Method for batch solving flood design standard at specific frequency in hydrology statistics

Abstract

The present application provides a method for batch solving flood design standard under specific frequency in hydrology statistics, which is used to solve the technical problems of the existing hydrology calculation, such as needing to be solved one by one, complicated calculation process and low efficiency. The steps of the present application are as follows: obtaining the longest hydrology series of the hydrology statistics project; constructing the Pearson III equation according to the data of the longest hydrology series; constructing the equation for batch solving runoff under specific frequency according to the constructed Pearson III equation; and batch solving the runoff equation under specific frequency to obtain the corresponding flood design standard under specific frequency. The present application can quickly and efficiently solve the corresponding flood design standard under specific frequency, and can be used in the field of flood control engineering design, which is very important for the field of flood control engineering design and water resources management, and has practicality and popularization value.

Description

Technical Field

[0001] This invention relates to the technical field of hydrological statistics, and in particular to a method for batch solving flood design standards at a specific frequency in hydrological statistics. Background Technology

[0002] The randomness of hydrological phenomena makes the magnitude of future floods uncertain. Therefore, to reduce flood risk, flood control engineering design often employs probabilistic analysis methods. Probabilistic analysis typically uses historical hydrological data, meteorological data, and topographical data to determine the probability distribution function of floods through statistical analysis. This allows for the estimation of the probability of floods of different magnitudes, and, based on the importance of the project and its risk tolerance, the selection of an appropriate flood design standard for flood control engineering design. Therefore, selecting an appropriate design standard through scientific and reasonable probabilistic analysis is of great practical significance for reducing flood risk and protecting people's lives and property.

[0003] Currently, the standards for hydrological calculations or design flood calculations in water conservancy and hydropower projects all adopt [the following]. The Pearson Type III distribution curve shows good correlation with the data, and its shape generally conforms to the general laws of hydrological phenomena, with reasonable extrapolation results. The process of solving the equations is largely based on examining the deviation coefficient of the Pearson Type III distribution. The value table solves for each solution one by one, but this method can usually only calculate the flood design standard for each frequency, making the calculation process cumbersome and inefficient. Furthermore, when encountering deviation coefficients... When the value table contains missing values, the difference method needs to be used to obtain a specific deviation coefficient from the mean. The limited values ​​result in low solution efficiency and a high susceptibility to errors. Therefore, a method is needed that can perform batch calculations of flood design standards at specific frequencies.

[0004] The invention patent with application number 201610680265.X discloses a method for constructing a regulation line for the confluence of sediment-rich rivers. The steps are as follows: (1) Determine an appropriate numerical model; (2) Introduce water flow into the second physical model according to the flood design standard flow to scour the sand particles laid in the second physical model. After the contour line of the scoured confluence stabilizes, record the position of the contour line of the scoured confluence and the water depth of the typical cross section of the confluence, and calculate the critical starting velocity of the sand particles in the prototype confluence; (3) Use the numerical model to calculate the velocity distribution of the prototype under the flood design standard flow conditions, and use the contour line of the critical starting velocity of the sand particles in the prototype confluence at the outer contour line of the confluence in the velocity distribution map as the regulation line of the confluence. The above invention can reasonably guide the regulation of the confluence of sediment-rich rivers, thereby alleviating the negative impact of the confluence confluence on the flood control safety of the river channel and improving the flood control safety of the river channel. However, the aforementioned invention requires a second physical model test and numerical model calculation, necessitating a significant investment of time and money, and presents certain technical barriers. This makes it difficult to popularize and promote in areas with small and medium-sized rivers or limited resources. Furthermore, this method is only applicable to the construction of regulation lines at the confluence of sediment-rich rivers. For other types of rivers or different regulation needs, redesign and experimental verification are required, lacking versatility and flexibility. Summary of the Invention

[0005] To address the technical problems of existing hydrological calculations requiring individual solutions, which are cumbersome and inefficient, this invention proposes a method for batch solving of flood design standards at specific frequencies in hydrological statistics. By employing the probability density function of a Pearson Type III distribution to construct a frequency curve equation conforming to a P-III distribution, an equation F(x) consisting of n products of G(x) - Pn is constructed to solve the runoff equation at a specific frequency. Through numerical calculation, the runoff equation F(x) at a specific frequency is solved quickly and accurately, yielding the required flood design standards. This method allows for rapid and accurate batch solving of flood design standards corresponding to different frequencies, providing crucial technical support for the design, construction, and management of water conservancy projects.

[0006] To achieve the above objectives, the technical solution of the present invention is as follows: a method for batch solving flood design standards at a specific frequency in hydrological statistics, comprising the following steps:

[0007] Step 1: Obtain the longest hydrological series for the hydrological statistics item;

[0008] Step 2: Construct the Pearson hydrological series based on the data from the longest hydrological series. Type equation;

[0009] Step 3: Based on the constructed Pearson Formulate equations to construct equations that can solve runoff at specific frequencies in batches;

[0010] Step 4: The runoff equations at specific frequencies can be solved in batches to obtain the flood design standards corresponding to those frequencies.

[0011] Preferably, the implementation method of step 1 is as follows: Based on the requirements, identify the hydrological stations for which statistical analysis needs to be conducted, and collect a series of hydrological statistical data for N years from the establishment of the hydrological station to the most recent year; the hydrological statistical data include, but are not limited to, annual runoff, time-period runoff, annual maximum flood peak, and time-period flood volume; statistically analyze the collected series of data according to the requirements by annual runoff, time-period runoff, annual maximum flood peak, time-period flood volume, or other required items; according to the principle of consistency of hydrological series in hydrological statistics, if the measurement and compilation accuracy control methods of hydrological stations are the same, and human activities have a small impact on hydrological statistical items, then the series of items statistically analyzed from the measured data of the hydrological station is determined as the longest hydrological series of hydrological statistical items; if the impact of human activities is large, based on the historical data comparison method, compare the data of different hydrological stations with the historical data of the same period and perform consistency correction, and determine the series after consistency correction as the longest hydrological series of hydrological statistical items.

[0012] Preferably, the step in step 2 of calculating the Pearson Type III equation G(x) corresponding to the longest hydrological series is as follows:

[0013] 1) Statistical analysis was performed on the data of the longest hydrological series to obtain statistical characteristics;

[0014] 2). Based on statistical characteristics, the longest hydrological series data is fitted using the probability density function of the Pearson Type III distribution to obtain the Pearson Type III equation G(x); where x represents the hydrological variable;

[0015] 3) Verify and optimize the Pearson type III equation G(x) to improve the fitting effect and accuracy.

[0016] Preferably, the longest hydrological series refers to hydrological data over a long period of time, including data on runoff, rainfall, and water level over many years;

[0017] The statistical characteristics include at least one of the following: mean, variance, skewness, or kurtosis; the hydrological variables include runoff, rainfall, or water level.

[0018] Preferably, in step 3, a frequency curve equation G(x) conforming to the Pearson type III distribution is constructed based on the input parameters of multi-year average runoff, coefficient of variation, and skewness coefficient; and an equation F(x) consisting of the product of n G(x)-Pn is constructed based on the input design frequency Pn.

[0019] Preferably, based on the parameters of the multi-year average runoff, coefficient of variation, and skewness coefficient, a frequency curve equation G(x) conforming to the P-III type distribution is constructed using the probability density function of the P-III type distribution; based on the input design frequency Pn, the frequency curve equation G(x) and the design frequency Pn are substituted into the equation G(x)-Pn to obtain a set of G(x)-Pn values; the above steps are repeated until n sets of G(x)-Pn values ​​are obtained; the n sets of G(x)-Pn values ​​are multiplied together to obtain the equation F(x) composed of the product of n G(x)-Pn values.

[0020] Preferably, in the design of flood control projects, the design frequency Pn is determined based on factors such as historical flood data, geographical environment, hydrological and meteorological conditions, and project requirements.

[0021] Preferably, in step 4, the minimum value min and maximum value max of the runoff are specified, as well as the accuracy eps. Mathematical methods can be used to solve the runoff equation F(x) at a specific frequency in batches. The mathematical method uses numerical integration to divide the interval of specified accuracy, and uses the bisection method to find the root within the interval.

[0022] Preferably, the method of dividing the interval with a specified precision using numerical integration is as follows:

[0023] Calculate the number of equally divided intervals m = (b - a) / eps;

[0024] Divide the solution interval [a, b] into m equal intervals, such that the length of each interval is dx, where dx is less than or equal to the required solution accuracy eps;

[0025] For each equally divided interval, the integral value of the equation F(x) in that interval is calculated using numerical integration; the composite trapezoidal rule is used to obtain several solution intervals.

[0026] Preferably, the implementation method of the bisection method for finding roots is as follows:

[0027] Determine the lower limit a and upper limit b of the solution interval;

[0028] By continuously bisecting the solution interval until the zero point of the equation F(x) is found, the solution of the runoff equation F(x) corresponding to a specific frequency can be obtained.

[0029] The method for finding the zeros of the equation F(x) is as follows:

[0030] i. Calculate the midpoint of the interval c = (a + b) / 2;

[0031] ii. Determine if the signs of the values ​​F(a) and F(c) are the same. If they are the same, the root is in the interval [c, b], and the value of the lower limit a is updated to c; otherwise, the root is in the interval [a, c], and the value of the upper limit b is updated to the upper limit c.

[0032] iii. Repeat steps i and ii until the difference between the upper limit a and the lower limit b is less than the solution accuracy eps.

[0033] Compared with existing technologies, this invention has the following advantages: By constructing an equation F(x) consisting of n products of G(x) - Pn, multiple solutions (flood design standards) can be obtained in a single solution. This allows for rapid and efficient solving of flood design standards corresponding to specific frequencies, and can be applied to fields such as flood control engineering design. It is crucial for flood control engineering design and water resource management, possessing practical value and potential for widespread application. Traditional methods require multiple calculations and experiments, while this invention can obtain multiple solutions in a single solution, significantly saving time and costs. Furthermore, it can be applied to various hydrological statistical problems, such as hydrological prediction and hydrological simulation. Therefore, this invention has broad application value and promising prospects for wider application. Attached Figure Description

[0034] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0035] Figure 1 This is a flowchart of the present invention. Detailed Implementation

[0036] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0037] like Figure 1 As shown, a method for batch solving flood design standards at a specific frequency in hydrological statistics includes the following steps:

[0038] Step 1: Obtain the longest hydrological series of hydrological statistics items.

[0039] Based on the requirements, identify the hydrological stations for which statistical analysis is needed, and collect a series of hydrological statistical data for N years, from the station's establishment to the most recent year. Hydrological statistical items include, but are not limited to, annual runoff, time-period runoff, annual maximum flood peak, and time-period flood volume. The collected data series will be statistically analyzed according to the requirements, categorized by annual runoff, time-period runoff, annual maximum flood peak, time-period flood volume, or other required items. The consistency of the hydrological series refers to the use of the same measurement and compilation accuracy control methods at the hydrological stations, and the impact of human activities on the hydrological statistical items. According to the principle of consistency in hydrological statistical requirements, if the measurement and compilation accuracy control methods at the hydrological stations are the same, and the impact of human activities on the hydrological statistical items is small, then the longest hydrological series is determined by the statistical data compiled from the hydrological station's measured data. Generally, if the natural environment of the area where the hydrological station is located remains stable, and the interference of human activities on hydrological processes is small, then the impact of human activities on the hydrological statistical items can be considered small. If human activities have a significant impact, consistency correction should be performed according to the methods specified in the "Specifications for Hydrological Calculation of Water Conservancy and Hydropower Projects" and the "Specifications for Design Flood Calculation of Water Conservancy and Hydropower Projects." The series after consistency correction should be determined as the longest hydrological series for the hydrological statistical item. Based on the historical data comparison method, data from different hydrological stations should be compared with historical data from the same period to eliminate inconsistencies.

[0040] Step 2: Construct the Pearson hydrological series based on the data from the longest hydrological series. Equations of the form.

[0041] Based on the longest hydrological series obtained, calculate the corresponding Pearson data. The equation is G(x). The longest hydrological series refers to hydrological data over a long period, typically including data on runoff, rainfall, and water level over many years. This data can be obtained through hydrological observation, hydrological models, or historical data.

[0042] The following steps are required to calculate the Pearson type III equation G(x) corresponding to the data:

[0043] 1. Perform statistical analysis on the data of the longest hydrological series, including calculating statistical characteristics such as mean, variance, skewness, and kurtosis.

[0044] 2. Based on statistical characteristics, the probability density function of the Pearson Type III distribution is used to fit the data of the longest hydrological series, resulting in the Pearson Type III equation G(x), where x represents hydrological variables such as runoff, rainfall, or water level.

[0045] 3. Verify and optimize the Pearson type III equation G(x) to improve the fitting effect and accuracy.

[0046] The benefits of this step include: it allows for a more accurate description of the distribution of hydrological variables such as runoff at different frequencies, improving the accuracy and reliability of hydrological statistical analysis; it provides more precise hydrological data for flood control engineering design, guiding the construction and management of flood control projects; and it provides fundamental data and scientific basis for fields such as water resources management, environmental protection, and climate change research.

[0047] Step 3: Based on the constructed Pearson Formulate equations to construct equations that can solve runoff at specific frequencies in batches.

[0048] Based on the constructed Pearson The P-III type equation G(x) is used to construct a runoff equation F(x) that can be solved in batches at a specific frequency. Based on the input multi-year average runoff, coefficient of variation, and skewness coefficient parameters, a frequency curve equation G(x) conforming to the P-III type distribution can be constructed. Then, based on the input design frequency Pn, an equation F(x) consisting of the product of n G(x) - Pn can be constructed. Specifically, this can be achieved by following these steps:

[0049] Based on parameters such as the multi-year average runoff, coefficient of variation, and skewness coefficient, a frequency curve equation G(x) conforming to the P-III type distribution is constructed using the probability density function of the P-III type distribution. Substituting G(x) and Pn into the equation G(x)-Pn according to the input design frequency Pn, a set of G(x)-Pn values ​​is obtained; this process is repeated until n sets of G(x)-Pn values ​​are obtained. Multiplying the n sets of G(x)-Pn values ​​yields the equation F(x), which is the product of n G(x)-Pn values. Obtaining equation F(x) allows for solving the runoff equation at a specific frequency, thus yielding the desired flood design standard.

[0050] The batch solution of the runoff equation F(x) at a specific frequency is achieved by sequentially inputting the specific frequency and G(x), i.e., the equation F(x) is composed of the product of n G(x) - Pn. The selection of the specific frequency is usually based on the needs of engineering or scientific research, such as selecting the frequency of the design flood standard according to actual needs. In flood control engineering design, the design frequency is usually determined based on factors such as historical flood data, geographical environment, hydrological and meteorological conditions, and engineering requirements to ensure the safety and economy of the project.

[0051] The construction of the Pearson Type III equation offers several benefits, including: more accurate runoff prediction: The Pearson Type III equation can more precisely describe the distribution of runoff at different frequencies, thus improving the accuracy of runoff prediction. More comprehensive hydrological statistical analysis: By constructing the Pearson Type III equation, various statistical characteristics of runoff can be analyzed, including mean, variance, skewness, and kurtosis, providing a more comprehensive understanding of runoff characteristics. More effective flood control engineering design: Hydrological statistical analysis and design flood standards based on the Pearson Type III equation can more effectively guide the design and construction of flood control projects, thereby improving flood control capacity and reducing flood losses. Scalability and flexibility: The Pearson Type III equation can be applied to different types of hydrological data, including runoff, rainfall, and water level, exhibiting good scalability and flexibility.

[0052] Step 4: Solve the runoff equations for specific frequencies in batches to obtain the flood design standards corresponding to those frequencies.

[0053] The minimum and maximum values ​​of the runoff to be solved (min and max) and the required precision (eps) need to be specified. A mathematical method is used to solve the runoff equation F(x) at a specific frequency in batches. This mathematical method employs numerical integration to divide the intervals into specified precision intervals and then uses the bisection method to find the roots within those intervals.

[0054] Based on flood statistics, the minimum and maximum runoff over a certain period are selected as the lower limit (minimum) and upper limit (maximum) of the solution interval. Generally, flood data from the most recent decades should be used as a reference to reflect the actual situation of the watershed. The accuracy of the solution should be set according to the required accuracy of the results. If high accuracy is required, a smaller accuracy value should be set; if lower accuracy is required, a larger accuracy value can be set.

[0055] The method for dividing intervals to a specified precision is as follows:

[0056] a. Use the formula m = (b - a) / eps to calculate the number of equal intervals m.

[0057] b. Divide the solution interval [a, b] into m equal intervals, such that the length of each interval is dx, where dx should be less than or equal to the required solution accuracy eps.

[0058] c. For each equally divided interval, calculate the integral value of the equation F(x) within that interval using numerical integration methods. Using the commonly used numerical integration method, the composite trapezoidal rule, several solution intervals are obtained.

[0059] The method of finding roots using the bisection method is as follows:

[0060] a. Determine the lower limit a and the upper limit b of the solution interval.

[0061] b. Repeatedly bisecte the solution interval until the zero of the equation F(x) is found. The specific process is as follows:

[0062] i. Calculate the midpoint of the interval c = (a + b) / 2.

[0063] ii. Determine if the signs of the values ​​F(a) and F(c) are the same. If they are the same, the root is in the interval [c, b], and the value of the lower limit a is updated to c; otherwise, the root is in the interval [a, c], and the value of the upper limit b is updated to the upper limit c.

[0064] iii. Repeat steps i and ii until the difference between the upper limit a and the lower limit b is less than the required solution accuracy eps.

[0065] c. Once the zeros of equation F(x) are found, the solution to the runoff equation F(x) at a specific frequency can be obtained. Since equation F(x) is monotonic in the solution interval, the root is unique. The bisection method for finding the root has a fast convergence speed and high accuracy.

[0066] According to hydrological statistics theory, flood frequency analysis refers to obtaining flood runoff at different frequencies through statistical analysis of historical hydrological data, thereby providing technical support for the design, construction, and management of water conservancy projects. Traditional one-by-one solution methods require calculations for each specific frequency, resulting in low efficiency. However, the batch solution method proposed in this invention, by constructing an equation F(x) consisting of n products of G(x) - Pn, can simultaneously solve the runoff equations at different frequencies, significantly improving computational efficiency and accuracy.

[0067] For a given specific frequency, the runoff equation obtained is the required flood design standard, because the runoff corresponding to this equation is the maximum flood volume occurring at that frequency. Therefore, by solving the runoff equations for a specific frequency in batches, flood design standards for multiple different frequencies can be obtained, providing important technical support for the design and management of water conservancy projects.

[0068] It will be apparent to those skilled in the art that the present invention is not limited to the details of the exemplary embodiments described above, and that the invention can be implemented in other specific forms without departing from its spirit or essential characteristics. Therefore, the embodiments should be considered in all respects as exemplary and non-limiting, and the scope of the invention is defined by the appended claims rather than the foregoing description. Thus, all variations falling within the meaning and scope of equivalents of the claims are intended to be included within the present invention. No markings in the claims should be construed as limiting the scope of the claims.

[0069] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A method for batch solving flood design standards at a specific frequency in hydrological statistics, characterized in that, The steps are as follows: Step 1: Obtain the longest hydrological series for the hydrological statistics item; Step 2: Construct the Pearson hydrological series based on the data from the longest hydrological series. Type equation; Step 3: Based on the constructed Pearson Formulate equations to construct equations that can solve runoff at specific frequencies in batches; Step 4: Solve the runoff equations at specific frequencies in batches to obtain the flood design standards corresponding to those frequencies; Based on the parameters of the multi-year average runoff, coefficient of variation, and skewness coefficient, a frequency curve equation G(x) conforming to the P-III type distribution is constructed using the probability density function of the P-III type distribution. Substituting the frequency curve equation G(x) and the design frequency Pn into the equation G(x)-Pn, a set of values ​​for G(x)-Pn is obtained. This process is repeated until n sets of values ​​for G(x)-Pn are obtained. Multiplying these n sets of values ​​yields the equation F(x) composed of the products of n G(x)-Pn values. In step 4, the minimum and maximum values ​​of the runoff are specified, as well as the required accuracy eps. Mathematical methods can be used to solve the runoff equation F(x) at a specific frequency in batches. The mathematical method uses numerical integration to divide the intervals of the specified accuracy and uses the bisection method to find the roots within the intervals.

2. The method for batch solving flood design standards at specific frequencies in hydrological statistics according to claim 1, characterized in that, The implementation method of step 1 is as follows: Based on the needs, identify the hydrological stations that need to be statistically analyzed, and collect a series of hydrological statistical items for N years from the establishment of the hydrological station to the most recent year; the hydrological statistical items include but are not limited to annual runoff, time-period runoff, annual maximum flood peak, and time-period flood volume; statistically analyze the collected series of data according to the needs by annual runoff, time-period runoff, annual maximum flood peak, time-period flood volume, or other required items; according to the principle of consistency of hydrological series required by hydrological statistics, if the measurement and compilation accuracy control methods of hydrological stations are the same, and the impact of human activities on hydrological statistical items is small, then the series of items statistically analyzed by the actual measurement data of the hydrological station is determined as the longest hydrological series of hydrological statistical items; if the impact of human activities is large, based on the historical data comparison method, the data of different hydrological stations are compared with the historical data of the same period and consistency correction is performed, and the series after consistency correction is determined as the longest hydrological series of hydrological statistical items.

3. The method for batch solving flood design standards at specific frequencies in hydrological statistics according to claim 1 or 2, characterized in that, The steps in step 2 for calculating the Pearson Type III equation G(x) corresponding to the longest hydrological series are as follows: 1) Statistical analysis was performed on the data of the longest hydrological series to obtain statistical characteristics; 2). Based on statistical characteristics, the longest hydrological series data is fitted using the probability density function of the Pearson Type III distribution to obtain the Pearson Type III equation G(x); where x represents the hydrological variable; 3) Verify and optimize the Pearson type III equation G(x) to improve the fitting effect and accuracy.

4. The method for batch solving flood design standards at specific frequencies in hydrological statistics according to claim 3, characterized in that, The longest hydrological series refers to hydrological data over a long period of time, including data on runoff, rainfall, and water level over many years. The statistical characteristics include at least one of the following: mean, variance, skewness, or kurtosis; the hydrological variables include runoff, rainfall, or water level.

5. The method for batch solving flood design standards at specific frequencies in hydrological statistics according to claim 3, characterized in that, In step 3, based on the input parameters of multi-year average runoff, coefficient of variation, and skewness coefficient, a frequency curve equation G(x) conforming to the Pearson type III distribution is constructed; based on the input design frequency Pn, an equation F(x) consisting of the product of n G(x)-Pn is constructed.

6. The method for batch solving flood design standards at specific frequencies in hydrological statistics according to claim 5, characterized in that, In flood control engineering design, the design frequency Pn is determined based on factors such as historical flood data, geographical environment, hydrological and meteorological conditions, and engineering requirements.

7. The method for batch solving flood design standards at specific frequencies in hydrological statistics according to claim 6, characterized in that, The method of dividing the interval to a specified precision using numerical integration is as follows: Calculate the number of equally divided intervals m = (b - a) / eps; Divide the solution interval [a, b] into m equal intervals, such that the length of each interval is dx, where dx is less than or equal to the required solution accuracy eps; For each equally divided interval, the integral value of the equation F(x) in that interval is calculated using numerical integration; the composite trapezoidal rule is used to obtain several solution intervals.

8. The method for batch solving flood design standards at specific frequencies in hydrological statistics according to claim 7, characterized in that, The implementation method of the bisection method for finding roots is as follows: Determine the lower limit 'a' and upper limit 'b' of the solution interval; By continuously bisecting the solution interval until the zero point of the equation F(x) is found, the solution of the runoff equation F(x) corresponding to a specific frequency can be obtained. The method for finding the zeros of the equation F(x) is as follows: i. Calculate the midpoint of the interval c = (a + b) / 2; ii. Determine whether the signs of the values ​​F(a) and F(c) are the same. If they are the same, the root is in the interval [c, b], and the value of the lower limit a is updated to c; otherwise, the root is in the interval [a, c], and the value of the upper limit b is updated to the upper limit c. iii. Repeat steps i and ii until the difference between the upper limit a and the lower limit b is less than the solution accuracy eps.

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