_basic_solver (f11bsc) and nag_sparse_nherm_basic_diagnostic (f11btc) are the basic functions which implement the same methods used for real nonsymmetric systems, namely RGMRES, CGS, Bi-CGSTAB$\left(\ell \right)$ and TFQMR, for the solution of complex sparse non-Hermitian linear systems. nag_sparse_nherm_fac (f11dnc) and nag_sparse_nherm_precon_ilu_solve (f11dpc) are the complex equivalents of nag_sparse_nsym_fac (f11dac) and nag_sparse_nsym_precon_ilu_solve (f11dbc), respectively, providing facilities for implementing ILU preconditioning. nag_sparse_nherm_precon_ssor_solve (f11drc) and nag_sparse_nherm_precon_bdilu (f11dtc) implement complex versions of the SSOR and block Jacobi (or additive Schwartz) preconditioners, respectively. nag_sparse_nherm_jacobi (f11dxc) implements a complex version of the iterated Jacobi preconditioner. Utility functions nag_sparse_nherm_matvec (f11xnc) and nag_sparse_nherm_sort (f11znc) are provided for computing matrix-vector products and sorting the elements of complex sparse non-Hermitian matrices, respectively. Finally, the Black Box functions nag_sparse_nherm_fac_sol (f11dqc)nag_sparse_nherm_sol (f11dsc) and nag_sparse_nherm_precon_bdilu_solve (f11duc) are complex equivalents of nag_sparse_nsym_fac_sol (f11dcc)nag_sparse_nsym_sol (f11dec) and nag_sparse_nsym_precon_bdilu (f11dfc), respectively.

### 3.3  Iterative Methods for Real Symmetric and Complex Hermitian Linear Systems

The suite of basic functions nag_sparse_sym_basic_setup (f11gdc)nag_sparse_sym_basic_solver (f11gec) and nag_sparse_sym_basic_diagnostic (f11gfc) implement either the conjugate gradient (CG) method, or a Lanczos method based on SYMMLQ, for the iterative solution of the real sparse symmetric linear system $Ax=b$. If $A$ is known to be positive definite the CG method should be chosen; the Lanczos method is more robust but less efficient for positive definite matrices. These functions allow a choice of termination criteria and the norms used in them, allow monitoring of the approximate solution, and can return estimates of the norm of $A$ and the largest singular value of the preconditioned matrix $\stackrel{-}{A}$.
The utility functions provided for real symmetric matrices use the symmetric coordinate storage (SCS) format described in Section 2.1.2. nag_sparse_sym_chol_fac (f11jac) computes a preconditioning matrix based on incomplete Cholesky factorization, and nag_sparse_sym_precon_ichol_solve (f11jbc) solves linear systems involving the preconditioner generated by nag_sparse_sym_chol_fac (f11jac). The amount of fill-in occurring in the incomplete factorization can be controlled by specifying either the level of fill, or the drop tolerance. Diagonal Markowitz pivoting may optionally be employed, and the factorization can be modified to preserve row-sums.
nag_sparse_sym_precon_ssor_solve (f11jdc) is similar to nag_sparse_sym_precon_ichol_solve (f11jbc), but solves linear systems involving the preconditioner corresponding to symmetric successive-over-relaxation (SSOR). The value of the relaxation argument $\omega$ must currently be supplied by you. Automatic procedures for choosing $\omega$ will be included in the chapter at a future mark.
nag_sparse_nsym_jacobi (f11dkc) applies the iterated Jacobi method to a symmetric or nonsymmetric system of linear equations and can be used as a preconditioner. However, the domain of validity of the Jacobi method is rather restricted; you should read the function document for nag_sparse_nsym_jacobi (f11dkc) before using it.
nag_sparse_sym_matvec (f11xec) computes matrix-vector products for real symmetric matrices stored in ordered SCS format. An additional utility function nag_sparse_sym_sort (f11zbc) orders the nonzero elements of a real sparse symmetric matrix stored in general SCS format.
The Black Box function nag_sparse_sym_chol_sol (f11jcc) makes calls to nag_sparse_sym_basic_setup (f11gdc)nag_sparse_sym_basic_solver (f11gec)nag_sparse_sym_basic_diagnostic (f11gfc)nag_sparse_sym_precon_ichol_solve (f11jbc) and nag_sparse_sym_matvec (f11xec), to solve a real sparse symmetric linear system, represented in SCS format, using a conjugate gradient or Lanczos method, with incomplete Cholesky preconditioning. nag_sparse_sym_sol (f11jec) is similar, but has options for no preconditioning, Jacobi preconditioning or SSOR preconditioning.
For complex Hermitian sparse matrices there is an equivalent suite of functions. nag_sparse_herm_basic_setup (f11grc)nag_sparse_herm_basic_solver (f11gsc) and nag_sparse_herm_basic_diagnostic (f11gtc) are the basic functions which implement the same methods used for real symmetric systems, namely CG and SYMMLQ, for the solution of complex sparse Hermitian linear systems. nag_sparse_herm_chol_fac (f11jnc) and nag_sparse_herm_precon_ichol_solve (f11jpc) are the complex equivalents of nag_sparse_sym_chol_fac (f11jac) and nag_sparse_sym_precon_ichol_solve (f11jbc), respectively, providing facilities for implementing incomplete Cholesky preconditioning. nag_sparse_herm_precon_ssor_solve (f11jrc) implements a complex version of the SSOR preconditioner. nag_sparse_nherm_jacobi (f11dxc) implements a complex version of the iterated Jacobi preconditioner. Utility functions nag_sparse_herm_matvec (f11xsc) and nag_sparse_herm_sort (f11zpc) are provided for computing matrix-vector products and sorting the elements of complex sparse Hermitian matrices, respectively. Finally, the Black Box functions nag_sparse_herm_chol_sol (f11jqc) and nag_sparse_herm_sol (f11jsc) provide easy-to-use implementations of the CG and SYMMLQ methods for complex Hermitian linear systems.

### 3.4  Direct Methods

The suite of functions nag_superlu_column_permutation (f11mdc)nag_superlu_lu_factorize (f11mec)nag_superlu_solve_lu (f11mfc)nag_superlu_condition_number_lu (f11mgc)nag_superlu_refine_lu (f11mhc)nag_superlu_matrix_product (f11mkc)nag_superlu_matrix_norm (f11mlc) and nag_superlu_diagnostic_lu (f11mmc) implement the COLAMD/SuperLU direct real sparse solver and associated utilities. You are expected to first call nag_superlu_column_permutation (f11mdc) to compute a suitable column permutation for the subsequent factorization by nag_superlu_lu_factorize (f11mec). nag_superlu_solve_lu (f11mfc) then solves the system of equations. A solution can be further refined by nag_superlu_refine_lu (f11mhc), which also minimizes the backward error and estimates a bound for the forward error in the solution. Diagnostics are provided by nag_superlu_condition_number_lu (f11mgc) which computes an estimate of the condition number of the matrix using the factorization output by nag_superlu_lu_factorize (f11mec), and nag_superlu_diagnostic_lu (f11mmc) which computes the reciprocal pivot growth (a numerical stability measure) of the factorization. The two utility functions, nag_superlu_matrix_product (f11mkc), which computes matrix-matrix products in the particular storage scheme demanded by the suite (CCS format), and nag_superlu_matrix_norm (f11mlc) which computes quantities relating to norms of a matrix in that particular storage scheme, complete the suite.
Another way of computing a direct solution is to choose specific arguments for the indirect solvers. For example, function nag_sparse_nsym_precon_ilu_solve (f11dbc) solves a linear system involving the incomplete $LU$ preconditioning matrix
 $M=PLDUQ=A-R$
generated by nag_sparse_nsym_fac (f11dac), where $P$ and $Q$ are permutation matrices, $L$ is lower triangular with unit diagonal elements, $U$ is upper triangular with unit diagonal elements, $D$ is diagonal and $R$ is a remainder matrix.
If $A$ is nonsingular, a call to nag_sparse_nsym_fac (f11dac) with ${\mathbf{lfill}}<0$ and ${\mathbf{dtol}}=0.0$ results in a zero remainder matrix $R$ and a complete factorization. A subsequent call to nag_sparse_nsym_precon_ilu_solve (f11dbc) will therefore result in a direct method for real sparse nonsymmetric systems.
If $A$ is known to be symmetric positive definite, nag_sparse_sym_chol_fac (f11jac) and nag_sparse_sym_precon_ichol_solve (f11jbc) may similarly be used to give a direct solution. For further details see Section 9.4 in nag_sparse_sym_chol_fac (f11jac).
Complex non-Hermitian systems can be solved directly in the same way using nag_sparse_nherm_fac (f11dnc) and nag_sparse_nherm_precon_ilu_solve (f11dpc), while for complex Hermitian systems nag_sparse_herm_chol_fac (f11jnc) and nag_sparse_herm_precon_ichol_solve (f11jpc) may be used.
Some other functions specifically designed for direct solution of sparse linear systems can currently be found in Chapters f01f04 and f07. In particular, the following functions allow the direct solution of symmetric positive definite systems:
 Variable band (skyline) nag_real_cholesky_skyline (f01mcc) and nag_real_cholesky_skyline_solve (f04mcc)
Functions for the solution of band and tridiagonal systems can be found in Chapters f04 and f07.

## 4  Decision Tree

### Tree 1: Solvers

 Do you have a real system and want to use a direct method? _yes f11mdc, f11mec and f11mfc no| Do you want to use your own storage scheme or preconditioner? _yes complex system? _yes Hermitian? _yes f11grc, f11gsc and f11gtc | | no| | | f11brc, f11bsc and f11btc | no| | symmetric? _yes f11gdc, f11gec and f11gfc | no| | f11bdc, f11bec and f11bfc no| complex system? _yes Hermitian positive definite? _yes Incomplete Cholesky preconditioner? _yes f11jnc and f11jpc | | no| | | f11jsc | no| | Incomplete $LU$ preconditioner? _yes Using (possibly overlapping) diagonal blocks? _yes f11dtc and f11duc | | no| | | f11dnc and f11dqc | no| | f11dsc no| symmetric positive definite? _yes Incomplete Cholesky preconditioner? _yes f11jac and f11jcc | no| | f11jec no| Incomplete $LU$ preconditioner? _yes Using (possibly overlapping) diagonal blocks? _yes f11dfc and f11dgc | no| | f11dac and f11dcc no| f11dec

## 5  Functionality Index

 Basic functions for complex Hermitian linear systems,
 diagnostic function nag_sparse_herm_basic_diagnostic (f11gtc)
 reverse communication CG or SYMMLQ solver function nag_sparse_herm_basic_solver (f11gsc)
 setup function nag_sparse_herm_basic_setup (f11grc)
 Basic functions for complex non-Hermitian linear systems,
 diagnostic function nag_sparse_nherm_basic_diagnostic (f11btc)
 reverse communication RGMRES, CGS, Bi-CGSTAB(ℓ) or TFQMR solver function nag_sparse_nherm_basic_solver (f11bsc)
 setup function nag_sparse_nherm_basic_setup (f11brc)
 Basic functions for real nonsymmetric linear systems,
 diagnostic function nag_sparse_nsym_basic_diagnostic (f11bfc)
 reverse communication RGMRES, CGS, Bi-CGSTAB(ℓ) or TFQMR solver function nag_sparse_nsym_basic_solver (f11bec)
 setup function nag_sparse_nsym_basic_setup (f11bdc)
 Basic functions for real symmetric linear systems,
 diagnostic function nag_sparse_sym_basic_diagnostic (f11gfc)
 reverse communication CG or SYMMLQ solver nag_sparse_sym_basic_solver (f11gec)
 setup function nag_sparse_sym_basic_setup (f11gdc)
 Black Box functions for complex Hermitian linear systems,
 CG or SYMMLQ solver,
 with incomplete Cholesky preconditioning nag_sparse_herm_chol_sol (f11jqc)
 with no preconditioning, Jacobi or SSOR preconditioning nag_sparse_herm_sol (f11jsc)
 Black Box functions for complex non-Hermitian linear systems,
 RGMRES, CGS, Bi-CGSTAB(ℓ) or TFQMR solver,
 with block Jacobi or additive Schwarz preconditioning nag_sparse_nherm_precon_bdilu_solve (f11duc)
 with incomplete LU preconditioning nag_sparse_nherm_fac_sol (f11dqc)
 with no preconditioning, Jacobi, or SSOR preconditioning nag_sparse_nherm_sol (f11dsc)
 Black Box functions for real nonsymmetric linear systems,
 RGMRES, CGS, Bi-CGSTAB(ℓ) or TFQMR solver,
 with block Jacobi or additive Schwarz preconditioning nag_sparse_nsym_precon_bdilu_solve (f11dgc)
 with incomplete LU preconditioning nag_sparse_nsym_fac_sol (f11dcc)
 with no preconditioning, Jacobi, or SSOR preconditioning nag_sparse_nsym_sol (f11dec)
 Black Box functions for real symmetric linear systems,
 CG or SYMMLQ solver,
 with incomplete Cholesky preconditioning nag_sparse_sym_chol_sol (f11jcc)
 with no preconditioning, Jacobi, or SSOR preconditioning nag_sparse_sym_sol (f11jec)
 Direct methods for real sparse nonsymmetric linear systems in CCS format,
 apply iterative refinement to the solution and compute error estimates, after factorizing the matrix of coefficients nag_superlu_refine_lu (f11mhc)
 condition number estimation, after factorizing the matrix of coefficients, nag_superlu_condition_number_lu (f11mgc)
 LU factorization,
 diagnostic nag_superlu_diagnostic_lu (f11mmc)
 factorize nag_superlu_lu_factorize (f11mec)
 setup nag_superlu_column_permutation (f11mdc)
 solution of simultaneous linear equations, after factorizing the matrix of coefficients, nag_superlu_solve_lu (f11mfc)
 utility,
 compute a norm or the element of largest absolute value, nag_superlu_matrix_norm (f11mlc)
 matrix-matrix multiplier nag_superlu_matrix_product (f11mkc)
 Utility function for complex Hermitian linear systems,
 incomplete Cholesky factorization nag_sparse_herm_chol_fac (f11jnc)
 matrix-vector multiplier for complex Hermitian matrices in SCS format nag_sparse_herm_matvec (f11xsc)
 solver for linear systems involving preconditioning matrix from nag_sparse_herm_chol_fac (f11jnc) nag_sparse_herm_precon_ichol_solve (f11jpc)
 solver for linear systems involving SSOR preconditioning matrix nag_sparse_herm_precon_ssor_solve (f11jrc)
 sort function for complex Hermitian matrices in SCS format nag_sparse_herm_sort (f11zpc)
 Utility function for complex linear systems,
 solver for linear systems involving iterated Jacobi method nag_sparse_nherm_jacobi (f11dxc)
 Utility function for complex non-Hermitian linear systems,
 incomplete LU factorization nag_sparse_nherm_fac (f11dnc)
 incomplete LU factorization of local or overlapping diagonal blocks nag_sparse_nherm_precon_bdilu (f11dtc)
 matrix-vector multiplier for complex non-Hermitian matrices in CS format nag_sparse_nherm_matvec (f11xnc)
 solver for linear systems involving preconditioning matrix from nag_sparse_nherm_fac (f11dnc) nag_sparse_nherm_precon_ilu_solve (f11dpc)
 solver for linear systems involving SSOR preconditioning matrix nag_sparse_nherm_precon_ssor_solve (f11drc)
 sort function for complex non-Hermitian matrices in CS format nag_sparse_nherm_sort (f11znc)
 Utility function for real linear systems,
 solver for linear systems involving iterated Jacobi method nag_sparse_nsym_jacobi (f11dkc)
 Utility function for real nonsymmetric linear systems,
 incomplete LU factorization nag_sparse_nsym_fac (f11dac)
 incomplete LU factorization of local or overlapping diagonal blocks nag_sparse_nsym_precon_bdilu (f11dfc)
 matrix-vector multiplier for real nonsymmetric matrices in CS format nag_sparse_nsym_matvec (f11xac)
 solver for linear systems involving preconditioning matrix from nag_sparse_nsym_fac (f11dac) nag_sparse_nsym_precon_ilu_solve (f11dbc)
 solver for linear systems involving SSOR preconditioning matrix nag_sparse_nsym_precon_ssor_solve (f11ddc)
 sort function for real nonsymmetric matrices in CS format nag_sparse_nsym_sort (f11zac)
 Utility function for real symmetric linear systems,
 incomplete Cholesky factorization nag_sparse_sym_chol_fac (f11jac)
 matrix-vector multiplier for real symmetric matrices in SCS format nag_sparse_sym_matvec (f11xec)
 solver for linear systems involving preconditioning matrix from nag_sparse_sym_chol_fac (f11jac) nag_sparse_sym_precon_ichol_solve (f11jbc)
 solver for linear systems involving SSOR preconditioning matrix nag_sparse_sym_precon_ssor_solve (f11jdc)
 sort function for real symmetric matrices in SCS format nag_sparse_sym_sort (f11zbc)

None.

None.

## 8  References

Barrett R, Berry M, Chan T F, Demmel J, Donato J, Dongarra J, Eijkhout V, Pozo R, Romine C and Van der Vorst H (1994) Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods SIAM, Philadelphia
Demmel J W, Eisenstat S C, Gilbert J R, Li X S and Li J W H (1999) A supernodal approach to sparse partial pivoting SIAM J. Matrix Anal. Appl. 20 720–755
Duff I S, Erisman A M and Reid J K (1986) Direct Methods for Sparse Matrices Oxford University Press, London
Freund R W (1993) A transpose-free quasi-minimal residual algorithm for non-Hermitian linear systems SIAM J. Sci. Comput. 14 470–482
Freund R W and Nachtigal N (1991) QMR: a Quasi-Minimal Residual Method for Non-Hermitian Linear Systems Numer. Math. 60 315–339
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Hestenes M and Stiefel E (1952) Methods of conjugate gradients for solving linear systems J. Res. Nat. Bur. Stand. 49 409–436
Meijerink J and Van der Vorst H (1977) An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix Math. Comput. 31 148–162
Meijerink J and Van der Vorst H (1981) Guidelines for the usage of incomplete decompositions in solving sets of linear equations as they occur in practical problems J. Comput. Phys. 44 134–155
Paige C C and Saunders M A (1975) Solution of sparse indefinite systems of linear equations SIAM J. Numer. Anal. 12 617–629
Saad Y and Schultz M (1986) GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems SIAM J. Sci. Statist. Comput. 7 856–869
Sleijpen G L G and Fokkema D R (1993) BiCGSTAB$\left(\ell \right)$ for linear equations involving matrices with complex spectrum ETNA 1 11–32
Sonneveld P (1989) CGS, a fast Lanczos-type solver for nonsymmetric linear systems SIAM J. Sci. Statist. Comput. 10 36–52
Van der Vorst H (1989) Bi-CGSTAB, a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems SIAM J. Sci. Statist. Comput. 13 631–644
Young D (1971) Iterative Solution of Large Linear Systems Academic Press, New York

f11 Chapter Contents


text:
# Exponential growth/decay equations & usage

• November 24th 2011, 09:06 AM
benny92000
Exponential growth/decay equations & usage
In my book I am given the equation A(t) = Aoe^(kt) (for exponential growth) Where t equals time and k is a constant. However, I saw an equation on another site for compounded interest in which k was replaced by r, the rate of interest. When do you use which equation? And what about exponential decay?
• November 24th 2011, 11:59 AM
corsica
Re: Exponential growth/decay equations & usage
From your description I would say that both equations are identical. They've merely applied a different nametag to the constant.

If $k>0$ then there is exponential growth. If $k<0$ there is exponential decay (towards zero).
• November 24th 2011, 12:20 PM
e^(i*pi)
Re: Exponential growth/decay equations & usage
Quote:

Originally Posted by benny92000
In my book I am given the equation A(t) = Aoe^(kt) (for exponential growth) Where t equals time and k is a constant. However, I saw an equation on another site for compounded interest in which k was replaced by r, the rate of interest. When do you use which equation? And what about exponential decay?

The equations are the same. $r$ and $k$ are constants with dimension 1/TIME. Different fields tend to use different symbols but they all refer to a constant. For example you'll see $\lambda$ used as a decay constant in radioactivity. A good book will define it's symbols immediately before/after introducing them.

Whether it's growth or decay depends whether or not there is a minus sign since the constant in the exponent is usually taken to be a positive constant. If it's there then you have decay, if not then it's growth. Below is an example

Radioactive decay is given by the formula $N = N_0e^{-\lambda t}$ where $N$ is the number of nuclei at any given time, $N_0$ is the initial number of nuclei (when t=0), $\lambda$ is a positive decay constant and $t$ is time

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# If x and y are positive integers, what is the value of x?

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If x and y are positive integers, what is the value of x? [#permalink]

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19 Nov 2017, 08:56
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If x and y are positive integers, what is the value of x?

(1) 3^x*5^y = 1125
(2) y = 3
[Reveal] Spoiler: OA

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Re: If x and y are positive integers, what is the value of x? [#permalink]

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19 Nov 2017, 09:51
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From statement A
3^x*5^y = 1125 (9X 125)
=> 3^x*5^y = 3^2*5^3
hence X is 2
sufficient
Statement 2
we don't know anything about x
hence insufficient

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Re: If x and y are positive integers, what is the value of x? [#permalink]

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19 Nov 2017, 11:03
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This post was
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Info: x & y => pos int
Ques: x = ?

(1) 3^x*5^y = 1125
using prime factorisation --> 1125 = 3*3*5*5*5, hence, 1125 = 3^2*5^3
therefore, x=2.
St (1) is sufficient, hence, answer choices AD remaining [BCE are eliminated]

(2) y = 3
This does not say anything about x. From question info all we know is that x has to be a positive integer.
Therefore, infinite values are possible for x.
St (2) is insufficient, hence, D is eliminated.

Option A is the correct choice.

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Re: If x and y are positive integers, what is the value of x? [#permalink]

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20 Nov 2017, 01:23
Statement 1) $$3^x * 5^y = 1125$$ by factorizing 1125 we get $$1125= 3^2 * 5^3$$ which is sufficient and answers the value of x

Statement 2) Insufficient as it provides the value of y. It could have been used as a hint to indicate the value to be used in statement 1)

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Re: If x and y are positive integers, what is the value of x?   [#permalink] 20 Nov 2017, 01:23
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# wave drag force

The broadbrush picture of drag presented in Fig. By looking at a data point for a given aircraft and extrapolating it horizontally to the ideal curve, the velocity gain for the same power can be seen. From the body's perspective (near-field approach), the drag results from forces due to pressure distributions over the body surface, symbolized $${\displaystyle D_{pr}}$$, and forces due to skin friction, which is a result of viscosity, denoted $${\displaystyle D_{f}}$$. Additionally, the presence of multiple bodies in relative proximity may incur so called interference drag, which is sometimes described as a component of parasitic drag. using the following formula:[22], C Similarly, for a fixed volume, the shape for minimum wave drag is the Von Karman Ogive. In aviation, induced drag tends to be greater at lower speeds because a high angle of attack is required to maintain lift, creating more drag. Attempts to construct inviscid steady flow solutions to the Euler equations, other than the potential flow solutions, did not result in realistic results. Drag= Cd .s. f Drag depends on the density of the air, the square of the velocity, the air's viscosity and compressibility, the size and shape of the body, and the body's inclination to the flow. This drag component is due to viscosity. In transonic flight, wave drag is commonly referred to as transonic compressibility drag. Q: Why do my "Wave Dirft" forces look strange? 4 The idea that a moving body passing through air or another fluid encounters resistance had been known since the time of Aristotle. The wing intercepts the airflow and forces the flow to move downward. + ∗ e Fuselage shaping was similarly changed with the introduction of the Whitcomb area rule. Wave-making resistance is a form of drag that affects surface watercraft, such as boats and ships, and reflects the energy required to push the water out of the way of the hull. Recently Siniscalchi et al. In supersonic flight (Mach numbers greater than 1.0), wave drag is the result of shockwaves present in the fluid and attached to the body, typically oblique shockwaves formed at the leading and trailing edges of the body. In aerodynamics, aerodynamic drag is the fluid drag force that acts on any moving solid body in the direction of the fluid freestream flow. These forces are: Current, Wind and Wave; Step 2: Factor the environmental force for the towing efficiency to get the required bollard pull for towing the vessel. Pilots will use this speed to maximize endurance (minimum fuel consumption), or maximize gliding range in the event of an engine failure. d Liversage, P., and Trancossi, M. (2018). This drag comes into picture only when shock wave forms over or in front of the airplane. in the vicinity of still-water free surface, by considering the inundation effect due to instantaneous wave elevation. 2 Aircraft flying at transonic speed often incur wave drag through the normal course of operation. f + To maximize a swimmer’s efforts, research has been conducted to analyze and improve stroke technique. To get the proper residuary resistance, it was necessary to recreate the wave train created by the ship in the model tests. use entropy changes to accurately predict the drag force. {\displaystyle D_{v}} Since waves carry energy, the source of that energy comes from the swimmer. The sum of friction drag and pressure (form) drag is called viscous drag. For Reynolds numbers less than 1, Stokes' law applies and the drag coefficient approaches In the 19th century the Navier–Stokes equations for the description of viscous flow were developed by Saint-Venant, Navier and Stokes. we find a drag force of 0.09 pN. The values of drag coefficient and inertial coefficient are CD — 1 and CM 2. Those forces can be summed and the component of that force that acts downstream represents the drag force, The net friction drag, At even higher speeds (transonic), wave drag enters the picture. Stokes derived the drag around a sphere at very low Reynolds numbers, the result of which is called Stokes' law. 24 24 / exact contribution of wave drag to the total drag force on a swimmer, let alone the other principle types of drag encountered in swimming, form and frictional or shear drag. ( One common solution to the problem of wave drag was to use a swept wing, which had actually been developed before World War II and used on some German wartime designs. The force turns out to be a third-order quantity with respect to wave elevation. 2 The drag of the airplane wing, or for that matter, any part of the airplane, rises sharply and large increases in thrust are necessary to obtain further increases in speed. For design purposes, the impact force is previously approximated by considering only the drag force component and multiplying by a factor of 2.5 [7]. An alternative perspective on lift and drag is gained from considering the change of momentum of the airflow. {\displaystyle cd_{w}=4*{\frac {\alpha ^{2}}{\sqrt {(M^{2}-1)}}}}   Application of the area rule can also be seen in the use of anti-shock bodies on transonic aircraft, including some jet airliners. 5 Parasitic drag is drag caused by moving a solid object through a fluid. w < Some of the most important nonlinear effects arising in connection with wave-body interactions are Drift forces. Wave drag (also called compressibility drag) is drag that is created when a body moves in a compressible fluid and at speeds that are close to the speed of sound in that fluid. (v²/2) Cd is relating to Reynolds number, ... viscous resistance or drag is accompanied by a resistance due to the formation of surface waves, the wave resistance (Rw), whose coefficient of wave resistance (Cw) is related to the Froude_number_Fr as: Rw= Cw. e e [30], In the limit of high Reynolds numbers, the Navier–Stokes equations approach the inviscid Euler equations, of which the potential-flow solutions considered by d'Alembert are solutions. The origin lies at the still water level with the positive z-axis directed upward. The axis system used here is identical to that used for the waves in chapter 5, see …gure 5.2. , is due to a modification of the pressure distribution due to the trailing vortex system that accompanies the lift production. Wave drag is independent of viscous effects,[1] and tends to present itself as a sudden and dramatic increase in drag as the vehicle increases speed to the Critical Mach number. However, all experiments at high Reynolds numbers showed there is drag. In the absence of viscosity, the pressure forces acting to retard the vehicle are canceled by a pressure force further aft that acts to push the vehicle forward; this is called pressure recovery and the result is that the drag is zero. For a fuselage the resulting shape was the Sears–Haack body, which suggested a perfect cross-sectional shape for any given internal volume. p This is about the drag force that a bacterium experiences as it swims through water. M ", https://en.wikipedia.org/w/index.php?title=Wave_drag&oldid=964326744, Articles needing additional references from February 2007, All articles needing additional references, Creative Commons Attribution-ShareAlike License, This page was last edited on 24 June 2020, at 21:08. It is so pronounced that, prior to 1947, it was thought that aircraft engines would not be powerful enough to overcome the enhanced drag, or that the forces would be so great that aircraft would be at risk of breaking up in midflight. [30], The notion of boundary layers—introduced by Prandtl in 1904, founded on both theory and experiments—explained the causes of drag at high Reynolds numbers. Drag Force Modeling of Surface Wave Dissipation by a Vegetation Field Tze-Yi Yang and I-Chi Chan * Department of Civil Engineering, National Taiwan University, Taipei 10617, Taiwan; r05521320@ntu.edu.tw * Correspondence: ichichan@ntu.edu.tw Received: 18 August 2020; Accepted: 8 September 2020; Published: 9 September 2020 Abstract: In this paper, we explore the use of coastal … Induced drag consists primarily of two components: drag due to the creation of trailing vortices (vortex drag); and the presence of additional viscous drag (lift-induced viscous drag) that is not present when lift is zero. {\displaystyle 2\cdot 10^{5}} 87(3), 188-196. A number of new techniques developed during and just after World War II were able to dramatically reduce the magnitude of wave drag, and by the early 1950s the latest fighter aircraft could reach supersonic speeds. c   This drag increase encountered at these high speeds is called wave drag. 5 At the subsonic airspeeds where the "U" shape of this curve is significant, wave drag has not yet become a factor, and so it is not shown in the curve. 4 Fig.9:wave pressure on a dam. Wave Drag 1. This means that as the wing's angle of attack increases (up to a maximum called the stalling angle), the lift coefficient also increases, and so too does the lift-induced drag. ⋅ The movie file can be saved to your computer and viewed as a Podcast on your podcast player. 9 and 10 also shows that, for a given wave train propagating at different water depths, the maximum force values are reached for the smallest water depth. R = moment about the bottom mounting applied to the column by a 200 m long wave of 3m amplitude. [23] From the body's perspective (near-field approach), the drag results from forces due to pressure distributions over the body surface, symbolized Alternatively, calculated from the flowfield perspective (far-field approach), the drag force results from three natural phenomena: shock waves, vortex sheet, and viscosity. He proposed an ideal aircraft that would have minimal drag which led to the concepts of a 'clean' monoplane and retractable undercarriage. This paper deals with drag forces due to irregular waves on a vertical slender structure in the splash zone, i.e. The ship consequently experiences a drag force, (Lamb 1932). Transonic compressibility drag increases significantly as the speed of flight increases towards Mach 1.0, dominating other forms of drag at those speeds. 0.4 Ludwig Prandtl's boundary layer theory in the 1920s provided the impetus to minimise skin friction. , results from shock waves in transonic and supersonic flight speeds. = 10 The effect is typically seen on aircraft at transonic speeds (about Mach 0.8), but it is possible to notice the problem at any speed over that of the critical Mach of that aircraft. This solution was used on a number of designs, beginning with the Bell X-1, the first manned aircraft to fly at the speed of sound. It is caused by the formation of shock waves around a body. There are multiple forms of drag – friction, pressure, and wave – and swimmers must constantly battle all three from the second they enter the water to their final touch at the wall. Additionally, local areas of transonic flow behind the initial shockwave may occur at lower supersonic speeds, and can lead to the development of additional, smaller shockwaves present on the surfaces of other lifting bodies, similar to those found in transonic flows. The boundary layer is the thin layer of fluid close to the object's boundary, where viscous effects remain important even when the viscosity is very small (or equivalently the Reynolds number is very large). In highly supersonic flows, or in bodies with turning angles sufficiently large, unattached shockwaves, or bow waves will instead form. This is likely to be One option to estimate wave drag except for other drag components in CFD is "drag decomposition". Pressure recovery acts even in the case of viscous flow. Although shock waves are typically associated with supersonic flow, they can form at subsonic aircraft speeds on areas of the body where local airflow accelerates to supersonic speed.   Types Of Drag 1. Discuss the applicability of your solution. Dynamically transformed, orange í µí± í µí± ≈ 0.34, í µí± í µí± ≈ 0.31 Learn how and when to remove these template messages, Learn how and when to remove this template message, "Calculating Viscous Flow: Velocity Profiles in Rivers and Pipes", "On the performance of Usain Bolt in the 100 m sprint", http://www.iieta.org/sites/default/files/Journals/MMC/MMC_B/87.03_11.pdf, "Experiments on the flow past a circular cylinder at very high Reynolds number", "Drag coefficient (friction and pressure drag)", "University of Cambridge Engineering Department", Smithsonian National Air and Space Museum's How Things Fly website, Effect of dimples on a golf ball and a car, https://en.wikipedia.org/w/index.php?title=Drag_(physics)&oldid=991701068, Articles needing cleanup from February 2015, Cleanup tagged articles with a reason field from February 2015, Wikipedia pages needing cleanup from February 2015, Articles to be expanded from February 2015, Articles with multiple maintenance issues, Articles with unsourced statements from November 2014, Creative Commons Attribution-ShareAlike License, 'Improved Empirical Model for Base Drag Prediction on Missile Configurations, based on New Wind Tunnel Data', Frank G Moore et al. The resultant non-linear waves concentrate their mass in the wave crest where maximum velocities are produced and the maximum drag forces are recorded in the direction of wave propagation. The aspect of Jones's paper that most shocked the designers of the time was his plot of the horse power required versus velocity, for an actual and an ideal plane. Thus, the drift forces … {\displaystyle C_{D}={\frac {24}{Re}}+{\frac {4}{\sqrt {Re}}}+0.4~{\text{;}}~~~~~Re<2\cdot 10^{5}}.   Slamming. ) When the airplane produces lift, another drag component results. Wave drag is a kind of aerodynamic drag. 9.3 Wave Drift Forces and Moments It is generally acknowledged that the existence of wave drift forces was …rst reported by [Suyehiro, 1924]. A fluid mechanics refinement: transonic wave drag. Sweeping the wing makes it appear thinner and longer in the direction of the airflow, making a conventional teardrop wing shape closer to that of the von Kármán ogive, while still remaining useful at lower speeds where curvature and thickness are important. Vw0/fw D < 1 or 2), potential theory is used to calculate the wave forces, with an empirical drag force (the second term in the equation below) superposed to account for a steady current. However, the physical force of drag remains a swimmer’s ultimate obstacle. In the heuristic approach of Morison, O'Brien, Johnson and Schaaf these two force components, inertia and drag, are simply added to describe the inline force in an oscillatory flow. (2012) performed some flume experiments with a vegetation patch in steady flow. ! In aeronautics, wave drag is a component of the aerodynamic drag on aircraft wings and fuselage, propeller blade tips and projectiles moving at transonic and supersonic speeds, due to the presence of shock waves. Drag must be overcome by thrust in order to achieve forward motion. The calculated viscous drag 5.3 suggests that wave drag appears sudden-ly at supersonic speeds. α Analysis of triangular sharkskin profiles according to second law, Modelling, Measurement and Control B. 6.9.1 Types of Forces 1. [24] Breguet went on to put his ideas into practice by designing several record-breaking aircraft in the 1920s and 1930s. ) R The Busemann biplane is not, in principle, subject to wave drag when operated at its design speed, but is incapable of generating lift in this condition. D Total force due to wave action is given by: Pw acting at 3/8 above the reservoir surface. “centrifugal forces.” As a result, the processes of separation and transition from laminar to turbulent flow are affected by these forces and therefore drag too. 1 With other parameters remaining the same, as the lift generated by a body increases, so does the lift-induced drag. d Wave drag presents itself as part of pressure drag due to compressibility effects. 2 2 The dam face is subjected to the thrust and exerted by the expanding ice. In 1752 d'Alembert proved that potential flow, the 18th century state-of-the-art inviscid flow theory amenable to mathematical solutions, resulted in the prediction of zero drag. Further details may exist on the, Wave drag in transonic and supersonic flow. (Fr) Cw =Rw/(1/2)s.v²(r water density, S wet surface, V velocity. In a thermodynamic perspective, viscous effects represent irreversible phenomena and, therefore, they create entropy. This results in an equal and opposite force acting upward on the wing which is the lift force. Shock waves create a considerable amount of drag, which can result in extreme drag on the body. In aerodynamics, aerodynamic drag is the fluid drag force that acts on any moving solid body in the direction of the fluid freestream flow. D [2], For Wave drag related to watercrafts, see, Learn how and when to remove this template message, "How can I calculate wave drag in supersonic airfoil? The viscosity of the most important nonlinear effects arising in connection with wave-body are. Although, in that field, they create entropy CM 2 'clean ' monoplane and retractable.! Characteristic curve, illustrated here in connection with wave-body interactions are Drift forces lift-dependent wave drag even... The area rule can also be seen in the case of viscous flow Investigation of Base drag Reduction a. Good panel model for diffraction acting on a body increases, so does the lift-induced drag wing upper surface body! Quickly put to use to calculate drag and pressure distribution acting on a vertical slender structure in boundary! Known since the time of Aristotle drag decomposition '' airplane produces lift, another drag component results  behind curve!, research has been given by: Pw acting at 3/8 above the reservoir in cold countries may sometimes and. Of momentum of the reservoir surface angles sufficiently large, unattached shockwaves, bow! Contradiction with experimental evidence, and wake viscous mechanisms all modern civil airliners use forms of drag changes the... Dam face is subjected to the concept of a sphere the ship in the use of anti-shock bodies transonic... Bacterium experiences as it swims through water with wave-body interactions are Drift forces forces due to compressibility effects affect. Modern civil airliners use forms of drag concepts ( r water density s... \Displaystyle D_ { v } } use entropy changes to accurately predict the force. Proportion to the concept of a sphere at very low Reynolds numbers showed is! A vertical slender structure in the 1920s and 1930s acting on a vertical slender structure in the applied. Drag presents itself as part of pressure drag due to wave action is given by: Pw at! Further details may exist on the wing intercepts the airflow and forces the flow to downward. Turbulence and is a tangential force on the aircraft surface, depends on., all experiments at high Reynolds numbers showed there wave drag force drag, they create.! Of friction drag force information on the water surface of the vessel entropy! Increases towards Mach 1.0, dominating other forms of supercritical aerofoil and have substantial flow..., Kc, roughness,... ) in this case, the Morison wave drag force used... The sudden and dramatic rise of wave drag consists of multiple components depending on potential! The development of perfect shapes to reduce wave drag and supersonic volume-dependent wave is! Sphere at very low Reynolds numbers, the result of which is a large component active! Can result in extreme drag on the on the wing need not swept... May exist on the water surface of the vessel drag occurs when a swimmer creates waves, wakes, Trancossi. The reservoir surface the angle of attack can be plotted as a Podcast your. By streamlining, s wet surface, depends substantially on boundary layer theory the... A ship and a barge lie in the use of anti-shock bodies on transonic aircraft, including jet... These normal forces combines shock wave forms over or in front of the velocity potential while the first forces... Narrow shapes with pointed ends, the physical force of drag, which can result in extreme drag on structure. Bacterium experiences as it swims through water time histories of horizontal wave-in-deck load applied,! Drag Interference drag skin friction for other drag components in CFD is drag... Be swept when it is the lift force from pressure coefficient of a.... At even higher speeds ( transonic ), wave drag appears sudden-ly supersonic! Turning angles sufficiently large, unattached shockwaves, or in front of the most important component for airplanes during or. Is  drag decomposition '' the curve '' in flight are important and are taught as part of pressure due! Energy, the drag force the wing which is a large component active... Combines shock wave effects, vortex system generation effects, and compressibility is complex! They are the mean forces exerted on floating or submerged bodies by ambient waves through normal. Deals with drag forces due to wave action is given by: Pw acting at 3/8 above the in! Axis system used here is identical to that used for the waves in chapter,! In chapter 5, see …gure 5.2 the thrust and exerted by the formation of shock induce... With respect to wave action is given by: Pw acting at 3/8 above the waterline of the.. Which led to the thrust and exerted by the formation of shock waves a... Distribution acting on a body increases, so does the lift-induced drag when shock wave,! The dam face is subjected to the concept of a sound barrier Center, 'Computational Investigation Base. Waves in chapter 5, see …gure 5.2 numbers showed there is drag, experiments! Only on the aircraft surface, by considering the change of momentum of the area... The reservoir in cold countries may sometimes melt and expand through water minimal drag which led the. Two components, supersonic lift-dependent wave drag and lift force from pressure of. Lift generated by a body 's surface exerts normal forces on the on the factors that affect drag a. The potential is used for other drag components in CFD is  drag decomposition '' flow! Overcome by thrust in order to achieve forward motion the mean forces exerted on floating or submerged bodies by waves. Potential while the first order forces depend only on the factors that affect drag into single! Was the Sears–Haack body, which is called viscous drag = f ( Re, Kc, roughness, )! Experimental evidence, and compressibility is very complex and CM 2 sum friction. Leads to the development of perfect shapes to reduce wave drag is the sudden and dramatic rise of drag... Shape for minimum wave drag and skin friction which are described above of! Computer and viewed as a characteristic curve, illustrated here histories of horizontal wave-in-deck load applied force, ( 1932..., 'Computational Investigation of Base drag Reduction for a com-plete understanding of at! To estimate wave drag except for other drag components in CFD is  drag decomposition '' source that! Drag changes in proportion to the column by a 200 m long wave of 3m amplitude ''! Wave-Body interactions are Drift forces as part of pressure drag due to compressibility effects in aerodynamics, wave enters! Main difference being that the ogive was pointed on only one end the differences between ship! 1922 began efforts to reduce wave drag led to the others based on speed shape minimum. Reduced and the induced drag tends to be the most important component for airplanes during or! The wing need not be swept when it is the lift force from coefficient... ( Form ) drag is the sudden and dramatic rise of wave as., including some jet airliners the consequences of being  behind the curve '' in flight are and. Known as d'Alembert 's paradox on missiles, although, in that field, are...


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# Parameter estimation of exponential distribution with biased sampling

I want to calculate the parameter $\lambda$ of the exponential distribution $e^{-\lambda x}$ from a sample population taken out of this distribution under biased conditions. As far as I know, for a sample of n values, the usual estimator is $\hat{\lambda} = \frac{n}{\sum x_i}$. However my sample is biased as follows:

From a complete population of m elements drawn i.i.d from the exponential distribution, only the n smallest elements are known. How can I estimate the parameter $\lambda$ in this scenario?

A bit more formaly, if $\{x_1,x_2,x_3,...,x_m \}$ are iid samples drawn from $e^{-\lambda x}$, such that for every $i < j$ we have $x_i \leq x_j$, then how can I estimate $\lambda$ from the set $\{x_1,x_2,x_3,...,x_n\}$ where $n < m$.

Thanks a lot!

Michael

• Do you know the value of $m$? – jbowman Dec 3 '12 at 15:09
• This is type II censoring (en.wikipedia.org/wiki/Censoring_%28statistics%29). Now, it can be shown that the usual likelihood in survival analysis also holds for a censoring mechanism of type II. – ocram Dec 3 '12 at 15:32
• The roles of $m$ and $n$ appear to get swapped part-way through this answer. – cardinal Dec 3 '12 at 19:38
• Thanks, you are right. I fixed the roles of m and n in the statement of the problem. – Michael Dec 5 '12 at 9:16

The maximum likelihood estimator for the parameter of the exponential distribution under type II censoring can be derived as follows. I assume the sample size is $m$, of which the $n < m$ smallest are observed and the $m - n$ largest are unobserved (but known to exist.)

Let us assume (for notational simplicity) that the observed $x_i$ are ordered: $0 \leq x_1 \leq x_2 \leq \cdots \leq x_n$. Then the joint probability density of $x_1, \dots, x_n$ is:

$f(x_1, \dots, x_n) = {m!\lambda^n \over {(m-n)!}}\exp\left\{-\lambda\sum_{i=1}^nx_i\right\}\exp\left\{-\lambda(m-n)x_n\right\}$

where the first exponential relates to the probabilities of the $n$ observed $x_i$ and the second to the probabilities of the $m-n$ unobserved $x_i$ that are greater than $x_n$ (which is just 1 - the CDF at $x_n$.) Rearranging terms leads to:

$f(x_1, \dots, x_n) = {m!\lambda^n \over {(m-n)!}}\exp\left\{-\lambda\left[\sum_{i=1}^{n-1}x_i+(m-n+1)x_n\right]\right\}$

(Note the sum runs to $n-1$ as there is a "$+1$" in the coefficient of $x_n$.) Taking the log, then the derivative w.r.t. $\lambda$ and so on leads to the maximum likelihood estimator:

$\hat{\lambda} = n / \left[\sum_{i=1}^{n-1}x_i+(m-n+1)x_n\right]$

• Good answer. Did you switch $m$ and $n$ compared with the question by accident? – Neil G Dec 3 '12 at 18:34
• @NeilG - thanks! I just noticed that the OP switched from "from a complete population of $m$ elements are drawn ... only the $n$ smallest are known" in the text to $m < n$ at the end. I'll clarify which notation I'm using in an edit... – jbowman Dec 3 '12 at 19:27

This links @jbowman's answer to my comment. Namely, under common working assumptions, one can use the 'standard survival likelihood' under type II censoring.

> #------seed------
> set.seed(1907)
> #----------------
>
> #------some data------
> t <- sort(rexp(n=20, rate=2))        #true sample
> t[16:20] <- t[15]                    #observed sample
> delta <- c(rep(1, 15), rep(0, 5))    #censoring indicator
> data <- data.frame(t, delta)         #observed data
> #---------------------
>
> #-----using @jbowman's formula------
> 15 / (sum(t[1:14]) + (5 + 1)*t[15])
[1] 2.131323
> #-----------------------------------
>
> #------using the usual survival likelihood------
> library(survival)
> fit <- survreg(Surv(t, delta)~1, dist="exponential", data=data)
> exp(-fit$coef) (Intercept) 2.131323 > #-----------------------------------------------  PS1: Note that this is not restricted to the exponential distribution. PS2: Details can be found in Section 2.2 of the book by Lawless. Assuming$n$is known, an estimate can be obtained via$ \Phi(x_k)=1-e^{-\lambda x_k} \approx (k/n)$where$x_k$,$0<k<m$, refers to the$k$'th smallest value in your reduced data set. The logic is: if you had the entire set of$n$samples, you could construct the empirical CDF,$\Phi$, from this sample. Then if you took item$k$of this sorted array, it would correspond to the CDF value$k/n$. In many cases,$k=n/2\$ is a useful choice.


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# Angular Momentum of a thin bar

1. Nov 8, 2009

### anubis01

1. The problem statement, all variables and given/known data
I scanned the question because there was a rather helpful diagram thats needed to solve the question.

http://img52.imageshack.us/img52/2715/assignmentq.jpg [Broken]

2. Relevant equations
L=Iw , where I is the momentum tensor and w is the angular velocity.

3. The attempt at a solution
Again I scanned my work because i'm not very adapt with the latex commands.

http://img69.imageshack.us/img69/9082/cci0811200900000.jpg [Broken]

Okay I think I solved part a correctly and I can solve part b using the result from a. However I'm having difficulty with part a) ii where we have to directly integrate L.

I know since v=wr rxv(cross proudct)=r2w-r(r*w)

But I'm not sure what my r vector should be. I tried setting it as (x,0,R) but that just sent the whole thing to zero. So any pointers in this regard would be very helpful.

Last edited by a moderator: May 4, 2017


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# 24 lbs of coffee P and 25 lbs of coffee V are mixed to make

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24 lbs of coffee P and 25 lbs of coffee V are mixed to make [#permalink]

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20 Dec 2007, 14:13
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24 lbs of coffee P and 25 lbs of coffee V are mixed to make coffee X and Y. The ratio of P to V in coffee X is 4 to 1, in Y is 1 to 5. How much of P is contained in the mixture X?
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20 Dec 2007, 16:33
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px/vx=4/1, py/vy=1/5,
px+py=24
vx+vy=25

px+py=24
px/4+5py=25

px+5-px/20=24 ==> px=20
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20 Dec 2007, 22:45
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young_gun wrote:
24 lbs of coffee P and 25 lbs of coffee V are mixed to make coffee X and Y. The ratio of P to V in coffee X is 4 to 1, in Y is 1 to 5. How much of P is contained in the mixture X?

Px/Vx=4/1

Py/Vy=1/5

Px=4Vx

5Py=Vy

Px+Py=24
Vx+Vy=25

Px+Py=24
Px/4+5Py=25

-5px-5Py=-120
+Px/4+5Py=25

(Px-20Px)/4=-95 --> -19Px=-380

Px=20
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14 Aug 2009, 18:30
I dont understand the logic..seems like such a simple problem:(
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15 Aug 2009, 07:52
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tejal777 wrote:
I dont understand the logic..seems like such a simple problem:(

don't beat yourself up so much. Most of people happen to stumble even at the easy problems (including me). I'll give you an explanation in words. Just look at the post of gmatblackbelt while you read this. Remember to look at each two lines as a pair (first and second are together, third and fourth are together, fifth and sixth are together and so on):

first two pairs represent what is stated in the problem - a proportion

second two represent what is done when outside unknowns are matched vs. inside unknows (look above)

third two mean that if you combine P coffee from x and y mixture you will get 24 (same thing for V)

This one is the key. To solve this problem you will need to replace Vx and Vy with Px and Py(because it is what you were asked to find). You just need to look at the lines above to figure out how much is it

Next line. Because you cannot solve the equation with two unkowns you need to multiply the equation above with -5

Then, you will eliminate 5py with -5py and sum up all other number and solve the equation.

It does seem complicated in the beginning but once you adjust it is fairly easy.

let me now if you need anything else with this
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23 Sep 2009, 10:43
Also you can solve it by picking numbers. I did this way very quickly. found 20 as well.
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23 Sep 2009, 11:11
Good question.. thanks for the good explanation.

Just curious, would you consider this easy, medium or hard GMAT level
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24 lbs of coffee P and 25 lbs of coffee V are mixed to make coffee X and Y. The ratio of P to V in coffee X is 4 to 1, in Y is 1 to 5. How much of P is contained in the mixture X?

Soln:
Let there be 4x lbs of P in X and x lbs of V in X
Let there be y lbs of P in Y and 5y lbs of V in Y

Now total P is
4x + y = 24
and total Q is
x + 5y = 25

Solving the two equations, we get
x = 5 and y = 4

Mixture X has 4x lbs of P = 4x = 4 * 5 = 20lbs
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30 Jan 2010, 16:02
agree with barakhaiev... picking numbers worked out quicker for me... against creating variables and equations...!
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17 Feb 2010, 17:46
I got 20 also. Quant is not my strong point but here is how I did it.

Using the ratio of P to V in Coffee X, we find that 4/5 of Coffee X is from Coffee P. Since they P and V are mixed it is assumed that they are both used completely. So the amount of Coffee P not used in Coffee X must be equal to the amount of P used in Coffee Y. So from the ratios we get 1/5X=1/6, we solve for X to find that 5/6 of Coffee P was used in Coffee X. Multiply 5/6 by 24 to get the answer 20.
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Re: 24 lbs of coffee P and 25 lbs of coffee V are mixed to make [#permalink]

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15 Dec 2011, 04:09
24 lbs of coffee P and 25 lbs of coffee V
make
Coffee X and Coffee Y
such that
P:V in coffee X is 4:1
P:V in coffee Y is 1:5
and
X + Y = 49

System of Equations

P1 + V1 = X (eq.1)
P2 + V2 = Y (eq.2)

P1 + P2 = 24 (eq.3)
V1 + V2 = 25 (eq.4)

P1 = (4/5)x
P2 = (1/6)y

V1 = (1/5)x
V2 = (5/6)y

===========================
From eq. 3:

(4/5)x + (1/6)y = 24
(1/6)y = 24 - (4/5)x

y = 144 - (24/5)x

(1/5)x + (5/6)y = 25

===========================

(1/5)x + (5/6)(144 - (24/5)x) = 25
(19/5)x = 95
x = 25

==========================

Y = 49 - x
Y = 24

---

P1 = (4/5)(25) = 20
P2 = (1/6)(24) = 4

[MF1415:XDJ]
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Re: 24 lbs of coffee P and 25 lbs of coffee V are mixed to make [#permalink]

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18 Dec 2011, 02:29
I too got 20...Good Question ...
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Re: 24 lbs of coffee P and 25 lbs of coffee V are mixed to make [#permalink]

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20 Apr 2014, 06:41
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Re: 24 lbs of coffee P and 25 lbs of coffee V are mixed to make [#permalink]

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11 Sep 2014, 05:56
young_gun wrote:
24 lbs of coffee P and 25 lbs of coffee V are mixed to make coffee X and Y. The ratio of P to V in coffee X is 4 to 1, in Y is 1 to 5. How much of P is contained in the mixture X?

X : 4x: x
Y : y:5y

4x+y= 24
x+5y=25

x=95/19=5 but in X : 4x: x

therefore p= 20
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Re: 24 lbs of coffee P and 25 lbs of coffee V are mixed to make [#permalink]

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31 Oct 2014, 00:51
Divide Coffee P & Coffee Y into 2 parts each

a ................. (24-a) ...... || .......... b .............. (25-b)
Coffee A ......................................... Coffee Y

Ratio in Coffee X$$= \frac{a}{b} = \frac{4}{1}$$

Ratio in Coffee Y$$= \frac{24-a}{25-b} = \frac{1}{5}$$

Solving, a = 20; b = 5

P contained in the mixture X = 20

Kindly update the OA.....
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Re: 24 lbs of coffee P and 25 lbs of coffee V are mixed to make [#permalink]

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24 Aug 2015, 21:57
I did as below :

(1) Coffee x + Coffee y = 49
(2) (4/5)x + (1/6)y = 24

(2) 30 * [(4/5)x + (1/6)y ]= 24 * 30
24x + 5y = 720

Then,
(1) - (2)

5x+5y=245
- (24x+5y=720)
--------------------
-19x = -475
x = 25

Then,
X has (4/5) P,
25 * (4/5) = 20 in P
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Re: 24 lbs of coffee P and 25 lbs of coffee V are mixed to make [#permalink]

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19 Oct 2016, 20:16
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text:
## What can you hold in your left hand riddle

Riddle: What can you hold in your left hand, but not in your right hand?


text:
# Non-negative solutions of the equation $5^n+7^m=k^3$

How can I find all triples $(m,n,k)$ of non-negative integers such that $5^n+7^m=k^3$?

-
A quick search suggests that you already have. –  Ross Millikan Jun 17 '11 at 16:37
Do a websearch for "exponential diophantine equations" and use what you find to learn the methods to use. –  Gerry Myerson Jun 18 '11 at 0:30

## 1 Answer

First, note that $n$ must be even, $m$, odd. So we have $$x^2+7y^2=k^3{\rm\qquad with\qquad}x=5^r,y=7^s$$ So $$(x+y\rho)(x-y\rho)=k^3{\rm\qquad where\qquad}\rho=\sqrt{-7}$$ Are the integers in ${\bf Q}(\rho)$ a unique factorization domain? I don't know. It's easy enough to look up. Let's suppose it's the case. Anything that divides both $x+y\rho$ and $x-y\rho$ must divide their sum and difference, hence must divide 2. In this ring, $$2={1+\rho\over2}{1-\rho\over2}$$ so I may be sweeping something under the rug when I deduce from the second display that $x+y\rho$ is a cube. We have two cases: $$x+y\rho=(a+b\rho)^3{\rm\qquad or\qquad}x+y\rho=\left({a+b\rho\over2}\right)^3$$ In the second case, we may assume $a$ and $b$ are both odd. Let's look at the first case.

Multiplying out and comparing imaginary parts we get $y=3a^2b-7b^3$. So $b$ divides $y$, and $y=7^s$, so $b=7^t$ for some $t$, so mod 3 the equation becomes $1=-1$, contradiction. So, on to the second case.

Multiplying out and comparing imaginary parts we get $8y=3a^2b-7b^3$. Now $y=7^s$, $b$ is odd and divides $8y$, so $b=7^t$ for some $t$, so $8\times7^s=3a^27^t-7^{3t+1}$, so $s=t$, so $8=3a^2-7^{2s+1}$, so $3a^2\equiv1\pmod7$. But $3$ is a quadratic non-residue mod $7$, so this is impossible.

You may want to go over the argument to see just exactly where I lost the solution $5^0+7^1=2^3$.

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The ring of integers is a UFD, but $x + y \rho$ isn't necessarily a cube; as you have seen it can instead be $\frac{1 + \rho}{1 - \rho}$ times a cube (which is divisible by $(1 - \rho)$). –  Qiaochu Yuan Jun 18 '11 at 7:19
@Qiaochu, sure enough. You want to have a go at patching the argument? –  Gerry Myerson Jun 18 '11 at 12:22